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Projection-operator methods for classical transport in magnetized plasmas. Part 2. Nonlinear response and the Burnett equations

Published online by Cambridge University Press:  09 November 2018

John A. Krommes*
Affiliation:
Princeton University, Plasma Physics Laboratory, P.O. Box 451, MS 28, Princeton, NJ 08543–0451, USA
*
Email address for correspondence: krommes@princeton.edu
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Abstract

The time-independent projection-operator formalism of Brey et al. (Physica A, vol. 109, 1981, pp. 425–444) for the derivation of Burnett equations is extended and considered in the context of multispecies and magnetized plasmas. The procedure provides specific formulas for transport coefficients in terms of two-time correlation functions involving both two and three phase-space points. It is shown how to calculate those correlation functions in the limit of weak coupling. The results are used to demonstrate, with the aid of a particular non-trivial example, that the Chapman–Enskog methodology employed by Catto & Simakov (CS) (Phys. Plasmas, vol. 11, 2004, pp. 90–102) to calculate the contributions to the parallel viscosity driven by temperature gradients is consistent with formulas previously derived from the two-time formalism by Brey (J. Chem. Phys., vol. 79, 1983, pp. 4585–4598). The work serves to unify previous work on plasma kinetic theory with formalism usually applied to turbulence. Additional contributions include discussions of (i) Braginskii-order interspecies momentum exchange from the point of view of two-time correlations; and (ii) a simple stochastic model, unrelated to many-body theory, that exhibits Burnett effects. Insights from that model emphasize the role of non-Gaussian statistics in the evaluation of Burnett transport coefficients, including the effects calculated by CS that stem from the nonlinear collision operator. Together, Parts 1 and 2 of this series provide an introduction to projection-operator methods that should be broadly useful in theoretical plasma physics.

Type
Research Article
Copyright
© Cambridge University Press 2018 

1 Introduction

The collisional fluid equations for weakly coupled and magnetized plasmas, often referred to as the Braginskii equations, are very well known (Braginskii Reference Braginskii and Leontovich1965). In Part 1 of this series (Krommes Reference Krommes2018b ), I discussed the derivation of the linearized Braginskii equations by means of the projection-operator formalism (Mori Reference Mori1965) that has been frequently used in the context of neutral gases. That formalism is appealing on both heuristic and technical levels. The use of projection operators fosters a clean separation between the hydrodynamic and orthogonal subspaces, and it provides an efficient construction of the subtracted fluxes whose autocorrelations determine the transport coefficients. It also leads to new results. For example, I showed in Part 1 that for linear response it leads to a straightforward derivation of fluctuating hydrodynamics, a topic not explicitly discussed by Braginskii.

Of course, the true fluid equations are nonlinear. The Braginskii equations are really a hybrid: the Euler parts of the equations, which close without approximation in terms of the fluid variables (density  $n$ , flow velocity  $\boldsymbol{u}$ and temperature  $T$ ), are nonlinear and valid to all orders in the macroscopic gradients, while the dissipative corrections are valid only to lowest order in the gradients or, in multicomponent systems, the momentum and energy exchange terms involving (in a quasineutral plasma with one ion species) $\unicode[STIX]{x0394}\boldsymbol{u}\doteq \boldsymbol{u}_{e}-\boldsymbol{u}_{i}$ and $\unicode[STIX]{x0394}T\doteq T_{e}-T_{i}$ . (I use $\doteq$ for definitions.) This is known as the Navier–Stokes order in non-equilibrium statistical mechanics; one obtains standard effects such as heat fluxes ( $\boldsymbol{q}=-n\unicode[STIX]{x1D705}\unicode[STIX]{x1D735}T$ , where $\unicode[STIX]{x1D705}$ is the thermal conductivity) or, in a plasma, frictional drag and collisional temperature equilibration. Equations valid to next order in the gradients are called the Burnett equations (Burnett Reference Burnett1935, Reference Burnett1936); a review with many references is provided by García-Colín, Velasco & Uribe (Reference García-Colín, Velasco and Uribe2008). They contain new linear dissipative fluxes such as $\unicode[STIX]{x1D705}s_{+}\prime \unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}$ ; nonlinear gradient–gradient interactions such as $\unicode[STIX]{x1D735}T\boldsymbol{\cdot }\unicode[STIX]{x1D735}T\equiv |\unicode[STIX]{x1D735}T|^{2}$ ; and, in a multicomponent system, additional terms such as $\unicode[STIX]{x0394}\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}T$ or $|\unicode[STIX]{x0394}\boldsymbol{u}|^{2}$ . Terms at Burnett order could be important in situations involving strong fluctuations and/or gradients (which may pertain to tokamak pedestals, for example). In the present paper, I employ projection-operator methods to discuss some aspects of the calculation of Burnett transport coefficients for plasmas. My focus is on the basic principles and technical methodology; I shall not consider practical implications, for which further research is required.

Although it is clearly desirable to have a description that is superior to Braginskii’s, there are daunting technical obstacles. Of course, when the gradients and/or nonlinearities are extremely large, perturbation theory loses its validity and a renormalized description involving terms of all orders becomes necessary. Such a theory is beyond the scope of this paper.Footnote 1 But even when those quantities are modest, so that one might expect that regular perturbation theory truncated at second order would give useful corrections to the Braginskii equations, there is a deeply troubling issue. It is unfortunately the case that the second-order (Burnett) description may not actually exist; coefficients such as  $\unicode[STIX]{x1D705}s_{+}\prime$ can be infinite, signifying an embarrassing breakdown of the standard perturbative and Markovian approach. The difficulty arises because of collisional excitation of long-lived and long-ranged hydrodynamic fluctuations that lead to slowly decaying tails on correlation functions and violate basic assumptions of locality. There is a large literature devoted to this topic, which regrettably is too extensive to review here. A fundamental paper was by Alder & Wainwright (Reference Alder and Wainwright1970), who discovered long-time tails in molecular-dynamics computer simulations of neutral fluids. The issue arises as well in plasma kinetic theory, where it was investigated by Krommes & Oberman (Reference Krommes and Oberman1976b ); see that paper for earlier references. The topic is also discussed in the books by Balescu (Reference Balescu1975, § 21.5), Reichl (Reference Reichl1998, § S11.A) and Zwanzig (Reference Zwanzig2001, chap. 9). Brey (Reference Brey1983) showed that the Burnett coefficients for neutral fluids were infinite. Although the specific generalization of Brey’s calculations to plasma kinetic theory at the Burnett level has not been done, there is no reason to believe that the situation is any better in plasmas. Why, then, should one proceed? This question was addressed by Wong et al. (Reference Wong, MacLennan, Lindenfeld and Dufty1978) as follows:Footnote 2

It is known that persistent correlations (Alder & Wainwright (Reference Alder and Wainwright1970), and the additional references cited by Wong et al. in their Ref. 11) lead to a divergence of the linear Burnett-order coefficients, which is to say that the expansion used is not justified; a well-defined expansion involving fractional powers of the wavenumber has been obtained by Ernst & Dorfman (Reference Ernst and Dorfman1972). For the nonlinear case a non-analytic dependence of stress on strain rate has been found (Kawasaki & Gunton (Reference Kawasaki and Gunton1973), and further references cited by Wong et al. in their Refs. 9 and 13) such that the corresponding nonlinear Burnett coefficients are also divergent. Thus an expansion of Chapman–Enskog type is actually not legitimate. However, it is possible to make a separation into a regular part (for which a Chapman–Enskog expansion is valid) plus a singular part, and in the cases considered the singular part is found to be small (Ernst & Dorfman Reference Ernst and Dorfman1972; Kawasaki & Gunton Reference Kawasaki and Gunton1973; Ernst et al. Reference Ernst, Cichocki, Dorfman, Sharma and van Beijeren1978). Hence the formulas $\ldots$ will give accurate results if evaluated in a way which suppresses persistent correlations $\,\ldots \,$ . Our point of view is therefore to ignore the effects of persistent correlations, but the results nevertheless shed some light on the nature of the divergences.

A summary of that same point was given slightly later by Brey (Reference Brey1983):

We want to point out that in spite of the problem of the divergencies, an expansion in gradients of the transport equations may still be useful. First, it may allow the calculation of the ‘regular’ or non-singular part of these coefficients and this may be interesting since the regular part is probably dominant for small, but not asymptotically small, gradients.

This discussion of divergences may confuse plasma physicists used to calculations based on the Landau (or Balescu–Lenard) collision operator. It is necessary to emphasize that no divergences arise in that context; those operators produce finite results for transport coefficients at both Navier–Stokes and Burnett orders. (See the discussion of the calculations of Catto & Simakov (Reference Catto and Simakov2004) in § 6.) However, although it is commonplace and frequently justifiable to begin with the Landau kinetic equation, it must not be forgotten that the usual plasma collision operators omit crucial many-body physics related to long-lived correlations. Although the strength of those effects naively scales with $\unicode[STIX]{x1D716}_{\text{p}}^{m}$ , where $m\geqslant 2$ ( $\unicode[STIX]{x1D716}_{\text{p}}\doteq 1/n\unicode[STIX]{x1D706}_{\text{D}}^{3}$ is the plasma discreteness parameter; $\unicode[STIX]{x1D706}_{\text{D}}$  is the Debye length), so they are ignored in the derivations of the usual operators, such fluctuations are excited for each of an infinity of long-wavelength modes, the superposition of which need not be negligible. Physically, the neglected higher-order terms are associated with collisional effects that lead to the excitation of long-wavelength, low-frequency hydrodynamic modes. While in three spatial dimensions those modes lead to small, finite corrections to transport coefficients at Navier–Stokes order, they can lead to infinities at Burnett order. In the language of the above two quotes, calculations based on the Landau operator lead to the regular parts of the transport coefficients in the limit of weak coupling.

Divergences of the transport coefficients aside, there are other well-known difficulties with the Burnett equations. For example, they may be mathematically ill posed in the sense that singularities in the predicted nonlinear fluid motions may arise for sufficiently large gradients, the proper choice of boundary conditions is an issue and they do not capture the physics of the Knudsen boundary layer.Footnote 3 There has been a great deal of work on these topics, which can be located through literature searches; the references in García-Colín et al. (Reference García-Colín, Velasco and Uribe2008) are a good place to start. Although these problems are obviously important, I shall not pursue them here.

In spite of these serious concerns, I shall proceed in the present paper to discuss the formal structure of the Burnett equations for multispecies and magnetized plasmas (ignoring any issues with long-time tails and non-locality); then I shall compare the results to earlier work for weakly coupled plasmas. In plasma physics, equations that contain (some) Burnett terms have been derived by Mikhaǐlovskiǐ (Reference Mikhaǐlovskiǐ1967), Mikhaǐlovskiǐ & Tsypin (Reference Mikhaǐlovskiǐ and Tsypin1971, Reference Mikhaǐlovskiǐ and Tsypin1984) and Catto & Simakov (Reference Catto and Simakov2004). None of those authors remarked on connections to the extensive work on Burnett equations in neutral gases. In particular, neither Mikhaǐlovskiǐ & Tsypin (Reference Mikhaǐlovskiǐ and Tsypin1984) nor Catto & Simakov (Reference Catto and Simakov2004) seem to have been aware of the earlier work of Wong et al. (Reference Wong, MacLennan, Lindenfeld and Dufty1978) and the more technically efficient subsequent work of Brey, Zwanzig & Dorfman (Reference Brey, Zwanzig and Dorfman1981), so they did not attempt to demonstrate consistency between their calculations and previously known general formulas. One goal of the present research is to show that the results of Catto & Simakov are, in fact, consistentFootnote 4 with a subset of the general formulas of Brey et al. as specialized to a one-component fluid by Brey (Reference Brey1983) and calculated to the lowest non-trivial order in weak coupling for a statistically homogeneous background. Another motivation, quite apart from the possible practical implications of Burnett-level effects (which are not addressed in this paper), is the elucidation of the proper treatment of plasma kinetic theory beyond linear order, which is generally instructive and has implications that transcend the specific application to second-order plasma transport. The discussion serves to unify several research threads, including traditional plasma kinetic theory and the results of Rose (Reference Rose1979) on non-Gaussian statistical closures that embrace both collective turbulence and discrete-particle effects.

The formulas recorded by Brey (Reference Brey1983) apply to an unmagnetized, one-component, neutral fluid. Adding a background magnetic field $\boldsymbol{B}^{\text{ext}}$ to the general formalism is formally trivial.Footnote 5 However, the resulting symmetry breaking leads to many additional transport coefficients relative to the unmagnetized case. I shall not give specific formulas for those extra terms at Burnett order. Although the way to do so is straightforward, in the limit of large  $B$ the second-order magnetic-field effects are very small; Catto & Simakov did not calculate them. The magnetized problem at Navier–Stokes order was treated from the projection-operator point of view in Part 1.

In a multicomponent system, additional interspecies momentum and energy exchange effects emerge. At Navier–Stokes order, those were treated in Part 1, and I shall give some further discussion in this paper. At Burnett order, the presence of exchange effects leads to yet more terms. Although I shall indicate where those terms arise in the general formalism, I shall not derive specific formulas for them because the first-order exchange effects are already small in the limit of small electron-to-ion mass ratio.

The discussion so far has implicitly assumed that regular perturbations, to several orders in the gradients, are made to a statistically homogeneous background. Clearly, the resulting transport coefficients, at either Navier–Stokes or Burnett order, will be obtained as well-defined functionals of the fluctuation level in that background. However, Kent & Taylor (Reference Kent and Taylor1969) have shown that such an assumption of homogeneity may be inadequate in practice. Specifically, in the presence of background gradients, regions of local instability lead to convective amplification of fluctuations. Even though the plasma is globally stable, the fluctuation level is overall enhanced. Such an effect increases the effective collisionality and requires a generalization of the usual plasma collision operator.

Fortunately, the actual method to be discussed in the body of the paper does not, in fact, perturb a homogeneous background; essentially, it calculates dissipative corrections to a locally Maxwellian state that includes gradients. Therefore, the general formulas, written for an arbitrarily inhomogeneous background, should include any effects related to convective amplification. However, when I illustrate the content of those formulas and especially when I compare with the calculations of Catto & Simakov, I shall assume that various correlation functions reflect statistical homogeneity. That is, a general spatially dependent correlation function $C(\boldsymbol{x},\boldsymbol{x}s_{+}\prime )$ will be assumed to depend only on the difference $\unicode[STIX]{x1D746}\doteq \boldsymbol{x}-\boldsymbol{x}s_{+}\prime$ , so it can be Fourier transformed according to $C(\unicode[STIX]{x1D746})=(2\unicode[STIX]{x03C0})^{-3}\int \text{d}\boldsymbol{k}\,\widehat{\mathit{C}}(\boldsymbol{k})$ . In such Fourier descriptions, which ultimately lead to the Balescu–Lenard or Landau collision operators in the limit of weak coupling, the effect of convective amplification is lost. I shall alert the reader at the relevant places.

Another possibly important effect that is omitted from the formulas derived in this paper, which focuses on collisional transport, is wave-induced transport enhancement. For example, Rosenbluth & Liu (Reference Rosenbluth and Liu1976) calculated the enhancement of cross-field energy transport due to plasma waves and found that it could exceed the classical electron thermal conductivity. I shall indicate at which point this mechanism is lost.

Given the restrictions listed in the last several paragraphs, one sees that the present paper does not present a complete description of Burnett or other transport effects in stable plasmas; nor does it discuss turbulence. Instead, the focus is on developing the basic ideas of collisional transport to second order in the gradients, illustrating the use of projection operators in a non-trivial context, and discussing the equivalence between the two-time formalism and Chapman–Enskog theory. Various other insights emerge along the way.

1.1 Introduction to Burnett effects

At first order in the gradients of a weakly coupled, unmagnetized, one-component plasma, there are just two non-vanishing dissipative coefficients, namely, the kinematic viscosity  $\unicode[STIX]{x1D707}$ and the thermal conductivity  $\unicode[STIX]{x1D705}$ ; see §§ 1:2 and 1:A.Footnote 6 For strong coupling, the bulk viscosity  $\unicode[STIX]{x1D701}$ is also required. At second order, there is an explosion of terms. The general theory of an unmagnetized, one-component plasma with arbitrary coupling involves, at Burnett order, 13 additional $\unicode[STIX]{x1D707}$  coefficients and 8 additional $\unicode[STIX]{x1D705}$  coefficients, collectively described in terms of 23 non-trivial integrals over two-time correlation functions involving two or three phase-space points (see § 4). The details are tedious. However, the basic idea is clear.

Figure 1. One mechanism leading to a Burnett contribution to the momentum flux that is proportional to $|\unicode[STIX]{x1D6FB}T|^{2}$ . The thick solid line depicts a temperature profile with constant gradient. The net second-order momentum flux across the $z=0$ plane arises from the unbalanced portion of the first-order viscous forces exerted on the velocity streams arriving from a mean free path  $\unicode[STIX]{x1D706}_{\text{mfp}}$ away. See the text for further discussion.

I shall illustrate by considering the scaling of the contribution of the unmagnetized (or parallel) momentum flux proportional to $(\unicode[STIX]{x1D6FB}T)^{2}$ (here I ignore tensorial properties, i.e. the distinction between $\unicode[STIX]{x1D735}T\,\unicode[STIX]{x1D735}T$ and $|\unicode[STIX]{x1D735}T|^{2}\unicode[STIX]{x1D644}$ , where $\unicode[STIX]{x1D644}$  is the unit tensor). One mechanism is illustrated in figure 1, which is in the spirit of some diagrams used by Braginskii (Reference Braginskii and Leontovich1965).Footnote 7 Posit a constant temperature gradient in the $-z$  direction, and assume zero net flow velocity ( $u_{z}=0$ ). Divide the distribution function into positive- and negative-going streams whose velocities are  $\unicode[STIX]{x1D701}_{+}\equiv \unicode[STIX]{x1D701}$ and  $\unicode[STIX]{x1D701}_{-}=-\unicode[STIX]{x1D701}$ ; those streams arrive from distances of the order of the mean free path  $\unicode[STIX]{x1D706}_{\text{mfp}}\doteq v_{\text{t}}/\unicode[STIX]{x1D708}$ , where $v_{\text{t}}\doteq (T/m)^{1/2}$ is the thermal velocity and $\unicode[STIX]{x1D708}$  is the collision frequency. Arguing heuristically, one can assert that at first order each stream experiences a viscous force $-nm\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FB}\unicode[STIX]{x1D701}$ . Because the streams arrive from regions of differing temperatures, that force is unbalanced; the net momentum flux  $\unicode[STIX]{x1D70B}$ across the plane $z=0$ is

(1.1) $$\begin{eqnarray}\unicode[STIX]{x1D70B}=-nm\unicode[STIX]{x1D707}_{+}\unicode[STIX]{x1D6FB}\unicode[STIX]{x1D701}_{+}-nm\unicode[STIX]{x1D707}_{-}\unicode[STIX]{x1D6FB}\unicode[STIX]{x1D701}_{-}\approx -nm(\unicode[STIX]{x1D707}_{+}-\unicode[STIX]{x1D707}_{-})\unicode[STIX]{x1D6FB}\unicode[STIX]{x1D701}.\end{eqnarray}$$

One has $\unicode[STIX]{x1D6FB}\unicode[STIX]{x1D701}=(\unicode[STIX]{x2202}_{T}\unicode[STIX]{x1D701})\unicode[STIX]{x1D6FB}T$ and $\unicode[STIX]{x1D707}\sim \langle \unicode[STIX]{x1D6FF}v^{2}\rangle /\unicode[STIX]{x1D708}$ , $\unicode[STIX]{x1D6FF}v$  being a thermal velocity fluctuation. Upon temporarily assuming that $\unicode[STIX]{x1D708}$  is constant (the temperature dependence of  $\unicode[STIX]{x1D708}$ is properly taken into account in the detailed calculations described later), one has

(1.2) $$\begin{eqnarray}\unicode[STIX]{x1D707}(z)\approx \unicode[STIX]{x1D707}_{0}+z(\unicode[STIX]{x1D708}^{-1}\unicode[STIX]{x2202}_{T}\langle \unicode[STIX]{x1D6FF}v^{2}\rangle )\unicode[STIX]{x1D6FB}T.\end{eqnarray}$$

The $\unicode[STIX]{x1D707}_{0}$ contributions cancel in (1.1). If one assumes that the streams arrive from distances  $\mp \unicode[STIX]{x1D706}_{\text{mfp}}$ , the remainder is

(1.3) $$\begin{eqnarray}\unicode[STIX]{x1D70B}\approx -nm[-2\unicode[STIX]{x1D706}_{\text{mfp}}(\unicode[STIX]{x1D708}^{-1}\unicode[STIX]{x2202}_{T}\langle \unicode[STIX]{x1D6FF}v^{2}\rangle )\unicode[STIX]{x1D6FB}T](\unicode[STIX]{x2202}_{T}\unicode[STIX]{x1D701})\unicode[STIX]{x1D6FB}T\sim nm\unicode[STIX]{x1D707}\left(\frac{\unicode[STIX]{x1D706}_{\text{mfp}}}{L}\right)\left(\frac{v_{\text{t}}}{L}\right).\end{eqnarray}$$

The small ratio $\unicode[STIX]{x1D706}_{\text{mfp}}/L$ signifies that this Burnett effect is of one higher order in the gradients than is the usual first-order viscous stress.

The combination $m\langle \unicode[STIX]{x1D6FF}v^{2}\rangle v_{\text{t}}$ , which appears in the above estimate, arises microscopically from the kinetic-energy flux $\boldsymbol{J}^{E}={\textstyle \frac{1}{2}}mv^{2}\boldsymbol{v}$ . In detail, it is shown later that the present effect arises from the cross-correlation between (subtracted versions ofFootnote 8 ) the microscopic momentum flux  $m\boldsymbol{v}\boldsymbol{v}$ and  $\boldsymbol{J}^{E}$ ; see formula (4.15e ), which according to (4.8e ) gives one contribution to the Burnett momentum flux proportional to $\unicode[STIX]{x1D735}T\,\unicode[STIX]{x1D735}T$ .

It is easy to see that an extension of the above argument to include a non-constant temperature gradient will lead to another Burnett term involving $\unicode[STIX]{x1D6FB}^{2}T$ . Various effects involving flow gradients follow as well. Detailed formulas for all of those are derived in the subsequent sections.

The concepts of velocity streams with definite speed, and of a mean free path of constant value, are useful for simple heuristic arguments. However, they do not provide a detailed description of the actual particle probability density function (PDF) and cannot be used if systematic and quantitative results are desired. In reality, both the particle velocities and the mean free path are random variables at the microscopic level. Consider, for example, the classical (first-order) viscosity $\unicode[STIX]{x1D707}\sim v_{\text{t}}^{2}/\unicode[STIX]{x1D708}=v_{\text{t}}\unicode[STIX]{x1D706}_{\text{mfp}}$ . More properly, with $\widetilde{\unicode[STIX]{x1D706}}=\widetilde{v}/\unicode[STIX]{x1D708}$ and $\langle \widetilde{v}\rangle =0$ , viscosity is expressed as the statistical average $\unicode[STIX]{x1D707}\sim \langle \widetilde{v}\widetilde{\unicode[STIX]{x1D706}}\rangle =\langle \unicode[STIX]{x1D6FF}v^{2}\rangle /\unicode[STIX]{x1D708}=v_{\text{t}}^{2}/\unicode[STIX]{x1D708}$ . ( $\unicode[STIX]{x1D708}$  also fluctuates, but that does not affect the basic scaling.) At second order, we know from the above physical picture that one needs to consider differences in the statistics of fluctuations in regions separated by a mean free path. This suggests a Taylor expansion of the velocity fluctuations and implies a second-order contribution to momentum flux of the form

(1.4) $$\begin{eqnarray}\unicode[STIX]{x0394}_{2}(nm\langle \unicode[STIX]{x1D6FF}v\,\unicode[STIX]{x1D6FF}v\rangle )\sim nm\langle [\widetilde{\unicode[STIX]{x1D706}}(\unicode[STIX]{x2202}_{T}\unicode[STIX]{x1D6FF}v)\unicode[STIX]{x1D6FB}T]\unicode[STIX]{x1D6FF}v\rangle .\end{eqnarray}$$

This involves a triplet correlation function, essentially $\langle \unicode[STIX]{x1D6FF}v\,\unicode[STIX]{x1D6FF}v\,\unicode[STIX]{x1D6FF}v\rangle$ . Whereas at first order in the gradients it is adequate to assume that $\unicode[STIX]{x1D6FF}v$  is Gaussian (only the second cumulant $\langle \unicode[STIX]{x1D6FF}v^{2}\rangle$ enters the above expression for  $\unicode[STIX]{x1D707}$ ), one must do better at second order because triplet correlations vanish for Gaussian statistics. Indeed, the presence of a temperature gradient distorts the distribution from Gaussian form, and on dimensional grounds it is reasonable to guess that

(1.5) $$\begin{eqnarray}\langle \unicode[STIX]{x1D6FF}v\,\unicode[STIX]{x1D6FF}v\,\unicode[STIX]{x1D6FF}v\rangle \sim \langle \unicode[STIX]{x1D6FF}v^{2}\rangle \langle \unicode[STIX]{x1D6FF}v^{2}\rangle ^{1/2}\unicode[STIX]{x1D706}_{\text{mfp}}\unicode[STIX]{x1D6FB}T.\end{eqnarray}$$

Upon inserting this estimate into (1.4), one recovers (1.3).

More quantitatively, the significance of non-Gaussian triplet correlations for Burnett-order effects can be demonstrated by a simple stochastic model discussed in appendix  A. That model has nothing to do with many-body physics per se, but it demonstrates the role of symmetry breaking in producing non-Gaussian statistics. Study of that appendix is not required in order to understand the remainder of this paper, but it adds additional perspective and may be of interest to people with backgrounds in statistical turbulence theory.

Ultimately, all of the Burnett coefficients will be expressed in terms of two-time correlation functions involving the Klimontovich phase-space microdensity (see (2.6) for the definition of that quantity). Although from the above argument those coefficients appear to be related to certain three-point correlation functions (which describe non-Gaussian symmetry-breaking effects), this (perhaps paradoxically) does not mean that in all cases one needs to evaluate Klimontovich correlations involving three phase-space points. The formalism to be described does a preliminary processing that expresses the non-Gaussian effects in terms of two-time Klimontovich correlations that involve either two or three phase-space points. For example, it turns out that the mechanism estimated above by (1.3) can be calculated from a two-point correlation function. (This is plausible because (1.5) is written in terms of $\langle \unicode[STIX]{x1D6FF}v^{2}\rangle$ , which is a second-order cumulant.) On the other hand, other non-Gaussian effects cannot be so expressed, and some contributions to the Burnett coefficients do require calculation of three-point phase-space correlations. See, for example, (4.8e ), which shows that a variety of effects contribute to the coefficient of $\unicode[STIX]{x1D735}T\,\unicode[STIX]{x1D735}T$ in the momentum equation. Roughly speaking, effects calculable from two-point Klimontovich correlations are related to the linearized collision operator, while ones calculable from three-point Klimontovich correlations involve the nonlinear collision operator. This point will become clearer as we work through the details.

1.2 The relationship between time-correlation formalism and Chapman–Enskog theory

In its most general form, the present procedure (based on two-timeFootnote 9 correlation functions) produces a plethora of transport effects, not all of which appear in the work of Catto & Simakov. It must be stated clearly that the absence of certain terms in that work is not related to the authors’ formalism or to algebraic mistakes; rather, it has to do with their ordering choices (made for relevance to drift-wave physics in magnetized plasmas). First, note that in a linear analysis (such as is presented in Part 1) it does not make sense to discuss the size of, say, the perturbed fluid velocity; it is formally infinitesimal. Thus, there is confusion already with the first sentence of Catto & Simakov (Reference Catto and Simakov2004), which reads

The short mean free path description of magnetized plasma as originally formulated by Braginskii $\ldots$ and Robinson and Bernstein $\ldots$ assumes an ordering in which the ion mean flow is on the order of the ion thermal speed.

While this is a familiar, often-repeated statement, it does not apply to linearized hydrodynamics because in that approximation the amplitudes can be scaled arbitrarily. Instead, the linearized Braginskii equations emerge by working to first order in $\unicode[STIX]{x0394}\doteq k_{\Vert }\unicode[STIX]{x1D706}_{\text{mfp}}\ll 1$ and $\unicode[STIX]{x1D6FF}\doteq k_{\bot }\unicode[STIX]{x1D70C}\ll 1$ . (Here I use the notation of Catto & Simakov for  $\unicode[STIX]{x0394}$ and  $\unicode[STIX]{x1D6FF}$ ; $k_{\Vert }\doteq |\unicode[STIX]{x1D735}_{\Vert }|\equiv L_{\Vert }^{-1}$ , $k_{\bot }\doteq |\unicode[STIX]{x1D735}_{\bot }|\equiv L_{\bot }^{-1}$ , and $\unicode[STIX]{x1D70C}$ is the gyroradius.) If one pursues an expansion to second order in  $\unicode[STIX]{x0394}$ and  $\unicode[STIX]{x1D6FF}$ with no further approximations, the complete set of Burnett effects will appear. For example, if at first (Navier–Stokes) order in the gradients one finds dissipative fluxes proportional to $\unicode[STIX]{x1D735}\boldsymbol{u}$ (momentum diffusion due to viscosity) or  $\unicode[STIX]{x1D735}T$ (heat diffusion due to thermal conductivity), at second (Burnett) order cross-effects in the fluxes such as $(\unicode[STIX]{x1D735}\boldsymbol{u})\boldsymbol{\cdot }(\unicode[STIX]{x1D735}T)$ will emerge. However, Catto & Simakov order  $u$ to be small, $u\sim \unicode[STIX]{x1D735}T$ . Then $(\unicode[STIX]{x1D735}u)\boldsymbol{\cdot }(\unicode[STIX]{x1D735}T)$ becomes formally of third order in the gradients and is neglected in their analysis. As they state, their ordering is appropriate ‘for most magnetic confinement and fusion devices in general, and the edge of many tokamaks in particular’, and I am not suggesting that they have made an error with this ordering. Of course, it is possible that the complete set of terms is required for some esoteric situations. If not, however, it is straightforward to apply the ordering of Catto & Simakov to the general results and attempt to reduce them to the smaller set of terms retained by those authors. I shall do that later in the paper.

Apart from subsidiary orderings, one can ask whether the end points of the several possible approaches should agree in general. If one focuses on the regular parts of the transport coefficients, one can proceed via either (i) Chapman–Enskog analysis of a $\unicode[STIX]{x1D707}$ -spaceFootnote 10 kinetic theory that includes a nonlinear collision operator (the approach used by Catto & Simakov); or (ii) a 𝛤-spaceFootnote 11 formalism based on multipoint, two-time correlation functions (the approach used by Brey et al.). This question was answered in the affirmative by Wong et al. (Reference Wong, MacLennan, Lindenfeld and Dufty1978) for the dilute neutral gas. I shall argue that the analogous correspondence holds as well for weakly coupled plasmas, and I shall illustrate by working out a certain example in detail. While it is clear on general principles that the procedures should be equivalent, a detailed demonstration is decidedly non-trivial, as it requires careful attention to the calculation and subsequent manipulation of certain correlation functions involving two times and as many as three phase-space points.

It is interesting to focus on the specific advance made by Catto & Simakov over the work of previous authors such as Mikhaǐlovskiǐ & Tsypin, which was to calculate heat-flow contributions to the plasma stress tensor due to the nonlinearity of the plasma collision operator. I shall show where those terms arise in the theory based on time-correlation functions. That provides an interesting perspective on the role of so-called nonlinear noise terms that are relevant not only in the many-body theory of discrete particles but also in continuum turbulence theory.

1.3 Gaussian and non-Gaussian statistics in kinetic theory

Before I delve into formal mathematics, I shall give a brief prelude to motivate the relationship between single-time, $\unicode[STIX]{x1D707}$ -space kinetic theory and the $\unicode[STIX]{x1D6E4}$ -space theory of two-time correlations. Consider a simple spatial random walk (Rudnick & Gaspari Reference Rudnick and Gaspari2004) and, for example, the associated density diffusion equation, a simplified version of which is $\unicode[STIX]{x2202}_{t}n=D\unicode[STIX]{x1D6FB}^{2}n$ . Similar equations for flow velocity  $\boldsymbol{u}$ (involving viscosity  $\unicode[STIX]{x1D707}$ ) and temperature  $T$ (involving thermal conductivity  $\unicode[STIX]{x1D705}$ ) can be derived by Chapman–Enskog expansion of the one-time Landau kinetic equation for plasmas. On the other hand, Taylor (Reference Taylor1921) showed in the context of turbulence theory that the spatial diffusion coefficient  $D$ for a test particle or fluid element can be obtained as the time integral of the two-time Lagrangian velocity-correlation function:Footnote 12

(1.6) $$\begin{eqnarray}D=\displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\,C_{vv}(\overline{\unicode[STIX]{x1D70F}}).\end{eqnarray}$$

Similar integrals (Kubo formulas) can be written for  $\unicode[STIX]{x1D707}$ and  $\unicode[STIX]{x1D705}$ . Thus, there is an intimate relationship between one- and two-time theory.

The technical problem with formulas such as (1.6) is that Lagrangian correlation functions are very difficult to calculate in the general case. However, it is well known that $D\sim \unicode[STIX]{x0394}x^{2}/\unicode[STIX]{x0394}t$ , where $\unicode[STIX]{x0394}x$  is the magnitude of the characteristic spatial step taken during the step time  $\unicode[STIX]{x0394}t$ . The steps $\unicode[STIX]{x0394}x$ and  $\unicode[STIX]{x0394}t$ represent two-point Eulerian information: two distinct spatial points are required in order to measure a spatial difference  $\unicode[STIX]{x0394}x$ , and two distinct time points are involved in the definition of  $\unicode[STIX]{x0394}t$ . Of course, these are statistical characteristics, so the relevant quantities are certain two-point Eulerian correlation functions in space and time that I shall here denote generically by $C(\boldsymbol{x},t,\boldsymbol{x}s_{+}\prime ,ts_{+}\prime )$ . One goal of theories of turbulence or non-equilibrium statistical mechanics is to establish the specific relationships between Lagrangian representations of transport coefficients and the relevant Eulerian correlation functions that expedite quantitative calculations.

It is the right-hand side of the equation $\unicode[STIX]{x2202}_{t}C=\cdots \,$ from which $C$  acquires its characteristic space and time scales. For classical spatial transport, those scales are (here I consider $\boldsymbol{B}=\mathbf{0}$ for simplicity) $\unicode[STIX]{x0394}t\sim \unicode[STIX]{x1D708}^{-1}$ and $\unicode[STIX]{x0394}x\sim \unicode[STIX]{x1D706}_{\text{mfp}}$ . However, in the usual weakly coupled plasma kinetic theory, those ‘kinetic’ scales do not appear in the lowest-order equation for two-point correlations, which is derived from the linearized Klimontovich equation. That contains only Vlasov physics; it describes the formation over a time scale of  $\unicode[STIX]{x1D714}_{\text{p}}^{-1}$ ( $\unicode[STIX]{x1D714}_{\text{p}}$  is the plasma frequency) of plasma shielding clouds whose spatial extent is the Debye length  $\unicode[STIX]{x1D706}_{\text{D}}$ . This is adequate for discussions of velocity-space diffusion and test-particle polarization effects (the content of the Balescu–Lenard and Landau collision operators), but it does not capture the kinetic scales associated with spatial transport. This implies that evaluation of spatial transport coefficients from two-time theory requires one to consider collisional corrections to the evolution equation for two-point correlations. By standard arguments, those corrections are represented by triplet correlation functions,Footnote 13 which describe (some) non-Gaussian effects. To repeat, those effects are not included in the derivation of the Landau collision operator, as they are of higher order in the plasma parameter  $\unicode[STIX]{x1D716}_{\text{p}}$ .

It is important to be clear about what ‘non-Gaussian’ means in this context. I am not referring to the fact that the shape of the one-particle distribution function  $f_{s}(\boldsymbol{x},\boldsymbol{v},t)\equiv f(\unicode[STIX]{x1D707},t)$ ( $s$  is a species label and $\unicode[STIX]{x1D707}\doteq \{\boldsymbol{x},\boldsymbol{v},s\}$ ) usually differs from a Maxwellian in velocity. In general, the statistics of a Gaussian time series  $\widetilde{\unicode[STIX]{x1D713}}(t)$ are completely specified by the mean $\langle \widetilde{\unicode[STIX]{x1D713}}(t)\rangle$ and the two-time correlation function $C(t,ts_{+}\prime )\doteq \langle \unicode[STIX]{x1D6FF}\widetilde{\unicode[STIX]{x1D713}}(t)\unicode[STIX]{x1D6FF}\widetilde{\unicode[STIX]{x1D713}}(ts_{+}\prime )\rangle$ , otherwise known as the first and second cumulants.Footnote 14 In the present discussion, the underlying random variable is the Klimontovich microdensity $\widetilde{f}(\unicode[STIX]{x1D707},t)$ , defined by (2.6) below. Its statistical mean is  $f(\unicode[STIX]{x1D707},t)$ , which is generally non-Maxwellian, and its two-point correlation function is $C(\unicode[STIX]{x1D707},t,\unicode[STIX]{x1D707}s_{+}\prime ,ts_{+}\prime )\doteq \langle \unicode[STIX]{x1D6FF}\widetilde{f}(\unicode[STIX]{x1D707},t)\unicode[STIX]{x1D6FF}\widetilde{f}(\unicode[STIX]{x1D707}s_{+}\prime ,ts_{+}\prime )\rangle$ . At equal times, one has

(1.8) $$\begin{eqnarray}C(\unicode[STIX]{x1D707},\unicode[STIX]{x1D707}s_{+}\prime ,t)=\overline{n}^{\,-1}\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D707}-\unicode[STIX]{x1D707}s_{+}\prime )f(\unicode[STIX]{x1D707}s_{+}\prime ,t)+g(\unicode[STIX]{x1D707},\unicode[STIX]{x1D707}s_{+}\prime ,t),\end{eqnarray}$$

where $\overline{n}$  is the mean density and $g$  is the pair correlation function. It is the approximate calculation of the long-time limit of that  $g$ to first order in  $\unicode[STIX]{x1D716}_{\text{p}}$ , producing a functional  $g[\,f]$ , that ultimately leads to the weakly coupled plasma collision operator. (This program is described in § G.1.) From this point of view, one might argue that the plasma kinetic equation with the nonlinear Landau collision operator involves a Gaussian approximation, since it involves only (special cases of) the first two cumulants of  $\widetilde{f}$ , namely the equal-time  $f$ and  $g$ .

The reader may recognize a possible paradox here. I have argued that determination of the characteristic random-walk steps from two-point correlations requires non-Gaussian corrections. Yet in Part 1 I showed from several points of view – classical Chapman–Enskog theory and projection-operator methods – that it is possible to calculate the correct Braginskii transport coefficients from the linearized Landau kinetic equation, which does not seem to be aware of non-Gaussian effects. How can this be reconciled?

The first part of the answer involves the reminder (see footnote 14 on page 12) that a random process is Gaussian only if all multiple-time cumulants vanish beyond second order. This is not the case for plasma kinetic theory due to the intrinsic nonlinearity of the Coulomb interaction.

Nevertheless, apparently it is possible to calculate transport effects from either a one-time theory involving just  $f(t)$ and  $g[\,f(t)]$ or a two-time theory of correlation functions. For consistency, there must therefore be a deep relationship between one- and two-time theory. This follows as a statistical generalization of the well-known fact that a linear ordinary differential equation (ODE) of first order in time can be solved by means of a two-time Green’s function. Namely, the ODE

(1.9) $$\begin{eqnarray}\frac{\text{d}\unicode[STIX]{x1D713}}{\text{d}t}+\text{i}\text{L}\unicode[STIX]{x1D713}=s(t),\end{eqnarray}$$

where $\text{L}$  is a linear operator and $s(t)$  is a given source, has the solution

(1.10) $$\begin{eqnarray}\unicode[STIX]{x1D713}(t)=R(t;0)\unicode[STIX]{x1D713}(0)+\displaystyle \int _{0}^{t}\!\text{d}\overline{t}\,R(t;\overline{t})s(\overline{t})\quad \text{for }t>0,\end{eqnarray}$$

where Green’s function  $R$ obeys

(1.11) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}R(t;ts_{+}\prime )+\text{i}\text{L}R(t;ts_{+}\prime )=\unicode[STIX]{x1D6FF}(t-ts_{+}\prime ),\quad R(ts_{+}\prime -\unicode[STIX]{x1D716};ts_{+}\prime )=0.\end{eqnarray}$$

Formally,

(1.12) $$\begin{eqnarray}R(t;ts_{+}\prime )=\frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713}(t)}{\unicode[STIX]{x1D6FF}\widehat{\unicode[STIX]{x1D702}}(ts_{+}\prime )},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}$  denotes functional differentiation and $\widehat{\unicode[STIX]{x1D702}}(t)$  is an arbitrary functionFootnote 15 that replaces  $s(t)$ in (1.9). That is, the functional variation of a one-time field with respect to a time-dependent perturbation obeys a two-time dynamics. The statistical generalization of this statement will be described later (see § 5), but in essence the basic result is that an $n$ -point cumulant function can be obtained from the appropriate functional derivative of an $(n-1)$ -point cumulant. Thus, functionally differentiating the (one-time) nonlinear Landau kinetic equation both linearizes that equation and produces a two-time dynamics that includes collisional corrections involving the linearized collision operator. In a direct derivation of the equation for the two-time correlation function (say from the Klimontovich equation), the collisional corrections arise from triplet correlations. But at first order the forms of the linearized equations in the $t$  variable are the same in both the two-time theoryFootnote 16 and the one-time Chapman–Enskog approach (or the equivalent projection-operator approach described in Part 1).

A more difficult question is what happens at second order. Upon pursuing the one-time Chapman–Enskog approach to that order, Catto & Simakov found contributions to the Burnett stress tensor from the nonlinear (in fact, bilinear when the Landau operator is used) collision operator evaluated with the first-order correction to the one-particle distribution function (i.e. $\text{C}[\,f_{1},\overline{f}_{1}]$ ).Footnote 17 For those contributions to be compatible with the general two-time theory, it has to be the case that certain triplet correlations are driven in a definite way by the nonlinear collision operator. Technically, it is by no means immediately obvious that this is so. In order to demonstrate it, we shall have to proceed systematically.

Representations of the unmagnetized Burnett transport coefficients in terms of time-correlation functions were obtained by Wong et al. (Reference Wong, MacLennan, Lindenfeld and Dufty1978). In the subsequent discussion, I shall instead closely follow the later paper of Brey et al. (Reference Brey, Zwanzig and Dorfman1981) because it fits more naturally into the projection-operator formalism that is explicated in this series of articles, and because it is in some ways more technically concise. This choice is in no way intended to detract from the importance of the earlier work by Wong et al., whose general results were correct and were reproduced by Brey et al.

1.4 Liouvillian dynamics versus kinetic equations; $\unicode[STIX]{x1D6E4}$ space versus $\unicode[STIX]{x1D707}$ space

The Chapman–Enskog formalism is explicitly a one-time theory. The following remarks pertain instead to the representation of transport coefficients in terms of two-time correlations. At least for weak coupling, the theory of two-time correlation functions can be developed within either a 𝛤-space or a $\unicode[STIX]{x1D707}$ -space formalism.

The work of Brey et al. proceeds directly from the Liouville equation

(1.13) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}P_{\mathscr{N}}(\unicode[STIX]{x1D6E4},t)+\text{i}\mathscr{L}P_{\mathscr{N}}=0,\end{eqnarray}$$

where $\unicode[STIX]{x1D6E4}$  denotes the complete set of phase-space variables for the $\mathscr{N}$ -particle system,Footnote 18   $P_{\mathscr{N}}$  is the $\mathscr{N}$ -particle PDF, and $\mathscr{L}$  is the (first-order, differential) Liouville operator:Footnote 19

(1.14) $$\begin{eqnarray}\text{i}\mathscr{L}\doteq \mathop{\sum }_{i=1}^{\mathscr{N}}\boldsymbol{v}_{i}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{i}+\mathop{\sum }_{i=1}^{\mathscr{N}}\left(\frac{q}{m}\right)_{i}(\boldsymbol{E}_{i}+c^{-1}\boldsymbol{v}_{i}\times \boldsymbol{B}_{i}^{\text{ext}})\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\boldsymbol{v}_{i}},\end{eqnarray}$$

where $\boldsymbol{E}_{i}\doteq \sum _{j\neq i}^{\mathscr{N}}\!\unicode[STIX]{x1D750}_{ij}q_{j}$ and $\unicode[STIX]{x1D750}_{ij}\doteq -\unicode[STIX]{x1D735}_{i}|\boldsymbol{x}_{i}-\boldsymbol{x}_{j}|^{-1}$ . I consider only the electrostatic approximation (no magnetic perturbations are allowed), so $\boldsymbol{B}_{i}^{\text{ext}}\equiv \boldsymbol{B}^{\text{ext}}(\boldsymbol{x}_{i})$  is a given external or ‘background’ magnetic field. (Brey et al. and others did not include a magnetic field.) The final formulas are valid, in principle, for arbitrarily strong coupling, although I shall not evaluate any terms for strongly coupled systems. For weak coupling, an alternative procedure that was used in Part 1 is to first derive the kinetic equation, then process that equation to obtain two-time correlation functions and, ultimately, transport coefficients. That is, while Brey et al. work in $\unicode[STIX]{x1D6E4}$ space, one could in principle proceed in 𝜇 space. The differences between the two approaches manifest in several ways, as I discuss in the next two subsections.

1.4.1 Fluxes in $\unicode[STIX]{x1D6E4}$ space and $\unicode[STIX]{x1D707}$ space

In the work of Brey et al., the transport coefficients are built from microscopic fluxes that appear in formulas involving averages over a 𝛤-space distribution function. Those formulas are difficult to evaluate because of the presence of potential-energy terms. For weak coupling those terms may be neglected,Footnote 20 but one must still perform a 𝛤-space average. If one instead works with a $\unicode[STIX]{x1D707}$ -space description as was done in Part 1, the kinetic fluxes ( $m\boldsymbol{v}\boldsymbol{v}$ and ${\textstyle \frac{1}{2}}mv^{2}\boldsymbol{v}$ ) appear directly, and the weakly coupled transport coefficients can be expressed as simple matrix elements in three-dimensional velocity space. For weak coupling, this simplicity would seem to argue in favour of a $\unicode[STIX]{x1D707}$ -space approach to the theory of two-time correlations that begins with the irreversible kinetic equation; however, see the next subsection for a technical difficulty.

1.4.2 Reversible versus irreversible operators

Because the Liouville operator  $\mathscr{L}$ is time reversible whereas the kinetic-equation approach involves the time-irreversible collision operator, there arise technical differences between 𝛤-space and $\unicode[STIX]{x1D707}$ -space analyses relating to the absence or presence of a dissipative collision operator in the kinetic equation. For example, $\text{e}^{-\text{i}\mathscr{L}\unicode[STIX]{x1D70F}}$ is a propagator that moves the phase-space variables back in time by an amount  $\unicode[STIX]{x1D70F}$ ,Footnote 21 so if $A(\unicode[STIX]{x1D6E4})$ and  $B(\unicode[STIX]{x1D6E4})$ are two arbitrary phase functions, one has the useful result

(1.15) $$\begin{eqnarray}\text{e}^{-\text{i}\mathscr{L}\unicode[STIX]{x1D70F}}(AB)=(\text{e}^{-\text{i}\mathscr{L}\unicode[STIX]{x1D70F}}A)(\text{e}^{-\text{i}\mathscr{L}\unicode[STIX]{x1D70F}}B).\end{eqnarray}$$

This property is used by Brey et al. in the course of various manipulations that lead to the final form of the Burnett coefficients. However, if $\widehat{\text{C}}$  is the linearized collision operatorFootnote 22 and if $a$  and  $b$ are two functions that live in $\unicode[STIX]{x1D707}$  space, it is not true thatFootnote 23   $\text{e}^{-\widehat{\text{C}}\unicode[STIX]{x1D70F}}(ab)=(\text{e}^{-\widehat{\text{C}}\unicode[STIX]{x1D70F}}a)(\text{e}^{-\widehat{\text{C}}\unicode[STIX]{x1D70F}}b)$ . Because this leads to technical complications, I shall abandon a $\unicode[STIX]{x1D707}$ -space approach in favour of the 𝛤-space one followed by Brey et al. That has the added advantage of leading to formulas that are applicable to arbitrarily strong coupling. Although in general those formulas are difficult to evaluate, it is not difficult to reduce them to the weakly coupled limit.

1.5 Modified and unmodified propagators

The interspecies coupling in a multispecies plasma introduces some complications that are absent in a one-component neutral gas. Thus, some generalizations and modifications of the pure-fluid formulas are required. Were it not for those modifications, the formulas of Brey (Reference Brey1983) could be used with only the straightforward addition of the Lorentz force term to the momentum equation and some generalizations of various symmetry arguments to allow for the anisotropy induced by a background magnetic field. In the multispecies case, however, one must be careful about the handling of the orthogonal projector  $\text{Q}$ , so one must repeat and extend the derivation, paying close attention at every step to the assumptions that Brey et al. make. To be more specific, I note that in linear-response theory a crucial role is played by the modified propagator $\text{G}_{\text{Q}}\doteq \exp (-\text{Q}\text{i}\text{L}\text{Q}t)$ , where $\text{L}$  is the linear operator in the perturbed kinetic equation. As is discussed in § 1:G.3 on the plateau phenomenon, an identity relates  $\text{G}_{\text{Q}}$ to the unmodified propagator $\text{G}\doteq \exp (-\text{i}\text{L}t)$ , and for the neutral-fluid case transport coefficients follow as the double (ordered) limit

(1.16) $$\begin{eqnarray}\unicode[STIX]{x1D63F}=\lim _{\unicode[STIX]{x1D714}\rightarrow 0}\lim _{k\rightarrow 0}k^{-2}\widehat{\unicode[STIX]{x1D731}}_{k}(\unicode[STIX]{x1D714}),\end{eqnarray}$$

where $\widehat{\unicode[STIX]{x1D731}}$  is the hydrodynamic transport matrix defined with  $\text{G}$ instead of  $\text{G}_{\text{Q}}$ . At linear order, Brey et al. (Reference Brey, Zwanzig and Dorfman1981) express the transport matrix  $\widehat{\unicode[STIX]{x1D6F4}}$ (defined with  $\text{G}_{\text{Q}}$ ) in terms of the equivalent quantity  $\widehat{\unicode[STIX]{x1D731}}$ defined with  $\text{G}$ , then argue that to second order in the gradient expansion the correction is negligible. That argument fails for the multispecies case because there the collision operator does not satisfy $\text{Q}\widehat{\text{C}}\text{Q}=\widehat{\text{C}}$ (see § 1:3.3) and because $\widehat{\text{C}}$  does not vanish with the gradients. Implications are that the Navier–Stokes and Burnett transport coefficients for a multispecies plasma are determined by matrix elements involving $(\text{Q}\widehat{\text{C}}\text{Q})^{-1}$ rather than  $\widehat{\text{C}}^{-1}$ , and also that the equations will contain extra terms due to the interspecies collisional coupling. The Navier–Stokes interspecies coupling terms were already treated in Part 1 for the special case of small electron-to-ion mass ratio  $\unicode[STIX]{x1D707}$ . Because for $\unicode[STIX]{x1D707}\ll 1$ (a standard ordering for magnetically confined fusion plasmas) those exchange terms are already small, I shall eschew discussion of the analogous Burnett effects in this work.

1.6 Plan of the paper

The remainder of the paper is organized as follows. In § 2 I introduce the general time-independent projection-operator formalism of Brey et al. (Reference Brey, Zwanzig and Dorfman1981). In § 3 I repeat their reduction of the general formulas to Burnett order, paying careful attention to modifications introduced by the multispecies and magnetized nature of plasmas. In § 4 I follow Brey (Reference Brey1983) and record the explicit results, written in terms of two-time correlation functions, for the special case of an unmagnetized one-component fluid. In § 5 I discuss some aspects of the theory of those correlation functions, and I show how to calculate them in the limit of weak coupling. In § 6 I compare the predictions of the two-time formalism to those of (the one-time) Chapman–Enskog theory; I conclude that they are equivalent in the limits of weak coupling and homogeneous background statistics. I summarize and discuss the paper in § 7. Several appendixes are included. In appendix A I present a simple stochastic model that shows that Burnett effects arise in contexts more general than classical many-body theory and emphasizes the importance of non-Gaussian statistics. In appendix B I discuss the time evolution of the microscopic momentum density, paying particular attention to modifications introduced by the long-range nature of the Coulomb force and the presence of multiple species. In appendix C I discuss the choice of conjugate variables, whose gradients serve as thermodynamic forces. In appendix D I describe the evaluation of the fundamental subtracted fluxes from which the transport coefficients are constructed. In appendix E I evaluate the hydrodynamic portion of the first-order momentum exchange term from its time-correlation representation and, in a non-trivial calculation, show that one obtains Braginskii’s result. Some technical manipulations leading to one of the non-Gaussian terms are described in appendix F. Derivations of both the nonlinear and linearized plasma collision operators are described in appendix G. Also discussed there is the relationship between certain so-called nonlinear noise terms and the nonlinear Balescu–Lenard operator. In appendix H I address some issues with one-sided correlations and the interpretation of nonlinear noise. Finally, I summarize the principal notation in appendix I. Some further details and supporting calculations are given in the online Part 2 Supplement available at https://doi.org/10.1017/S0022377818000892 (Krommes Reference Krommes2018c ). Both Part 2 and its Supplement rely on a basic understanding of the material in Part 1.

2 The time-independent projection-operator formalism of Brey et al.

In this section I shall review the time-independent projection-operator formalism of Brey et al. (Reference Brey, Zwanzig and Dorfman1981), paying particular attention to the generalizations needed to include a background magnetic field and especially multiple species.

2.1 Statistical ensembles and distribution functions

We shall consider several kinds of statistical ensemble.

The goal of the present work is to consider transport coefficients dependent on fluctuation spectra driven by discrete particles. I assume that the system is globally stable. One does not want to impose temporally constant gradients of macroscopic quantities such as temperature across the entire system since they would serve as infinite sources of free energy that could drive instabilities and turbulence. Therefore, it is adequate to work in a very large, fixed, periodic box of volume  $\mathscr{V}$ that is in thermal contact with a heat reservoir at temperature  $T$ but is impervious to particles. In this case the total particle number  $\mathscr{N}$ does not change and the mean density $\overline{n}\doteq \mathscr{N}/\mathscr{V}$ is spatially constant. This system is described in thermal equilibrium by the standard canonical ensemble. I shall consider the thermodynamic limit $\mathscr{N}\rightarrow \infty$ , $\mathscr{V}\rightarrow \infty$ , $\overline{n}=\text{const}$ . I shall not consider an external electric field  $\boldsymbol{E}^{\text{ext}}$ , but I shall allow for an external constant magnetic field  $\boldsymbol{B}^{\text{ext}}$ .

Now consider the time evolution of an arbitrary non-equilibrium initial state. In principle, all statistical properties of the system follow from solution of the Liouville equation (1.13) or of multiple-time generalizations of that equation (Krommes & Oberman Reference Krommes and Oberman1976a ). In the general case, direct solution of the Liouville equation is extremely difficult. In order to focus on transport theory, consider long-wavelength sinusoidal perturbations with characteristic wavenumber  $k$ . Then one can imagine dividing  $\mathscr{V}$ into a large collection of cubic cells of fixed volume  $\unicode[STIX]{x0394}\mathscr{V}$ , where $(\unicode[STIX]{x0394}\mathscr{V})^{1/3}$  is much larger than a collisional correlation length  $\unicode[STIX]{x1D706}_{\text{mfp}}$ but much smaller than a macroscopic gradient scale length  $L\sim k^{-1}$ . Each cell will contain the (variable) number of particles  $\widetilde{\mathscr{N}}_{\unicode[STIX]{x0394}\mathscr{V}}(\boldsymbol{r},t)$ , so will be of random density $\widetilde{n}(\boldsymbol{r},t)\doteq \widetilde{\mathscr{N}}_{\unicode[STIX]{x0394}\mathscr{V}}(\boldsymbol{r},t)/\unicode[STIX]{x0394}\mathscr{V}$ . Thus, on the average and at lowest order in the gradients a cell has a local particle density  $n(\boldsymbol{r},t)$ , is moving with a local flow velocity  $\boldsymbol{u}(\boldsymbol{r},t)$ , and has a local temperature  $T(\boldsymbol{r},t)$ (i.e. it is approximately in local thermodynamic equilibrium).Footnote 24 One is interested in working to second order in the gradients of those fluid quantities.Footnote 25

There are technical difficulties with implementing such a program. First, a division into intermediate-sized cells fails if there are long-ranged correlations at the microscopic level. But even if that issue is ignored, as I shall do (see the previous quotes from Wong et al. and Brey et al. in § 1, and also recall that non-local wave-induced transport is neglected), one must face the problem of connecting the cells together in a way that is compatible with the solution of the Liouville equation in the entire box. At lowest order, one popular way of doing that begins with the introduction of the so-called local equilibrium distribution Footnote 26    $F_{\boldsymbol{B}}(\unicode[STIX]{x1D6E4};t)$ . This will be defined and further discussed in § 2.4. Here, it is necessary to clarify the use of the word ‘local’, which may be confusing. In fact, the local-equilibrium ensemble applies to the entire box; it is not parametrized by any particular spatial location. Instead, ‘local’ refers to the fact that each cell is approximately in a local equilibrium. Corrections to  $F_{\boldsymbol{B}}$ define the dissipative fluxes and, through the transport equations, provide the means of connecting the cells together.

Unfortunately, direct use of the local-equilibrium distribution as a basis for a local gradient expansion turns out to be technically somewhat unwieldy because $F_{\boldsymbol{B}}$  contains a spatial integral over the entire box. Brey et al. therefore employ a reference distribution  $F_{0}(\unicode[STIX]{x1D6E4};\boldsymbol{r},t)$ that is intermediate between the two distributions introduced above. It is essentially a canonical ensemble, but with local parameters  $\unicode[STIX]{x1D6FD}(\boldsymbol{r},t)$ and $(\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707})(\boldsymbol{r},t)$ . (Here $\unicode[STIX]{x1D6FD}$  is the inverse temperature and $\unicode[STIX]{x1D707}$  is the chemical potential per particle.) The reference distribution will be defined and further discussed in § 2.5.

Following Brey et al., I shall employ two sets of space–time arguments: the position and time at which the evolution of the hydrodynamic variables is desired are denoted as above by $(\boldsymbol{r},t)$ , while dependence on space and time at an arbitrary space–time point is denoted by $(\boldsymbol{x},s)$ . The distinction between dependence on species  $s$ (a subscript) and arbitrary time  $s$ (an argument) should be clear. Thus, for example, one can discuss the number density $n_{s}(\boldsymbol{x},s)$ ; its hydrodynamic evolution equation will be written $\unicode[STIX]{x2202}_{t}n_{s}(\boldsymbol{r},t)=\cdots \,$ .

Although the paper of Brey et al. is fairly self-contained, it is succinct and might not serve as a suitable starting point for someone who is not familiar with the long history and extensive technical developments of the field of non-equilibrium statistical mechanics. A very clear explanation of the basic issues is given by Piccirelli (Reference Piccirelli1968), who cites earlier fundamental papers. In particular, he discusses the philosophy of the Bogoliubov–Chapman–Enskog asymptotic methods for the development of transport equations, he explains the interpretation of the local-equilibrium ensemble, and he shows how an appropriate projection operation can be employed to usefully rearrange the content of the full Liouville distribution. Some of his appendices and technical footnotes may also be useful for the following discussion.

2.2 Fundamental microscopic variables

I shall denote a random variable by a tilde. Given a random time series  $\widetilde{\unicode[STIX]{x1D713}}(t)$ and a PDF $P(\unicode[STIX]{x1D713},t)$ , the average of a function  $g\boldsymbol{(}\widetilde{\unicode[STIX]{x1D713}}(t)\boldsymbol{)}$ is given by $\langle g\boldsymbol{(}\widetilde{\unicode[STIX]{x1D713}}(t)\boldsymbol{)}\rangle \doteq \int \text{d}\unicode[STIX]{x1D713}\,g(\unicode[STIX]{x1D713})P(\unicode[STIX]{x1D713},t)$ (Schrödinger representation). The PDF itself is given by

(2.1) $$\begin{eqnarray}P(\unicode[STIX]{x1D713},t)=\langle \unicode[STIX]{x1D6FF}\boldsymbol{(}\unicode[STIX]{x1D713}-\widetilde{\unicode[STIX]{x1D713}}(t)\boldsymbol{)}\rangle .\end{eqnarray}$$

The non-random variable  $\unicode[STIX]{x1D713}$ is called the observer coordinate. The representation (2.1) is trivial if the average is taken with the PDF at time  $t$ , but is non-trivial if $\widetilde{\unicode[STIX]{x1D713}}(t)$ has evolved from random variables at some earlier time  $t_{0}$ and the average is evaluated with  $P(\unicode[STIX]{x1D713}_{0},t_{0})$ (Heisenberg representation). It is a common abuse of notation to drop the tildes on random variables inside expectations and thus to write $\langle g\boldsymbol{(}\widetilde{\unicode[STIX]{x1D713}}(t)\boldsymbol{)}\rangle \equiv \langle g(\unicode[STIX]{x1D713})\rangle _{t}\equiv \langle g(\unicode[STIX]{x1D713})\rangle$ , the evaluation time  $t$ being understood implicitly in the last form.

Brey et al. considered only a single species of particle (i.e. a one-component fluid). I shall allow $S$  species ( $S=2$ being the most popular case for plasma physicists), but shall consider interspecies coupling effects only at lowest (Braginskii) order. The fundamental random variables are then chosen to be

(2.2) $$\begin{eqnarray}\widetilde{\boldsymbol{A}}_{s}(\boldsymbol{r},t)=(\widetilde{N}_{s}\;\;\widetilde{\boldsymbol{P}}_{s}\!\text{}^{\text{T}}\;\;\widetilde{E}_{s})^{\text{T}}\end{eqnarray}$$

(T denotes transpose), where $\widetilde{N}_{s}(\boldsymbol{r},t)$  is the microscopic number density, $\widetilde{\boldsymbol{P}}_{s}(\boldsymbol{r},t)$  is the microscopic momentum density,Footnote 27 and $\widetilde{E}_{s}(\boldsymbol{r},t)$  is the microscopic energy density. (All of these are defined below.) A script notation will be used for the total (volume-integrated) amounts of these and other quantities in the system. That is, for an arbitrary variable  $\widetilde{A}$ , the total amount of  $\widetilde{A}$ is

(2.3) $$\begin{eqnarray}\widetilde{\mathscr{A}}(t)\doteq \displaystyle \int \text{d}\boldsymbol{r}\,\widetilde{A}(\boldsymbol{r},t).\end{eqnarray}$$

Such total quantities will be seen to appear in the reference distribution  $F_{0}$ ((2.37) below), which is a spatially dependent generalization of the Gibbs distribution.

The averages of the microscopic densities will be denoted by lower-case quantities:

(2.4) $$\begin{eqnarray}\boldsymbol{a}_{s}(\boldsymbol{r},t)\doteq \langle \widetilde{\boldsymbol{A}}_{s}\rangle =(n_{s}\;\;\boldsymbol{p}_{s}^{\text{T}}\;\;e_{s})^{\text{T}}.\end{eqnarray}$$

Lower case will also be used for other intensive quantities such as the pressure  $p$ . (Temperature  $T$ is an exception, as lower case would conflict with time  $t$ .)

I shall use Greek letters to denote the components of the hydrodynamic vectors: $\widetilde{\boldsymbol{A}}\rightarrow \widetilde{A}^{\unicode[STIX]{x1D707}}$ and $\boldsymbol{a}\rightarrow a^{\unicode[STIX]{x1D707}}$ , where $\unicode[STIX]{x1D707}\in \{n,\;\boldsymbol{p},\;e\}$ . (Brey et al. used upper case for those quantities.) The Einstein summation convention will apply for repeated indices. Species labels will be subsumed into the field indices unless they are written explicitly for emphasis.

2.2.1 Number density $\widetilde{N}$

The microscopic number density is

(2.5) $$\begin{eqnarray}\widetilde{N}_{s}(\boldsymbol{r},t)\doteq \mathop{\sum }_{i\in s}\unicode[STIX]{x1D6FF}\boldsymbol{(}\boldsymbol{r}-\widetilde{\boldsymbol{x}}_{i}(t)\boldsymbol{)},\end{eqnarray}$$

where $\sum _{i\in s}$ denotes summation over all of the $\mathscr{N}_{s}$  particles of species  $s$ . Obviously, $\widetilde{N}_{s}$  depends on (some of) the random phase-space coordinates $\widetilde{\unicode[STIX]{x1D6E4}}(t)$ , but that dependence will be indicated only by the tilde. In terms of the Klimontovich microdensity  $\widetilde{f}_{s}$ as usually defined (such that its velocity integral is dimensionless),

(2.6) $$\begin{eqnarray}\widetilde{f}_{s}(\boldsymbol{r},\boldsymbol{v},t)\doteq \frac{1}{\overline{n}_{s}}\mathop{\sum }_{i\in s}\unicode[STIX]{x1D6FF}\boldsymbol{(}\boldsymbol{r}-\widetilde{\boldsymbol{x}}_{i}(t)\boldsymbol{)}\unicode[STIX]{x1D6FF}\boldsymbol{(}\boldsymbol{v}-\widetilde{\boldsymbol{v}}_{i}(t)\boldsymbol{)}\end{eqnarray}$$

(where $\overline{n}_{s}\doteq \mathscr{N}_{s}/\mathscr{V}$ is the mean density of species  $s$ ), one has

(2.7) $$\begin{eqnarray}\widetilde{N}_{s}(\boldsymbol{r},t)=\overline{n}_{s}\displaystyle \int \text{d}\boldsymbol{v}\,\widetilde{f}_{s}(\boldsymbol{r},\boldsymbol{v},t).\end{eqnarray}$$

The average over a pure (statistically homogeneous) thermal-equilibrium ensemble is

(2.8) $$\begin{eqnarray}\langle \widetilde{N}_{s}\rangle =\mathop{\sum }_{i\in s}\displaystyle \int \frac{\text{d}\boldsymbol{x}_{i}}{\mathscr{V}}\,\unicode[STIX]{x1D6FF}(\boldsymbol{r}-\boldsymbol{x}_{i})=\mathscr{N}_{s}/\mathscr{V}=\overline{n}_{s}=\text{const}.\end{eqnarray}$$

In an arbitrary ensemble, the average of (2.7) leads to a general formula for the mean particle density:

(2.9) $$\begin{eqnarray}n_{s}(\boldsymbol{r},t)=\langle \widetilde{N}_{s}(\boldsymbol{r},t)\rangle =\overline{n}_{s}\displaystyle \int \text{d}\boldsymbol{v}\,f_{s}(\boldsymbol{r},\boldsymbol{v},t),\end{eqnarray}$$

where $f\doteq \langle \,\widetilde{f}\rangle$  is the one-particle distribution function in a convenient normalization.Footnote 28 The total particle number is

(2.10) $$\begin{eqnarray}\widetilde{\mathscr{N}}_{s}(t)\doteq \displaystyle \int \text{d}\boldsymbol{r}\mathop{\sum }_{i\in s}\unicode[STIX]{x1D6FF}\boldsymbol{(}\boldsymbol{r}-\widetilde{\boldsymbol{x}}_{i}(t)\boldsymbol{)}=\mathscr{N}_{s}.\end{eqnarray}$$

Note that although in principle $\widetilde{\mathscr{N}}_{s}$   is random, it is constant in the ensemble considered here.

2.2.2 Momentum density $\widetilde{\boldsymbol{P}}$

The microscopic momentum density is

(2.11) $$\begin{eqnarray}\widetilde{\boldsymbol{P}}_{s}(\boldsymbol{r},t)\doteq \mathop{\sum }_{i\in s}m_{s}\widetilde{\boldsymbol{v}}_{i}(t)\unicode[STIX]{x1D6FF}\boldsymbol{(}\boldsymbol{r}-\widetilde{\boldsymbol{x}}_{i}(t)\boldsymbol{)}.\end{eqnarray}$$

The total momentum is

(2.12) $$\begin{eqnarray}\widetilde{\pmb{\pmb{\mathscr{G}}}}_{s}(t)\doteq \displaystyle \int \text{d}\boldsymbol{r}\,\widetilde{\boldsymbol{P}}_{s}(\boldsymbol{r},t)=\mathop{\sum }_{i\in s}m_{s}\widetilde{\boldsymbol{v}}_{i}(t)=\mathbf{0}.\end{eqnarray}$$

The last equality is an assumption.Footnote 29 The mean momentum density is

(2.13) $$\begin{eqnarray}\boldsymbol{p}_{s}(\boldsymbol{r},t)\doteq \langle \widetilde{\boldsymbol{P}}_{s}(\boldsymbol{r},t)\rangle =n_{s}(\boldsymbol{r},t)m_{s}\boldsymbol{u}_{s}(\boldsymbol{r},t),\end{eqnarray}$$

which defines the mean flow velocity  $\boldsymbol{u}_{s}$ .

2.2.3 Energy density $\widetilde{E}$

The microscopic energy density is

(2.14) $$\begin{eqnarray}\widetilde{E}_{s}(\boldsymbol{r},t)\doteq \mathop{\sum }_{i\in s}\widetilde{E}_{i}(t)\unicode[STIX]{x1D6FF}\boldsymbol{(}\boldsymbol{r}-\widetilde{\boldsymbol{x}}_{i}(t)\boldsymbol{)},\end{eqnarray}$$

where

(2.15) $$\begin{eqnarray}\widetilde{E}_{i}\doteq {\textstyle \frac{1}{2}}m\widetilde{v}_{i}^{2}(t)+\widetilde{U}_{i}(t)\end{eqnarray}$$

and the microscopic potential energy is

(2.16) $$\begin{eqnarray}\widetilde{U}_{i}(t)\doteq \frac{1}{2}\mathop{\sum }_{j\neq i}^{\mathscr{N}}\widetilde{U}_{ij}(t),\end{eqnarray}$$

with $\widetilde{U}_{ij}(t)\equiv U_{ij}\boldsymbol{(}\widetilde{\boldsymbol{x}}_{i}(t)-\widetilde{\boldsymbol{x}}_{j}(t)\boldsymbol{)}$ being the two-particle potential energy. (In the last expression, the subscripts on  $U_{ij}$ denote dependence on the charges  $q_{i}$ and  $q_{j}$ . By assumption, there is no external potential.) Note that the $j$  sum in (2.16) is extended over all particles, so $\widetilde{U}_{i}$  contains contributions from multiple species. Equation (2.14) can be written as

(2.17) $$\begin{eqnarray}\widetilde{E}_{s}(\boldsymbol{r},t)={\textstyle \frac{1}{2}}m_{s}u_{s}^{2}(\boldsymbol{r},t)\widetilde{N}_{s}(\boldsymbol{r},t)+\widetilde{E}_{0,s}(\boldsymbol{r},t),\end{eqnarray}$$

where, with $\widetilde{\boldsymbol{w}}_{i}\doteq \widetilde{\boldsymbol{v}}_{i}-\boldsymbol{u}_{s}(\boldsymbol{r},t)$ ,

(2.18) $$\begin{eqnarray}\widetilde{E}_{0,s}(\boldsymbol{r},t)\doteq \mathop{\sum }_{i\in s}\left(\frac{1}{2}m_{s}\widetilde{w}_{i}^{2}(t)+\widetilde{U}_{i}(t)\right)\unicode[STIX]{x1D6FF}\boldsymbol{(}\boldsymbol{r}-\widetilde{\boldsymbol{x}}_{i}(t)\boldsymbol{)}\end{eqnarray}$$

is the total thermal energy density with respect to the local moving frame. The total amount of particle energy (mechanical plus thermal) isFootnote 30

(2.19) $$\begin{eqnarray}\widetilde{\mathscr{E}}_{s}(t)\doteq \mathop{\sum }_{i\in s}\left(\frac{1}{2}m_{s}\widetilde{v}_{i}^{2}(t)+\widetilde{U}_{i}(t)\right).\end{eqnarray}$$

The mean energy density, including mechanical motion, is

(2.20) $$\begin{eqnarray}e_{s}(\boldsymbol{r},t)=\langle \widetilde{E}_{s}(\boldsymbol{r},t)\rangle ={\textstyle \frac{1}{2}}m_{s}u_{s}^{2}(\boldsymbol{r},t)n_{s}(\boldsymbol{r},t)+\langle \widetilde{E}_{0,s}(\boldsymbol{r},t)\rangle .\end{eqnarray}$$

Upon performing the velocity average according to ${\textstyle \frac{1}{2}}m\langle w^{2}\rangle ={\textstyle \frac{3}{2}}T$ , one findsFootnote 31

(2.21) $$\begin{eqnarray}\langle \widetilde{E}_{0,s}(\boldsymbol{r},t)\rangle ={\textstyle \frac{3}{2}}n_{s}(\boldsymbol{r},t)T_{s}(\boldsymbol{r},t)+u_{s}^{\text{int}}(\boldsymbol{r},t),\end{eqnarray}$$

where $u_{s}^{\text{int}}$  is the internal energy density.

In spite of my convention that a tilde denotes a random variable, I shall frequently drop it for the particle variables  $\boldsymbol{x}_{i}$ and  $\boldsymbol{v}_{i}$ since the presence of a particle index also indicates a random nature. I shall also drop the time argument for those variables when there is no possibility of confusion between the Schrödinger and Heisenberg representations of phase-space averages. Thus, I shall write formulas such as $\widetilde{N}_{s}(\boldsymbol{r})\doteq \sum _{i\in s}\unicode[STIX]{x1D6FF}(\boldsymbol{r}-\boldsymbol{x}_{i})$ .

2.3 The microscopic fluxes

For a one-component, unmagnetized system with short-ranged forces, it is well known that the time derivatives of the microscopic densities can be written as the divergences of microscopic fluxes or currents according to

(2.22) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\widetilde{\boldsymbol{A}}(\boldsymbol{r},t)=-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\widetilde{\boldsymbol{J}}(\boldsymbol{r},t),\end{eqnarray}$$

where $\widetilde{\boldsymbol{J}}$  is a column vector of random currents. Specifically,

(2.23a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}\widetilde{N} & = & \displaystyle -\unicode[STIX]{x1D735}\boldsymbol{\cdot }(m^{-1}\widetilde{\boldsymbol{P}}),\end{eqnarray}$$
(2.23b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}\widetilde{\boldsymbol{P}} & = & \displaystyle -\unicode[STIX]{x1D735}\boldsymbol{\cdot }\widetilde{\unicode[STIX]{x1D749}},\end{eqnarray}$$
(2.23c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}\widetilde{E} & = & \displaystyle -\unicode[STIX]{x1D735}\boldsymbol{\cdot }\widetilde{\boldsymbol{J}}\text{}^{E},\end{eqnarray}$$
where the Fourier transforms of the microscopic stress tensorFootnote 32   $\widetilde{\unicode[STIX]{x1D749}}$ and energy current  $\widetilde{\boldsymbol{J}}\text{}^{E}$ are
(2.24a ) $$\begin{eqnarray}\displaystyle \widetilde{\unicode[STIX]{x1D749}}(\boldsymbol{k}) & \doteq & \displaystyle \mathop{\sum }_{i=1}^{\mathscr{N}}[m\boldsymbol{v}_{i}\boldsymbol{v}_{i}+\unicode[STIX]{x0394}\widetilde{\unicode[STIX]{x1D749}}_{i}(\boldsymbol{k})]\text{e}^{-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{i}},\end{eqnarray}$$
(2.24b ) $$\begin{eqnarray}\displaystyle \widetilde{\boldsymbol{J}}\text{}^{E}(\boldsymbol{k}) & \doteq & \displaystyle \mathop{\sum }_{i=1}^{\mathscr{N}}[E_{i}\boldsymbol{v}_{i}+\unicode[STIX]{x0394}\widetilde{\unicode[STIX]{x1D749}}_{i}(\boldsymbol{k})\boldsymbol{\cdot }\boldsymbol{v}_{i}]\text{e}^{-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{i}},\end{eqnarray}$$
with (appendix B)
(2.25) $$\begin{eqnarray}\unicode[STIX]{x0394}\widetilde{\unicode[STIX]{x1D749}}_{i}(\boldsymbol{k})\doteq -\frac{1}{2}\mathop{\sum }_{ij}^{\mathscr{N}}\frac{\unicode[STIX]{x2202}\widetilde{U}_{ij}}{\unicode[STIX]{x2202}\boldsymbol{x}_{ij}}\,\boldsymbol{x}_{ij}\left(\frac{\text{e}^{\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{ij}}-1}{\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{ij}}\right).\end{eqnarray}$$

For multiple species, the density continuity equation (2.23a ) holds separately for each species. However, for a plasma the momentum and energy equations must be revisited in order to account for the long-ranged nature of the Coulomb force, the Lorentz force, and random interspecies coupling or exchange terms. As shown in appendix B, one findsFootnote 33

(2.26a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}\widetilde{N}_{s}(\boldsymbol{r},t) & = & \displaystyle -\unicode[STIX]{x1D735}\boldsymbol{\cdot }(m_{s}^{-1}\widetilde{\boldsymbol{P}}_{s}),\end{eqnarray}$$
(2.26b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}\widetilde{\boldsymbol{P}}_{s}(\boldsymbol{r},t) & = & \displaystyle (nq)_{s}\boldsymbol{E}+\unicode[STIX]{x1D714}_{\text{c}s}\widetilde{\boldsymbol{P}}_{s}\times \widehat{\boldsymbol{b}}-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\widetilde{\unicode[STIX]{x1D749}}_{s}^{\prime }+\dot{\widetilde{\boldsymbol{P}}}_{\unicode[STIX]{x0394},s}\text{}s_{+}\prime ,\end{eqnarray}$$
(2.26c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}\widetilde{E}_{s}(\boldsymbol{r},t) & = & \displaystyle -\unicode[STIX]{x1D735}\boldsymbol{\cdot }\widetilde{\boldsymbol{J}}\text{}_{s}^{E}+\dot{\widetilde{E}}_{\unicode[STIX]{x0394},s},\end{eqnarray}$$
where $\boldsymbol{E}(\boldsymbol{r},t)$  is the macroscopic electric field determined via Poisson’s equation (an external electric field is assumed to vanish), $\widehat{\boldsymbol{b}}$  is the unit vector in the direction of the magnetic field, $\unicode[STIX]{x1D714}_{\text{c}s}\doteq (qB/mc)_{s}$ , a prime denotes removal of any contribution from the long-ranged mean potential, $\widetilde{\unicode[STIX]{x1D749}}_{s}^{\prime }$  is discussed in appendix B, and $\dot{\widetilde{\boldsymbol{P}}}_{\unicode[STIX]{x0394},s}\text{}s_{+}\prime$ and  $\dot{\widetilde{E}}_{\unicode[STIX]{x0394},s}$ represent the random momentum and energy exchanges due to interactions with other species. Ultimately, the terms with a $\unicode[STIX]{x0394}$  subscript lead to collisional interspecies equilibrations of momentum and energy, which I shall often call exchange effects. For weakly coupled plasmas, the consequences of those effects are already well known to first order in the interspecies velocity and temperature differences  $\unicode[STIX]{x0394}\boldsymbol{u}$ and  $\unicode[STIX]{x0394}T$ , as discussed in Part 1. At second order, a plethora of additional terms will arise involving $\unicode[STIX]{x0394}\boldsymbol{u}\,\unicode[STIX]{x0394}\boldsymbol{u}$ , $\unicode[STIX]{x0394}\boldsymbol{u}\,\unicode[STIX]{x0394}T$ , and  $\unicode[STIX]{x0394}T^{2}$ as well as cross-terms involving products of the differences and gradients, such as $\unicode[STIX]{x0394}\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}T$ . I shall neglect all such effects (as did Catto & Simakov). In the general, strongly coupled case, analysis of those terms ultimately requires the derivation of a collision operator that goes beyond the Landau and Balescu–Lenard forms, but that is beyond the scope of this paper.

For general manipulations, it is useful to write (2.26) succinctly as

(2.27) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\widetilde{\boldsymbol{A}}_{s}=\widetilde{\boldsymbol{F}}_{\text{EM},s}-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\widetilde{\boldsymbol{J}}_{s}+\dot{\widetilde{\boldsymbol{A}}}_{\unicode[STIX]{x0394},s}\text{}s_{+}\prime .\end{eqnarray}$$

This involves the random electromagnetic forceFootnote 34   $\widetilde{\boldsymbol{F}}_{\text{EM},s}$ , which has only a momentum component; the generalized random currents  $\widetilde{\boldsymbol{J}}_{s}$ ; and the random exchange terms  $\dot{\widetilde{\boldsymbol{A}}}_{\unicode[STIX]{x0394},s}\text{}s_{+}\prime$ .

2.4 The local-equilibrium distribution

The local-equilibrium distributionFootnote 35   $F_{\boldsymbol{B}}(\unicode[STIX]{x1D6E4};t)$ generalized to include multiple species is defined by

(2.28) $$\begin{eqnarray}F_{\boldsymbol{B}}(\unicode[STIX]{x1D6E4};t)\doteq [Z_{\boldsymbol{B}}(t)]^{-1}\exp \left(\mathop{\sum }_{\overline{s}}\displaystyle \int \text{d}\overline{\boldsymbol{r}}\,\boldsymbol{A}_{\overline{s}}(\unicode[STIX]{x1D6E4};\overline{\boldsymbol{r}})\boldsymbol{\cdot }\boldsymbol{B}_{\overline{s}}(\overline{\boldsymbol{r}},t)\right),\end{eqnarray}$$

where

(2.29) $$\begin{eqnarray}Z_{\boldsymbol{B}}(t;[\boldsymbol{B}])\doteq \displaystyle \int \text{d}\unicode[STIX]{x1D6E4}\,\exp \left(\mathop{\sum }_{\overline{s}}\displaystyle \int \text{d}\overline{\boldsymbol{r}}\,\boldsymbol{A}_{\overline{s}}(\unicode[STIX]{x1D6E4};\overline{\boldsymbol{r}})\boldsymbol{\cdot }\boldsymbol{B}_{\overline{s}}(\overline{\boldsymbol{r}},t)\right).\end{eqnarray}$$

The choice of the conjugate variables Footnote 36   $\boldsymbol{B}_{s}$ will be discussed below and in appendix C. $F_{\boldsymbol{B}}(\unicode[STIX]{x1D6E4};t)$ is the generalization of an equilibrium Gibbs distribution to the general case in which the thermodynamic variables vary with time and assume different values in different local regions of space. (Think of the $\overline{\boldsymbol{r}}$  integration as a Riemann sum.) A useful shorthand notation is to define a $\star$  operationFootnote 37 by

(2.30) $$\begin{eqnarray}\mathop{\sum }_{\overline{s}}\displaystyle \int \text{d}\overline{\boldsymbol{r}}\,\boldsymbol{A}_{\overline{s}}(\overline{\boldsymbol{r}},t)\boldsymbol{\cdot }\boldsymbol{B}_{\overline{s}}(\overline{\boldsymbol{r}},t)\equiv \boldsymbol{A}\star \boldsymbol{B};\end{eqnarray}$$

then one can write

(2.31) $$\begin{eqnarray}F_{\boldsymbol{B}}(\unicode[STIX]{x1D6E4};t)=\frac{\text{e}^{\boldsymbol{A}\star \boldsymbol{B}}}{\displaystyle \int \text{d}\unicode[STIX]{x1D6E4}\,\text{e}^{\boldsymbol{A}\star \boldsymbol{B}}}.\end{eqnarray}$$

The local-equilibrium partition function  $Z_{\boldsymbol{B}}[t;\boldsymbol{B}]$ , a time-independent functional of $\boldsymbol{B}_{s}(\boldsymbol{r},t)$ at a particular time  $t$ , is a generating function for the equal-time moments of  $\widetilde{\boldsymbol{A}}_{s}$ in the local-equilibrium ensemble; similarly, its logarithm is a cumulant generating function. For example, upon suppressing the time arguments,

(2.32a ) $$\begin{eqnarray}\displaystyle \boldsymbol{a}_{s,\boldsymbol{B}}(\boldsymbol{r})\doteq \langle \widetilde{\boldsymbol{A}}_{s}(\boldsymbol{r})\rangle _{\boldsymbol{B}} & = & \displaystyle \frac{\unicode[STIX]{x1D6FF}\ln Z_{\boldsymbol{B}}}{\unicode[STIX]{x1D6FF}\boldsymbol{B}_{s}(\boldsymbol{r})},\end{eqnarray}$$
(2.32b ) $$\begin{eqnarray}\displaystyle \langle \widetilde{\boldsymbol{A}}\text{}s_{+}\prime _{s,\boldsymbol{B}}(\boldsymbol{r})\widetilde{\boldsymbol{A}}\text{}s_{+}\prime _{s^{\prime },\boldsymbol{B}}(\boldsymbol{r}s_{+}\prime )\rangle _{\boldsymbol{B}} & = & \displaystyle \frac{\unicode[STIX]{x1D6FF}^{2}\ln Z_{\boldsymbol{ B}}}{\unicode[STIX]{x1D6FF}\boldsymbol{B}_{s}(\boldsymbol{r})\unicode[STIX]{x1D6FF}\boldsymbol{B}_{ss_{+}\prime }(\boldsymbol{r}s_{+}\prime )}=\frac{\unicode[STIX]{x1D6FF}\langle \widetilde{\boldsymbol{A}}_{s}(\boldsymbol{r})\rangle _{\boldsymbol{B}}}{\unicode[STIX]{x1D6FF}\boldsymbol{B}_{ss_{+}\prime }(\boldsymbol{r}s_{+}\prime )},\end{eqnarray}$$
where $\widetilde{\boldsymbol{A}}\text{}s_{+}\prime _{s,\boldsymbol{B}}\doteq \widetilde{\boldsymbol{A}}_{s}-\langle \widetilde{\boldsymbol{A}}_{s}\rangle _{\boldsymbol{B}}$ is the fluctuation with respect to the local mean. Note that the multipoint correlation functions so defined are not quite the true ones for the non-equilibrium $\mathscr{N}$ -particle system because the expectations are calculated in the local-equilibrium ensemble, not the exact one described by $F(\unicode[STIX]{x1D6E4},t)$ . The latter contains the dissipative transport fluxes, while the former does not.

It is technically convenient to choose the $\boldsymbol{B}$ variables such that the components of $\boldsymbol{a}$ are the true fluid variables; thus, one enforces the constraint

(2.33) $$\begin{eqnarray}\boldsymbol{a}_{s}(\boldsymbol{r},t)=\boldsymbol{a}_{s,\boldsymbol{B}}(\boldsymbol{r},t).\end{eqnarray}$$

This choice is discussed at length by Piccirelli (Reference Piccirelli1968). The basic idea is that the local equilibrium ensemble is supposed to provide a good zeroth-order starting point for a subsequent expansion in the gradients of  $\boldsymbol{B}$ (or, more generally, in the differences  $\unicode[STIX]{x0394}\boldsymbol{B}$ defined by (2.35) below, which include exchange effects). This choice is not unique, and Piccirelli points out that a variety of forms for  $F_{\boldsymbol{B}}$ will lead to similar expansions in terms of other quantities  $\unicode[STIX]{x0394}\boldsymbol{B}s_{+}\prime$ . However, if the present choice is not made, various difficulties may ensue: $\unicode[STIX]{x0394}\boldsymbol{B}s_{+}\prime$  may not have a correspondence to natural thermodynamic forces that can be measured physically, or there may be difficulty in the preparation of an appropriate initial state.Footnote 38

For the one-component fluid, Piccirelli showed that the conjugate variables  $\boldsymbol{B}$ are

(2.34) $$\begin{eqnarray}\boldsymbol{B}(\boldsymbol{r},t)=\boldsymbol{(}\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D707}-{\textstyle \frac{1}{2}}mu^{2})\;\;\hspace{2.22198pt}\unicode[STIX]{x1D6FD}\boldsymbol{u}^{\text{T}}\;\;\hspace{2.22198pt}-\unicode[STIX]{x1D6FD}\boldsymbol{)}^{\text{T}},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FD}\doteq T^{-1}$ and all of the quantities on the right-hand side are functions of  $\boldsymbol{r}$ and  $t$ . In appendix C, I give some details of the argument extended to the multispecies case and conclude that this choice remains valid with the mere addition of species labels to all quantities in (2.34).

The method to be described shortly will express the time evolution of  $\boldsymbol{a}$ in terms of  $\boldsymbol{B}$ ; it does not depend on the definition of  $\boldsymbol{B}$ . However, in order to obtain a closed set of transport equations, one must express $\boldsymbol{B}$ in terms of  $\boldsymbol{a}$ . In a collision-dominated state, the energy density  $e$ will be some function of  $n$ , $\boldsymbol{u}$ and  $T$ . Thus, one can use  $T$ instead of  $e$ as one of the fluid variables. Then, given the result (2.34), the system $\{\unicode[STIX]{x2202}_{t}n=f_{n}(\boldsymbol{B}),\;\unicode[STIX]{x2202}_{t}\boldsymbol{u}=f_{\boldsymbol{u}}(\boldsymbol{B}),\;\unicode[STIX]{x2202}_{t}T=f_{T}(\boldsymbol{B})\}$ (where the $f$  functions depend on  $\boldsymbol{B}$ and/or  $\unicode[STIX]{x1D735}\boldsymbol{B}$ ) is closed in the variables  $\boldsymbol{u}$ and  $T=\unicode[STIX]{x1D6FD}^{-1}$ ; the only remaining closure problem is to determine the chemical potential  $\unicode[STIX]{x1D707}(n,T)$ . That can be done by using thermodynamic relations valid in local thermal equilibrium. See § 2-S:1.1 for further details.

2.5 The reference distribution

Consider a particular reference space–time point $(\boldsymbol{r},t)$ and reference species  $s$ , then defineFootnote 39

(2.35) $$\begin{eqnarray}\unicode[STIX]{x0394}\boldsymbol{B}_{\overline{s}s}(\overline{\boldsymbol{r}},\boldsymbol{r},t)\doteq \boldsymbol{B}_{\overline{s}}(\overline{\boldsymbol{r}},t)-\boldsymbol{B}_{s}(\boldsymbol{r},t).\end{eqnarray}$$

Equation (2.28) can then be written as

(2.36) $$\begin{eqnarray}F_{\boldsymbol{B}}(\unicode[STIX]{x1D6E4};t)=Z_{\boldsymbol{B}}^{-1}\exp \left(\vphantom{\left(\sum \right)}\right.\!\!\displaystyle \mathop{\sum }_{\overline{s}}\pmb{\pmb{\mathscr{A}}}_{\overline{s}}\boldsymbol{\cdot }\boldsymbol{B}_{\overline{s}}(\boldsymbol{r},t)+\mathop{\sum }_{\overline{s}}\displaystyle \int \text{d}\overline{\boldsymbol{r}}\,\boldsymbol{A}_{\overline{s}}(\overline{\boldsymbol{r}})\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{B}_{\overline{s}\,\overline{s}}(\overline{\boldsymbol{r}},\boldsymbol{r},t)\!\!\left.\vphantom{\left(\sum \right)}\right).\end{eqnarray}$$

(This quantity is independent of  $\boldsymbol{r}$ even though the individual terms depend on  $\boldsymbol{r}$ .) The reference distribution $F_{0}$ is defined by omitting the terms in  $\unicode[STIX]{x0394}\boldsymbol{B}$ . (The effects due to  $\unicode[STIX]{x0394}\boldsymbol{B}$ will be seen to be small when appropriate averages are taken, so they can be treated as perturbative corrections to  $F_{0}$ .) Thus,

(2.37) $$\begin{eqnarray}F_{0}(\unicode[STIX]{x1D6E4};\boldsymbol{r},t)\doteq [Z_{0}(\boldsymbol{r},t)]^{-1}\exp \left(\vphantom{\left(\sum \right)}\right.\!\!\displaystyle \mathop{\sum }_{\overline{s}}\pmb{\pmb{\mathscr{A}}}_{\overline{s}}(\unicode[STIX]{x1D6E4})\boldsymbol{\cdot }\boldsymbol{B}_{\overline{s}}(\boldsymbol{r},t)\!\!\left.\vphantom{\left(\sum \right)}\right),\end{eqnarray}$$

where the local partition function is

(2.38) $$\begin{eqnarray}Z_{0}(\boldsymbol{r},t)\doteq \displaystyle \int \text{d}\unicode[STIX]{x1D6E4}\,\exp \left(\vphantom{\left(\sum \right)}\right.\!\!\displaystyle \mathop{\sum }_{\overline{s}}\pmb{\pmb{\mathscr{A}}}_{\overline{s}}(\unicode[STIX]{x1D6E4})\boldsymbol{\cdot }\boldsymbol{B}_{\overline{s}}(\boldsymbol{r},t)\!\!\left.\vphantom{\left(\sum \right)}\right).\end{eqnarray}$$

Here $\pmb{\pmb{\mathscr{A}}}_{s}(\unicode[STIX]{x1D6E4})\doteq \pmb{\pmb{\mathscr{A}}}_{s}\boldsymbol{(}\widetilde{\unicode[STIX]{x1D6E4}}(t)\boldsymbol{)}|_{\widetilde{\unicode[STIX]{x1D6E4}}(t)=\unicode[STIX]{x1D6E4}}$ . The order of the arguments in $F_{0}(\unicode[STIX]{x1D6E4};\boldsymbol{r},t)$ emphasizes that $F_{0}$  is fundamentally a function of  $\unicode[STIX]{x1D6E4}$ but is parametrized by  $\boldsymbol{r}$ and  $t$ . By rearranging the dot product and using the definition (2.34) of  $\boldsymbol{B}$ , one can see that

(2.39) $$\begin{eqnarray}F_{0}(\unicode[STIX]{x1D6E4})=Z_{0}^{-1}\exp \left(\vphantom{\left(\sum \right)}\right.\!\!\displaystyle \mathop{\sum }_{\overline{s}}N_{\overline{s}}\unicode[STIX]{x1D6FD}_{\overline{s}}\unicode[STIX]{x1D707}_{\overline{s}}-\unicode[STIX]{x1D6FD}_{\overline{s}}\mathscr{E}_{0\overline{s}}(\unicode[STIX]{x1D6E4})\!\!\left.\vphantom{\left(\sum \right)}\right),\end{eqnarray}$$

where

(2.40) $$\begin{eqnarray}\mathscr{E}_{0s}(\unicode[STIX]{x1D6E4})\doteq \mathop{\sum }_{i\in s}\left(\frac{1}{2}m_{s}w_{i}^{2}+U_{i}\right)\end{eqnarray}$$

is the total energy with respect to the local reference frame moving with the velocity  $\boldsymbol{u}_{s}$ . Note that although the parameters  $\unicode[STIX]{x1D6FD}$ , $\unicode[STIX]{x1D707}$ , and  $\boldsymbol{u}$ depend on  $\boldsymbol{r}$ and  $t$ , the spatial phase-space dependence of  $F_{0}$ enters only through the internal energy, so $F_{0}$  is invariant under translations of the particle positions.

Differentiations with respect to the parameters  $(\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707})_{s}$ and  $-\unicode[STIX]{x1D6FD}_{s}$ generate the system-mean number and thermal-energy densities:

(2.41a ) $$\begin{eqnarray}\displaystyle \left(\frac{\unicode[STIX]{x2202}\ln Z_{0}}{\unicode[STIX]{x2202}[(\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707})_{s}(\boldsymbol{r},t)]}\right)_{\unicode[STIX]{x1D6FD}} & = & \displaystyle \langle \widetilde{\mathscr{N}}_{s}\rangle =\mathscr{N}_{s}=\mathscr{V}\overline{n}_{s},\end{eqnarray}$$
(2.41b ) $$\begin{eqnarray}\displaystyle \left(\frac{\unicode[STIX]{x2202}\ln Z_{0}}{\unicode[STIX]{x2202}[-\unicode[STIX]{x1D6FD}_{s}(\boldsymbol{r},t)]}\right)_{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}} & = & \displaystyle \langle \widetilde{\mathscr{E}}_{0}\rangle =\mathscr{V}\left(e_{0s}(\boldsymbol{r},t)-\frac{1}{2}m_{s}\overline{n}_{s}u_{s}^{2}(\boldsymbol{r},t)\right).\end{eqnarray}$$
Note that these are ordinary partial derivatives, not functional derivatives. I shall write, for example, $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FD}|_{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}}\equiv \unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FD}$ when it is clear which variable is being held fixed. One also holds  $\boldsymbol{u}$ fixed when performing those derivatives. Note that $\overline{n}$  does not depend on  $\boldsymbol{r}$ and  $t$ , while the reference energy density  $e_{0}$ does (e.g. through its dependence on the local temperature).

With the definition

(2.42) $$\begin{eqnarray}\unicode[STIX]{x0394}\boldsymbol{B}_{\overline{s}}(\overline{\boldsymbol{r}},\boldsymbol{r},t)\doteq \unicode[STIX]{x0394}\boldsymbol{B}_{\overline{s}\,\overline{s}}(\overline{\boldsymbol{r}},\boldsymbol{r},t)\end{eqnarray}$$

(subsequently, a  $\unicode[STIX]{x0394}\boldsymbol{B}$ without subscripts will denote this species-diagonal part), it can be seen from (2.36) that expectations of an arbitrary quantity  $\widetilde{G}$ with the local-equilibrium distribution  $F_{\boldsymbol{B}}$ can be reexpressed in terms of the reference distribution  $F_{0}$ as

(2.43) $$\begin{eqnarray}\langle \widetilde{G}\rangle _{B}(t)=\frac{\langle \widetilde{G}\,\text{e}^{\widetilde{\boldsymbol{A}}\star \unicode[STIX]{x0394}\boldsymbol{B}}\rangle _{0}}{\langle \text{e}^{\widetilde{\boldsymbol{A}}\star \unicode[STIX]{x0394}\boldsymbol{B}}\rangle _{0}}.\end{eqnarray}$$

This important result will be used later. Note that the left-hand side of (2.43) does not depend on  $\boldsymbol{r}$ even though both  $F_{0}$ and  $\unicode[STIX]{x0394}\boldsymbol{B}$ do. Clearly, the local-equilibrium average $\langle \ldots \rangle _{B}$ and the reference average $\langle \ldots \rangle _{0}$ are equivalent to zeroth order in  $\unicode[STIX]{x0394}\boldsymbol{B}$ .

2.6 Projection operators and subtracted fluxes

The ultimate goal is the derivation of closed evolution equations for the hydrodynamic variables: $\unicode[STIX]{x2202}_{t}\boldsymbol{a}_{s}=\cdots \,$ . The strategy is to evaluate the full expectation $\boldsymbol{a}_{s}=\langle \widetilde{\boldsymbol{A}}_{s}\rangle$ , which is taken with respect to the Liouville distribution  $F$ , by referring it to the reference distribution  $F_{0}$ :

(2.44) $$\begin{eqnarray}\boldsymbol{a}_{s}(\boldsymbol{r},t)=\displaystyle \int \text{d}\unicode[STIX]{x1D6E4}\,\boldsymbol{A}_{s}(\unicode[STIX]{x1D6E4};\boldsymbol{r})\left(\frac{F(\unicode[STIX]{x1D6E4},t)}{F_{0}(\unicode[STIX]{x1D6E4};\boldsymbol{r},t)}\right)F_{0}(\unicode[STIX]{x1D6E4};\boldsymbol{r},t).\end{eqnarray}$$

The ratio $F/F_{0}$ will be expanded to second order in the gradients with the aid of projection operators. Obviously, the utility of the manipulations depends on an apt choice of those operators.

2.6.1 The fundamental projection operator $\text{P}$

Given the definition (2.30) of the $\star$  operator, the form of the projection operator  $\text{P}$ used by Brey et al. holds as well for multiple species and is defined for arbitrary phase function  $\widetilde{\unicode[STIX]{x1D712}}$ in terms of averages over the reference distribution by

(2.45) $$\begin{eqnarray}\text{P}\widetilde{\unicode[STIX]{x1D712}}\doteq \langle \widetilde{\unicode[STIX]{x1D712}}\rangle _{0}+\widetilde{\boldsymbol{A}}\text{}s_{+}\prime ^{\text{T}}\star \unicode[STIX]{x1D648}_{}^{-1}\star \langle \widetilde{\boldsymbol{A}}\text{}s_{+}\prime \,\widetilde{\unicode[STIX]{x1D712}}s_{+}\prime \rangle _{0},\end{eqnarray}$$

where

(2.46a ) $$\begin{eqnarray}\displaystyle \boldsymbol{A}s_{+}\prime (\boldsymbol{r},t;\unicode[STIX]{x1D6E4}) & \doteq & \displaystyle \widetilde{\boldsymbol{A}}(\boldsymbol{r})|_{\widetilde{\unicode[STIX]{x1D6E4}}(t)=\unicode[STIX]{x1D6E4}}-\langle \widetilde{\boldsymbol{A}}\rangle _{0}\equiv \widetilde{\boldsymbol{A}}\text{}s_{+}\prime (\boldsymbol{r},t),\end{eqnarray}$$
(2.46b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D614}_{sss_{+}\prime }^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D707}s_{+}\prime }(\boldsymbol{x},\boldsymbol{x}s_{+}\prime ) & \doteq & \displaystyle \langle \widetilde{A}_{s}^{\prime \unicode[STIX]{x1D707}}(\boldsymbol{x},t)\widetilde{A}_{ss_{+}\prime }^{\prime \unicode[STIX]{x1D707}s_{+}\prime }(\boldsymbol{x}s_{+}\prime ,t)\rangle _{0}\equiv \unicode[STIX]{x1D614}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D707}s_{+}\prime }(\boldsymbol{x},\boldsymbol{x}s_{+}\prime ).\end{eqnarray}$$
In the last form, the species indices have been subsumed into the field indices. Note that $\unicode[STIX]{x1D648}_{}$  is an equal-time cumulant and that averaging with respect to  $F_{0}(\unicode[STIX]{x1D6E4};\boldsymbol{r},t)$ introduces a dependence on  $\boldsymbol{r}$ and  $t$ , which is indicated implicitly through the subscript 0 or the prime that indicates a fluctuation from the reference state. Strictly speaking, each of  $\text{P}$ , $\text{Q}\doteq 1-\text{P}$ , $\boldsymbol{A}s_{+}\prime$ , $\unicode[STIX]{x1D648}_{}$ , and various other quantities to be introduced below should be adorned with a subscript 0, as Brey et al. do. I shall omit those subscripts with the goal of somewhat uncluttering the notation. In more detail, equation (2.45) means
(2.47) $$\begin{eqnarray}\text{P}\widetilde{\unicode[STIX]{x1D712}}=\langle \widetilde{\unicode[STIX]{x1D712}}\rangle _{0}+\mathop{\sum }_{\overline{\unicode[STIX]{x1D707}},\overline{\unicode[STIX]{x1D707}}s_{+}\prime }\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,\text{d}\overline{\boldsymbol{x}}s_{+}\prime \,A^{\prime \,\overline{\unicode[STIX]{x1D707}}}(\overline{\boldsymbol{x}},t;\unicode[STIX]{x1D6E4})\unicode[STIX]{x1D614}_{\overline{\unicode[STIX]{x1D707}}\,\overline{\unicode[STIX]{x1D707}}s_{+}\prime }^{-1}(\overline{\boldsymbol{x}},\overline{\boldsymbol{x}}s_{+}\prime )\langle \widetilde{A}^{\prime \,\overline{\unicode[STIX]{x1D707}}s_{+}\prime }(\overline{\boldsymbol{x}}s_{+}\prime )\widetilde{\unicode[STIX]{x1D712}}\rangle _{0}(\boldsymbol{r},t),\end{eqnarray}$$

with $\unicode[STIX]{x1D707}$  denoting both the index for the components of the hydrodynamic column vector as well as species dependence. Note that $\text{P}=\text{P}(\unicode[STIX]{x1D6E4};\boldsymbol{r},t)$ , where the $(\boldsymbol{r},t)$ arises from the various averages over the reference distribution.

The significance of  $\text{P}$ is that it projects onto the hydrodynamic subspace defined by the reference state. Namely, upon suppressing $t$  arguments for brevity,

(2.48a ) $$\begin{eqnarray}\displaystyle \langle \widetilde{A}^{\unicode[STIX]{x1D70E}}(\boldsymbol{r})\text{P}\widetilde{\unicode[STIX]{x1D712}}\rangle _{0} & = & \displaystyle \langle \widetilde{A}^{\unicode[STIX]{x1D70E}}(\boldsymbol{r})\rangle _{0}\langle \widetilde{\unicode[STIX]{x1D712}}\rangle _{0}+\mathop{\sum }_{\overline{\unicode[STIX]{x1D707}},\overline{\unicode[STIX]{x1D707}}s_{+}\prime }\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,\text{d}\overline{\boldsymbol{x}}s_{+}\prime \,\langle \widetilde{A}^{\unicode[STIX]{x1D70E}}(\boldsymbol{r})\widetilde{A}^{\prime \,\overline{\unicode[STIX]{x1D707}}}(\overline{\boldsymbol{x}})\rangle \unicode[STIX]{x1D614}_{\overline{\unicode[STIX]{x1D707}}\,\overline{\unicode[STIX]{x1D707}}s_{+}\prime }^{-1}(\overline{\boldsymbol{x}},\overline{\boldsymbol{x}}s_{+}\prime )\langle \widetilde{A}^{\prime \,\overline{\unicode[STIX]{x1D707}}s_{+}\prime }(\overline{\boldsymbol{x}}s_{+}\prime )\widetilde{\unicode[STIX]{x1D712}}\rangle _{0}\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
(2.48b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle \langle \widetilde{A}^{\unicode[STIX]{x1D70E}}(\boldsymbol{r})\rangle _{0}\langle \widetilde{\unicode[STIX]{x1D712}}\rangle _{0}+\langle \widetilde{A}^{\prime \unicode[STIX]{x1D70E}}(\boldsymbol{r})\widetilde{\unicode[STIX]{x1D712}}\rangle _{0}\end{eqnarray}$$
(2.48c ) $$\begin{eqnarray}\displaystyle & = & \displaystyle \langle \widetilde{A}^{\unicode[STIX]{x1D70E}}(\boldsymbol{r})\widetilde{\unicode[STIX]{x1D712}}\rangle _{0}.\end{eqnarray}$$
If $\unicode[STIX]{x1D712}\doteq F/F_{0}$ , the average (2.48c ) produces the hydrodynamic variables $a^{\unicode[STIX]{x1D70E}}$ . (See (2.44); note that  $\unicode[STIX]{x1D70E}$ incorporates both a species index and a field index.)

The 𝛤-space projection operation $\text{P}\equiv _{\unicode[STIX]{x1D6E4}}\text{P}$ defined here generalizes the $\unicode[STIX]{x1D707}$ -space one

(2.49) $$\begin{eqnarray}_{\unicode[STIX]{x1D707}}\text{P}\doteq |A^{\overline{\unicode[STIX]{x1D707}}}\!\rangle \unicode[STIX]{x1D614}_{\overline{\unicode[STIX]{x1D707}}\,\overline{\unicode[STIX]{x1D707}}\prime }\langle \!A^{\overline{\unicode[STIX]{x1D707}}\prime }|\end{eqnarray}$$

that was used in Part 1 to project the weakly coupled kinetic equation onto the hydrodynamic subspace; see (1:3.20). That operation is local in space (potential-energy terms are neglected in the definitions of the $\boldsymbol{A}$  variables used for  $_{\unicode[STIX]{x1D707}}\text{P}$ ). It can be confusing to work with the $\unicode[STIX]{x1D707}$ -space Dirac notation because the scalar product implied by the bra involves a species summation, but (as explained in Part 1) that summation must be inhibited when a specific component  $A^{\unicode[STIX]{x1D707}}$ appears under the expectation. No such difficulty arises with  $_{\unicode[STIX]{x1D6E4}}\text{P}$ . The difference arises because $_{\unicode[STIX]{x1D6E4}}\text{P}$  is built from expectations taken with the $\mathscr{N}$ -particle distribution  $F(\unicode[STIX]{x1D6E4})$ , which has no preferred species dependence. On the other hand, $_{\unicode[STIX]{x1D707}}\text{P}$  is built from expectations taken with the one-particle distribution  $f_{s}(\boldsymbol{v})$ , which intrinsically involves a particular species. Given the choice between the 𝛤-space and the $\unicode[STIX]{x1D707}$ -space projection routes, the 𝛤-space one used in the present paper is the more general, the cleaner and the easier to understand.

Nevertheless, both routes must lead to the same  $\boldsymbol{a}_{s}$ for the same physics situation. It is therefore instructive to contemplate the subtle differences in the definitions (2.45) of  $_{\unicode[STIX]{x1D6E4}}\text{P}$ and (2.49) of  $_{\unicode[STIX]{x1D707}}\text{P}$ , and of the associated $\boldsymbol{A}$  variables. In the former, the mean is broken out separately from the remainder of the projection, defined with $_{\unicode[STIX]{x1D6E4}}\boldsymbol{A}_{s}^{\prime }\doteq (\widetilde{N}_{s}^{\prime },\;\widetilde{\boldsymbol{P}}\text{}s_{+}\prime _{\!s}\text{}^{\text{T}},\;\widetilde{E}_{s}^{\prime })^{\text{T}}$ (see (2.2)); in the latter, the mean is not broken out and the mixed vector $_{\unicode[STIX]{x1D707}}\boldsymbol{A}_{s}=\boldsymbol{(}1,\;\boldsymbol{P}\text{}s_{+}\prime _{\!s}\text{}^{\text{T}}(\boldsymbol{v}),\;K_{s}^{\prime }(\boldsymbol{v})\boldsymbol{)}^{\text{T}}$ (see (1:2.25)) is used. It is left as an exercise to convince oneself that both of these projections produce equivalent results for linear response in a weakly coupled plasma.

A general question is, how does one know that either  $_{\unicode[STIX]{x1D707}}\text{P}$ or  $_{\unicode[STIX]{x1D6E4}}\text{P}$ is ‘correct’? In fact, as was discussed in appendix  1:G, this question is unanswerable because, with appropriate care, any projection operator can be used; if there is a slow hydrodynamic manifold at all (Gorban & Karlin Reference Gorban and Karlin2014), its physics must be invariant to the choice of mathematical description. However, some projection operators are easier to work with than others in the course of constructing a gradient expansion. The $_{\unicode[STIX]{x1D707}}\text{P}$ employed in Part 1 and the $_{\unicode[STIX]{x1D6E4}}\text{P}$ used in the present paper are efficient because they project into the subspace related to globally conserved quantities. In the $\unicode[STIX]{x1D707}$ -space description used in Part 1, this is easy to see; the components of  $_{\unicode[STIX]{x1D707}}\boldsymbol{A}$ are built from the null eigenfunctions of the collision operator (here I focus on the one-component case for simplicity). The situation is more subtle in the present Part 2 formalism because no explicit collision operator has been written down or even assumed. But one still knows the quantities that are conserved globally; the components of  $_{\unicode[STIX]{x1D6E4}}\boldsymbol{A}$ are built from those. A beautiful discussion of the issues and general program is given by Piccirelli (Reference Piccirelli1968).

2.6.2 The basic subtracted fluxes

The quantities that will be shown to appear in the expressions for the transport coefficients are the subtracted fluxes  $\widehat{\boldsymbol{J}}\doteq \text{Q}\widetilde{\boldsymbol{J}}$ , a concept already familiar from the discussions in Part 1. (Note that a hatted quantity is also random, and of course $\text{P}\text{Q}=\text{Q}\text{P}=0$ .) The physical meaning of a subtracted flux is that it is the residual, gradient-driven portion of the total flux, over and above the microscopic part that already exists in local thermal equilibrium.

The projections of the basic fluxes are worked out in appendix D; one finds

(2.50a ) $$\begin{eqnarray}\displaystyle \widehat{\boldsymbol{J}}_{s}\text{}\!^{N} & = & \displaystyle 0,\end{eqnarray}$$
(2.50b ) $$\begin{eqnarray}\displaystyle \widehat{\boldsymbol{J}}_{s}\text{}\!^{\boldsymbol{P}}\equiv \widehat{\unicode[STIX]{x1D749}}_{s} & = & \displaystyle \widetilde{\unicode[STIX]{x1D749}}_{s}^{\prime }-\unicode[STIX]{x1D644}\left[\vphantom{\left(\frac{1}{2}\right)}\right.\!\!\,p_{s}+\mathop{\sum }_{\overline{s}}N_{\overline{s}}^{\prime }\left(\frac{\unicode[STIX]{x2202}p_{s}}{\unicode[STIX]{x2202}n_{\overline{s}}}\right)_{e}+\mathop{\sum }_{\overline{s}}E_{\overline{s}}^{\prime }\left(\frac{\unicode[STIX]{x2202}p_{s}}{\unicode[STIX]{x2202}e_{\overline{s}}}\right)_{n}\!\!\left.\vphantom{\left(\frac{1}{2}\right)}\right],\end{eqnarray}$$
(2.50c ) $$\begin{eqnarray}\displaystyle \widehat{\boldsymbol{J}}_{s}\text{}\!^{E} & = & \displaystyle \widetilde{\boldsymbol{J}}_{s}\text{}\!^{E}-\left(\frac{h}{mn}\right)_{s}\widetilde{\boldsymbol{P}}_{s}.\end{eqnarray}$$
Here $h\doteq e+p$  is the enthalpy density. (For an ideal gas, $h={\textstyle \frac{5}{2}}nT$ .) As is clear from appendix D, the subtracted fluxes are to be evaluated in the local frame with $\boldsymbol{u}_{s}=\mathbf{0}$ .

2.7 Formal solution of the Liouville equation using time-independent projection operators

In this section I shall describe the process of constructing a formal solution to the Liouville equation by using projection operators. Because the  $\text{P}$ as defined above depends on time due to its dependence on the reference distribution, straightforward application of the projection procedures described in Part 1 leads to technical complications involving time-ordered propagators.Footnote 40 To circumvent that, Brey et al. use a clever trick. They describe their strategy as follows:

The technical advantage of our method [over the ones used by Wong et al. (Reference Wong, MacLennan, Lindenfeld and Dufty1978) and earlier workers] will consist of using a reference state that is determined by the local properties of the system at the position and time of interest. We refer the evolution of the system for all times  $s$ prior to the chosen  $t$ to the reference distribution function $F_{0}(\unicode[STIX]{x1D6E4};\boldsymbol{r},t)$ ; i.e. the reference state is the same for the whole past evolution of the system. The motivation for this choice is clear, if we remember that we expect $F_{0}(\unicode[STIX]{x1D6E4};\boldsymbol{r},t)$ to carry the main information about the system at $(\boldsymbol{r},t)$ .

Following Brey et al., I write the solution of the Liouville equation (1.13) as

(2.51) $$\begin{eqnarray}F(\unicode[STIX]{x1D6E4},s)=\text{e}^{W(\unicode[STIX]{x1D6E4},s;\boldsymbol{r},t)}F_{0}(\unicode[STIX]{x1D6E4};\boldsymbol{r},t)\quad (s<t).\end{eqnarray}$$

This exact representation of  $F$ is not unique, but it will turn out to be convenient. It is the 𝛤-space generalization of the decomposition of the one-particle distribution function  $f$ used in Part 1, $f_{s}=f_{\text{M},s}+\unicode[STIX]{x1D712}_{s}f_{\text{M},s}+\cdots \,$ . Note that the  $\boldsymbol{r}$ and  $t$ dependence of  $W$ must turn out to be such that $F(\unicode[STIX]{x1D6E4},s)$ does not depend on  $\boldsymbol{r}$ and  $t$ . This is possible because the functions are related according to

(2.52) $$\begin{eqnarray}\ln F(\unicode[STIX]{x1D6E4},s)=W(\unicode[STIX]{x1D6E4},s;\boldsymbol{r},t)+\ln F_{0}(\unicode[STIX]{x1D6E4};\boldsymbol{r},t).\end{eqnarray}$$

To find an evolution equation for  $W$ , apply the linear operator $\unicode[STIX]{x2202}_{s}+\text{i}\mathscr{L}$ to (2.52) (note that the time derivative is with respect to  $s$ , not  $t$ ) and use (1.13) and (2.37):

(2.53) $$\begin{eqnarray}0=(\unicode[STIX]{x2202}_{s}+\text{i}\mathscr{L})W+\mathop{\sum }_{\overline{s}}(\text{i}\mathscr{L}\pmb{\pmb{\mathscr{A}}}_{\overline{s}})\boldsymbol{\cdot }\boldsymbol{B}_{\overline{s}}.\end{eqnarray}$$

From footnote 21 on page 15 and the spatial integrals of the microscopic evolution equations (2.26), one has

(2.54) $$\begin{eqnarray}\text{i}\mathscr{L}\pmb{\pmb{\mathscr{A}}}_{s}=\unicode[STIX]{x2202}_{t}\pmb{\pmb{\mathscr{A}}}_{s}=\displaystyle \int \text{d}\boldsymbol{r}\,[(nq)_{s}(\boldsymbol{E}+c^{-1}\boldsymbol{u}_{s}\times \boldsymbol{B}^{\text{ext}})\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x1D707}\boldsymbol{p}}-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{J}_{s}+\dot{\boldsymbol{A}}s_{+}\prime _{\unicode[STIX]{x0394},s}].\end{eqnarray}$$

(The Kronecker symbol $\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x1D707}\boldsymbol{p}}$ restricts the electromagnetic force term to the momentum equation.) The divergence term integrates away. So does the $(nq\boldsymbol{u})_{s}$ term by virtue of the assumption (2.12). In the one-component case the $\dot{\boldsymbol{A}}_{\unicode[STIX]{x0394}}$ term is absent and the $\boldsymbol{E}$  term can be shown to vanish,Footnote 41 so one concludes that $W$  obeys the homogeneous Liouville equation (Brey et al. Reference Brey, Zwanzig and Dorfman1981). More generally, total conservation of momentum and energy gives $\sum _{\overline{s}}\dot{\pmb{\pmb{\mathscr{A}}}}_{\unicode[STIX]{x0394},\overline{s}}^{\prime }=\mathbf{0}$ , and $\sum _{\overline{s}}\int \text{d}\boldsymbol{r}\,(nq)_{\overline{s}}\boldsymbol{E}=\int \text{d}\boldsymbol{r}\,\unicode[STIX]{x1D70C}\boldsymbol{E}$ vanishes by the same manipulation used in footnote 41. If one refers the  $\boldsymbol{B}_{\overline{s}}$ to a fixed  $\boldsymbol{B}_{s}$ , one has

(2.55a ) $$\begin{eqnarray}\displaystyle S_{\unicode[STIX]{x0394}}(\boldsymbol{r},t)\doteq -\mathop{\sum }_{\overline{s}}\dot{\pmb{\pmb{\mathscr{A}}}}_{\overline{s}}\boldsymbol{\cdot }\boldsymbol{B}_{\overline{s}}(\boldsymbol{r},t) & = & \displaystyle -\left[\vphantom{\left(\sum \right)}\right.\!\!\displaystyle \mathop{\sum }_{\overline{s}}\dot{\pmb{\pmb{\mathscr{A}}}}_{\overline{s}}\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{B}_{\overline{s}s}(\boldsymbol{r},t)+\left(\vphantom{\left(\sum \right)}\right.\!\!\displaystyle \mathop{\sum }_{\overline{s}}\dot{\pmb{\pmb{\mathscr{A}}}}_{\overline{s}}\!\!\left.\vphantom{\left(\sum \right)}\right)\displaystyle \boldsymbol{\cdot }\,\boldsymbol{B}_{s}\!\!\left.\vphantom{\left(\sum \right)}\right]\qquad\end{eqnarray}$$
(2.55b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle -\mathop{\sum }_{\overline{s}}\dot{\pmb{\pmb{\mathscr{A}}}}_{\unicode[STIX]{x0394},\overline{s}}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{B}_{\overline{s}s}(\boldsymbol{r},t).\end{eqnarray}$$
In obtaining the last result, I required $\int \text{d}\boldsymbol{r}\,(nq)_{s}\boldsymbol{E}=\mathbf{0}$ in order to ensure the bulk motion constraint (2.12). Thus, one needs to solve
(2.56) $$\begin{eqnarray}\unicode[STIX]{x2202}_{s}W+\text{i}\mathscr{L}W=S_{\unicode[STIX]{x0394}},\end{eqnarray}$$

where the effect of  $S_{\unicode[STIX]{x0394}}$ is no larger than first order. The source $S_{\unicode[STIX]{x0394}}$ will ultimately give rise to the momentum and energy exchange terms; for example, $\unicode[STIX]{x0394}\boldsymbol{B}_{ie}^{\boldsymbol{u}_{e}}(\boldsymbol{r},t)\doteq (\unicode[STIX]{x1D6FD}\boldsymbol{u})_{i}(\boldsymbol{r},t)-(\unicode[STIX]{x1D6FD}\boldsymbol{u})_{e}(\boldsymbol{r},t)\approx -T^{-1}[\boldsymbol{u}_{e}(\boldsymbol{r},t)-\boldsymbol{u}_{i}(\boldsymbol{r},t)]$ $=-\text{T}^{-1}\unicode[STIX]{x0394}\boldsymbol{u}(\boldsymbol{r},t)$ . Note that although $\unicode[STIX]{x0394}\boldsymbol{B}_{\overline{s}s}$ depends on  $s$ , $S_{\unicode[STIX]{x0394}}$  is independent of  $s$ .

It may seem that one has merely traded one very difficult problem – solution of the original, homogeneous Liouville equation (1.13) – for one of at least equal difficulty, namely the inhomogeneous Liouville equation (2.56) for  $W$ . However, as we shall see, the fact that both the reference distribution  $F_{0}$ and the projection operator  $\text{P}$ are built from  $\widetilde{\boldsymbol{A}}$ and  $\boldsymbol{B}$ enables one to make progress when an expansion in the gradients is desired. The strategy of Brey et al. is to solve for  $W$ by calculating  $\text{Q}W$ , then adding the result to the formal representation of  $\text{P}W$ . Because the required projection operations depend only on  $t$ , not  $s$ , they can be passed through the $s$  derivative in (2.56) and one has

(2.57a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{s}\text{P}W+\text{P}\text{i}\mathscr{L}\text{P}W+\text{P}\text{i}\mathscr{L}\text{Q}W & = & \displaystyle \text{P}S_{\unicode[STIX]{x0394}},\end{eqnarray}$$
(2.57b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{s}\text{Q}W+\text{Q}\text{i}\mathscr{L}\text{Q}W+\text{Q}\text{i}\mathscr{L}\text{P}W & = & \displaystyle \text{Q}S_{\unicode[STIX]{x0394}}.\end{eqnarray}$$
The solution of (2.57b ) is
(2.58) $$\begin{eqnarray}\text{Q}W(s)=\text{U}(s)W(0)-\displaystyle \int _{0}^{s}\!\text{d}\overline{s}\,\text{U}(s-\overline{s})\text{i}\mathscr{L}\text{P}W(\overline{s})+\unicode[STIX]{x1D6F6}_{\unicode[STIX]{x0394}}(s),\end{eqnarray}$$

where the modified propagator is

(2.59) $$\begin{eqnarray}\text{U}(s)\doteq \text{Q}\text{e}^{-\text{Q}\text{i}\mathscr{L}\text{Q}s}\text{Q}\end{eqnarray}$$

and

(2.60) $$\begin{eqnarray}\unicode[STIX]{x1D6F6}_{\unicode[STIX]{x0394}}(s)\doteq \left(\displaystyle \int _{0}^{s}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\,\text{U}(\overline{\unicode[STIX]{x1D70F}})\right)S_{\unicode[STIX]{x0394}}=\text{Q}\left(\displaystyle \int _{0}^{s}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\,\text{e}^{-\text{Q}\text{i}\mathscr{L}\overline{\unicode[STIX]{x1D70F}}}\right)\text{Q}S_{\unicode[STIX]{x0394}}.\end{eqnarray}$$

From the definition (2.45) of the projection operation, one finds the representation

(2.61) $$\begin{eqnarray}\text{P}W(s)=\unicode[STIX]{x1D714}(s)+\boldsymbol{A}s_{+}\prime ^{\text{T}}\star \boldsymbol{b}(s),\end{eqnarray}$$

where $\unicode[STIX]{x1D714}(s)\doteq \langle W\rangle _{0}(s)$ and

(2.62) $$\begin{eqnarray}\boldsymbol{b}_{s}(s)\doteq \unicode[STIX]{x1D648}_{}^{-1}\star \langle \boldsymbol{A}_{s}^{\prime }\,W(s)\rangle _{0}.\end{eqnarray}$$

Equation (2.58) can thus be written as

(2.63) $$\begin{eqnarray}\text{Q}W(s)=\text{U}(s)W(0)+\displaystyle \int _{0}^{s}\!\text{d}\overline{s}\,\boldsymbol{F}^{\text{T}}(s-\overline{s})\star \boldsymbol{b}(\overline{s})+\unicode[STIX]{x1D6F6}_{\unicode[STIX]{x0394}}(s),\end{eqnarray}$$

where

(2.64) $$\begin{eqnarray}\boldsymbol{F}_{s}(s)\doteq -\text{U}(s)\text{i}\mathscr{L}\boldsymbol{A}_{s}^{\prime }.\end{eqnarray}$$

Brey et al. argue that it is legitimate to assert the initial condition $\text{Q}W(0)=0$ . That removes the initial-condition term in (2.58). Then, upon adding (2.61) and (2.63), one obtains

(2.65) $$\begin{eqnarray}W(s)=\unicode[STIX]{x1D714}(s)+\boldsymbol{A}s_{+}\prime ^{\text{T}}\star \boldsymbol{b}(s)+\displaystyle \int _{0}^{s}\!\text{d}\overline{s}\,\boldsymbol{F}^{\text{T}}(s-\overline{s})\star \boldsymbol{b}(\overline{s})+\unicode[STIX]{x1D6F6}_{\unicode[STIX]{x0394}}(s).\end{eqnarray}$$

(Note that the $\text{P}$ -projected equation (2.57a ) has not been used at this point. Its role will ultimately be to determine the proper representations of the dissipative fluxes in terms of  $\text{Q}W$ .) Finally, for arbitrary vector  $\unicode[STIX]{x1D738}(s)$ define

(2.66) $$\begin{eqnarray}\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D738}}(s)\doteq \displaystyle \int _{0}^{s}\!\text{d}\overline{s}\,\boldsymbol{F}^{\text{T}}(s-\overline{s})\star \unicode[STIX]{x1D738}(\overline{s}).\end{eqnarray}$$

Upon taking the limit $s\rightarrow t$ , one then finds that (2.51) can be written as

(2.67) $$\begin{eqnarray}F(\unicode[STIX]{x1D6E4},t)=\exp [\unicode[STIX]{x1D714}(t)+\boldsymbol{A}s_{+}\prime ^{\text{T}}\star \boldsymbol{b}(t)+\unicode[STIX]{x1D713}_{\boldsymbol{b}}(t)+\unicode[STIX]{x1D6F6}_{\unicode[STIX]{x0394}}(t)]F_{0}(\unicode[STIX]{x1D6E4};\boldsymbol{r},t).\end{eqnarray}$$

This generalizes formula (36) of Brey et al. to include interspecies exchange effects (through the $\unicode[STIX]{x1D6F6}_{\unicode[STIX]{x0394}}$  term). The value of  $\unicode[STIX]{x1D714}(t)$ , which plays the role of a normalization factor, will not be required.

As Brey et al. emphasize, the representation (2.67) is an exact consequence of the Liouville equation, given the particular choice of initial condition. Note from (2.62) that $\boldsymbol{b}$ contains  $W$ , thus is still unknown. That is, equation (2.67) is an implicit representation of the phase-space distribution; it is not an explicit solution for it. The clever advance of Brey et al. was to show how to usefully exploit that representation in order to develop hydrodynamics as a gradient expansion. I shall now sketch that procedure. Again, my discussion closely follows that of Brey et al. except for the extra exchange term $\unicode[STIX]{x1D6F6}_{\unicode[STIX]{x0394}}$ and the implicit sum over species that resides in the $\star$  operation.

Consider the phase-space average of any random function $\widetilde{G}$ . From (2.67), one has

(2.68) $$\begin{eqnarray}\langle \widetilde{G}\rangle (t)=\displaystyle \int \text{d}\unicode[STIX]{x1D6E4}\,G(\unicode[STIX]{x1D6E4})F(\unicode[STIX]{x1D6E4};t)=\frac{\langle \widetilde{G}\exp [\widetilde{\boldsymbol{A}}\text{}s_{+}\prime \text{}^{\text{T}}\star \boldsymbol{b}(t)+\widetilde{\unicode[STIX]{x1D713}}_{\boldsymbol{b}}(t)+\widetilde{\unicode[STIX]{x1D6F6}}_{\unicode[STIX]{x0394}}(t)]\rangle _{0}}{\langle \exp [\widetilde{\boldsymbol{A}}\text{}s_{+}\prime \text{}^{\text{T}}\star \boldsymbol{b}(t)+\widetilde{\unicode[STIX]{x1D713}}_{\boldsymbol{b}}(t)+\widetilde{\unicode[STIX]{x1D6F6}}_{\unicode[STIX]{x0394}}(t)]\rangle _{0}}.\end{eqnarray}$$

(The normalization of  $F$ was used to eliminate the $\exp [\unicode[STIX]{x1D714}(t)]$ factor in (2.67).) As a special case of (2.68), the hydrodynamic variables are

(2.69) $$\begin{eqnarray}\boldsymbol{a}_{s}(\boldsymbol{r},t)=\frac{\langle \widetilde{\boldsymbol{A}}_{s}(\boldsymbol{r},t)\exp [\widetilde{\boldsymbol{A}}\text{}s_{+}\prime \text{}^{\text{T}}\star \boldsymbol{b}(t)+\widetilde{\unicode[STIX]{x1D713}}_{\boldsymbol{b}}(t)+\widetilde{\unicode[STIX]{x1D6F6}}_{\unicode[STIX]{x0394}}(t)]\rangle _{0}}{\langle \exp [\widetilde{\boldsymbol{A}}\text{}s_{+}\prime \text{}^{\text{T}}\star \boldsymbol{b}(t)+\widetilde{\unicode[STIX]{x1D713}}_{\boldsymbol{b}}(t)+\widetilde{\unicode[STIX]{x1D6F6}}_{\unicode[STIX]{x0394}}(t)]\rangle _{0}}.\end{eqnarray}$$

An independent expression for  $\boldsymbol{a}_{s}(\boldsymbol{r},t)$ follows from the constraint (2.33) that the conjugate variables  $\boldsymbol{B}_{s}(\boldsymbol{r},t)$ are supposed to be defined such that the mean hydrodynamic variables are obtained exactly by averaging over the local-equilibrium distribution:

(2.70) $$\begin{eqnarray}\boldsymbol{a}_{s}(t)=\displaystyle \int \text{d}\unicode[STIX]{x1D6E4}\,\boldsymbol{A}_{s}(\unicode[STIX]{x1D6E4};\boldsymbol{r})F_{\boldsymbol{B}}(\unicode[STIX]{x1D6E4};t),\end{eqnarray}$$

where $F_{\boldsymbol{B}}$  is given by (2.28). From (2.43), one can reexpress (2.70) as an average over the reference distribution as

(2.71) $$\begin{eqnarray}\boldsymbol{a}_{s}(\boldsymbol{r},t)=\frac{\langle \widetilde{\boldsymbol{A}}_{s}(\boldsymbol{r},t)\exp (\widetilde{\boldsymbol{A}}\text{}s_{+}\prime \text{}^{\text{T}}\star \unicode[STIX]{x0394}\boldsymbol{B})\rangle _{0}}{\langle \exp (\boldsymbol{A}s_{+}\prime ^{\text{T}}\star \unicode[STIX]{x0394}\boldsymbol{B})\rangle _{0}},\end{eqnarray}$$

where again $\unicode[STIX]{x0394}\boldsymbol{B}_{\overline{s}}$  is defined by (2.42) as being the diagonal part of $\unicode[STIX]{x0394}\boldsymbol{B}_{\overline{s}s}$ . Because the average limits the range of the  $\overline{\boldsymbol{r}}$ in the $\star$  operation to lie within a correlation length of  $\boldsymbol{r}$ , $\unicode[STIX]{x0394}\boldsymbol{B}_{\overline{s}s}(\overline{\boldsymbol{r}},\boldsymbol{r},t)$ will be assumed to be small. Thus, when a Markovian approximation is invoked, one has

(2.72) $$\begin{eqnarray}\unicode[STIX]{x0394}\boldsymbol{B}_{\overline{s}s}(\overline{\boldsymbol{r}},\boldsymbol{r},t)\approx \unicode[STIX]{x0394}\boldsymbol{B}_{\overline{s}s}(\boldsymbol{r},\boldsymbol{r},t)+(\overline{\boldsymbol{r}}-\boldsymbol{r})\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{B}_{\overline{s}}(\boldsymbol{r},t)+\cdots \,.\end{eqnarray}$$

The first term vanishes for $s=\overline{s}$ , leaving only a small gradient contribution; that is the effect studied by Brey et al. to second order in the gradients. For $s\neq \overline{s}$ , the first term does not vanish; it appears in  $\unicode[STIX]{x1D6F6}_{\unicode[STIX]{x0394}}$ , (2.60), through the definition (2.55b ) of  $S_{\unicode[STIX]{x0394}}$ and will lead to exchange effects. As a fundamental assumption, I shall order the exchange effects to be small and of the same order as the gradient terms. If gradients are symbolically represented by  $\unicode[STIX]{x1D6FB}$ and exchange effects by  $\unicode[STIX]{x0394}$ , this ordering implies that a general theory of Burnett-order transport will include terms of order $\unicode[STIX]{x1D6FB}^{2}$ , $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FB}$ , and  $\unicode[STIX]{x0394}^{2}$ . However, although I shall indicate where the latter two kinds of terms arise, in this work I shall not calculate terms of order $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FB}$ and $\unicode[STIX]{x0394}^{2}$ .

It will be seen that the unknown  $\boldsymbol{b}_{s}$ is approximately equal to  $\unicode[STIX]{x0394}\boldsymbol{B}_{s}$ and therefore is also small. Thus, one can expand (2.69) in powers of  $\boldsymbol{b}$ , expand (2.71) in powers of  $\unicode[STIX]{x0394}\boldsymbol{B}$ , then equate the two equivalent representations to find an expression for  $\boldsymbol{b}$ in terms of  $\unicode[STIX]{x0394}\boldsymbol{B}$ . The result of this exercise is

(2.73) $$\begin{eqnarray}\boldsymbol{b}(t)=\unicode[STIX]{x0394}\boldsymbol{B}(t)+\unicode[STIX]{x0394}\boldsymbol{B}^{(2)}(t)+\text{third-order terms},\end{eqnarray}$$

where

(2.74) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}\boldsymbol{B}^{(2)}(t) & \doteq & \displaystyle -\unicode[STIX]{x1D648}_{}^{-1}\star [\langle \boldsymbol{A}s_{+}\prime \unicode[STIX]{x1D713}_{\unicode[STIX]{x0394}\boldsymbol{B}}(t)\boldsymbol{A}s_{+}\prime ^{\text{T}}\rangle _{0}\star \unicode[STIX]{x0394}\boldsymbol{B}(t)+{\textstyle \frac{1}{2}}\langle \boldsymbol{A}s_{+}\prime \unicode[STIX]{x1D713}_{\unicode[STIX]{x0394}\boldsymbol{B}}^{2}(t)\rangle _{0}]\nonumber\\ \displaystyle & & \displaystyle +\,O(\unicode[STIX]{x1D6F6}_{\unicode[STIX]{x0394}}^{2},\unicode[STIX]{x1D6F6}_{\unicode[STIX]{x0394}}\unicode[STIX]{x0394}\boldsymbol{B}).\end{eqnarray}$$

Since $\unicode[STIX]{x1D6F6}_{\unicode[STIX]{x0394}}(t)$ is additive to  $\unicode[STIX]{x1D713}_{b}$ in (2.68), the form of the unwritten second-order terms in (2.74) involving  $\unicode[STIX]{x1D6F6}_{\unicode[STIX]{x0394}}$ can be obtained from the explicit terms by replacing $\unicode[STIX]{x1D713}_{\unicode[STIX]{x0394}\boldsymbol{B}}$ by $\unicode[STIX]{x1D713}_{\unicode[STIX]{x0394}\boldsymbol{B}}+\unicode[STIX]{x1D6F6}_{\unicode[STIX]{x0394}}$ .

These results can now be used to expand (2.68) to the desired order. I shall retain gradient effects to second orderFootnote 42 but exchange effects only to first order. The result is then

(2.75) $$\begin{eqnarray}\displaystyle \langle G\rangle (t) & {\approx} & \displaystyle \underbrace{\langle G\rangle _{B}(t)}_{\text{(i)}}+\underbrace{\langle Gs_{+}\prime \unicode[STIX]{x1D713}_{\unicode[STIX]{x0394}\boldsymbol{B}}(t)\rangle _{0}}_{\text{(ii}_{\unicode[STIX]{x1D735}}\text{)}}+\underbrace{\langle Gs_{+}\prime \unicode[STIX]{x1D6F6}_{\unicode[STIX]{x0394}}(t)\rangle _{0}}_{\text{(ii}_{\unicode[STIX]{x0394}}\text{)}}\nonumber\\ \displaystyle & & \displaystyle \quad +\underbrace{\langle Gs_{+}\prime \text{Q}\unicode[STIX]{x1D713}_{\unicode[STIX]{x0394}\boldsymbol{B}}(t)\boldsymbol{A}s_{+}\prime \rangle _{0}\star \unicode[STIX]{x0394}\boldsymbol{B}(t)}_{\text{(iii}_{\text{a}}\text{)}}+\underbrace{{\textstyle \frac{1}{2}}\langle Gs_{+}\prime \text{Q}\unicode[STIX]{x1D713}_{\unicode[STIX]{x0394}\boldsymbol{B}}^{2}(t)\rangle _{0}}_{\text{(iii}_{\text{b}}\text{)}}+\underbrace{\langle Gs_{+}\prime \unicode[STIX]{x1D713}_{\unicode[STIX]{x0394}\boldsymbol{B}^{(2)}}\rangle _{0}}_{\text{(iv)}}.\qquad\end{eqnarray}$$

This generalizes formula (51) of Brey et al. to include the term (ii $_{\unicode[STIX]{x0394}}$ ). See Brey et al. for a discussion of how some terms have been combined into term (i), the local-equilibrium average $\langle G\rangle _{B}$ . Terms (ii)–(iv) will ultimately lead to the dissipative fluxes and exchange effects.

3 The gradient expansion to Burnett order

I shall now discuss and simplify each of the terms in (2.75). The discussion follows Brey et al. closely, but generalizations are necessary at various steps in order to deal with the magnetic field and especially the interspecies coupling.

3.1 Term $(\text{i})$

As was noted above, term (i) is the local-equilibrium average. For a consistency check, one can see that when $G$ is replaced by  $\boldsymbol{A}$ all terms except for term (i) vanish because of either explicit or implicit $\text{Q}$  operators; note that $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D6FE}}$  contains a factor of  $\text{Q}$ on the left because of definitions (2.66), (2.64) and (2.59). In that case, term (i) generates the right-hand side of the Euler equations.

3.2 Term $(\text{i}\text{i}_{\unicode[STIX]{x1D735}})$

Upon using the definitions (2.66) and (2.64), one has

(3.1) $$\begin{eqnarray}\text{term }(\text{i}\text{i}_{\unicode[STIX]{x1D735}})\doteq \langle Gs_{+}\prime \unicode[STIX]{x1D713}_{\unicode[STIX]{x0394}\boldsymbol{B}}(t)\rangle _{0}=-\displaystyle \int _{0}^{t}\!\text{d}s\,\langle Gs_{+}\prime \text{U}(t-s)\text{i}\mathscr{L}\boldsymbol{A}s_{+}\prime ^{\text{T}}\rangle _{0}\star \unicode[STIX]{x0394}\boldsymbol{B}(s).\end{eqnarray}$$

This form is somewhat schematic because only the time dependence is displayed. In order to process it further, it is necessary to be more explicit. Recall that the transport equations will be evaluated at the reference point $(\boldsymbol{r},t)$ , while contributions to those equations (e.g. the transport coefficients) will involve integrals over the past state of the system at $(\boldsymbol{x},s)$ . In the following, I shall adopt a somewhat more symmetrical notation in which the reference variables are unbarred while the integration variables are barred, i.e. $(\boldsymbol{x},s)\rightarrow (\overline{\boldsymbol{x}},\overline{t})$ . With $\unicode[STIX]{x1D707}\doteq \{\boldsymbol{r},s\}$ (here $s$  is a species index) and upon noting that the propagator  $\text{U}$ depends on the time difference  $\overline{\unicode[STIX]{x1D70F}}\doteq t-\overline{t}$ , one can write $\text{term }(\text{i}\text{i}_{\unicode[STIX]{x1D735}})$ as

(3.2) $$\begin{eqnarray}\langle Gs_{+}\prime (\unicode[STIX]{x1D707})\unicode[STIX]{x1D713}_{\unicode[STIX]{x0394}\boldsymbol{B}}(t)\rangle _{0}=-\displaystyle \int _{0}^{t}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}\,\langle Gs_{+}\prime (\unicode[STIX]{x1D707})\text{U}(\overline{\unicode[STIX]{x1D70F}})\text{i}\mathscr{L}As_{+}\prime ^{\unicode[STIX]{x1D6FD}}(\overline{\unicode[STIX]{x1D707}})\rangle _{0}\unicode[STIX]{x0394}B_{\unicode[STIX]{x1D6FD}}(\overline{\unicode[STIX]{x1D707}},t-\overline{\unicode[STIX]{x1D70F}}),\end{eqnarray}$$

where $\int \text{d}\overline{\unicode[STIX]{x1D707}}\equiv \sum _{\overline{s}}\int \text{d}\overline{\boldsymbol{x}}$ , $\boldsymbol{A}s_{+}\prime (\overline{\unicode[STIX]{x1D707}})\doteq \boldsymbol{A}_{\overline{s}}(\overline{\boldsymbol{x}})-\langle \boldsymbol{A}\rangle _{0\overline{s}}(\boldsymbol{r},t)$ and $\unicode[STIX]{x0394}\boldsymbol{B}(\overline{\unicode[STIX]{x1D707}},\overline{t})\doteq \boldsymbol{B}_{\overline{s}}(\overline{\boldsymbol{x}},\overline{t})-\boldsymbol{B}_{\overline{s}}(\boldsymbol{r},t)$ . The $(\boldsymbol{r},t)$ dependence on the reference state is implicit in the notation $\boldsymbol{A}s_{+}\prime (\overline{\unicode[STIX]{x1D707}})$ and  $\unicode[STIX]{x0394}\boldsymbol{B}(\overline{\unicode[STIX]{x1D707}},\overline{t})$ .

Expression (3.2) displays two difficulties: it involves the modified propagator  $\text{U}$ (see (2.59)), which is difficult to work with because of the $\text{Q}$  projections in the operator $\text{Q}\text{i}\mathscr{L}\text{Q}$ ; and it is non-local in space and time. The non-locality is relatively straightforward to deal with by means of a Markovian approximation. Thus, the characteristic correlation length introduced by  $\text{U}$ is, for the case of the weakly coupled plasma, either the collisional mean free path or the gyroradius; both are assumed to be of short range relative to the macroscopic gradient scale length. (It is at this point that one ignores the possibility of long-ranged collisional correlations.) Then, with $\overline{\unicode[STIX]{x1D746}}\doteq \boldsymbol{r}-\overline{\boldsymbol{x}}$ and upon noting that $\unicode[STIX]{x0394}\boldsymbol{B}_{\overline{s}\,\overline{s}}(\boldsymbol{r},t)=0$ , one has

(3.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}\boldsymbol{B}(\overline{\unicode[STIX]{x1D707}},t-\overline{\unicode[STIX]{x1D70F}}) & = & \displaystyle -\overline{\unicode[STIX]{x1D746}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{B}_{\overline{s}}(\boldsymbol{r},t)-\overline{\unicode[STIX]{x1D70F}}\,\unicode[STIX]{x2202}_{t}\boldsymbol{B}_{\overline{s}}^{(1)}(\boldsymbol{r},t)\nonumber\\ \displaystyle & & \displaystyle +\,{\textstyle \frac{1}{2}}(\overline{\unicode[STIX]{x1D746}}\,\overline{\unicode[STIX]{x1D746}})\boldsymbol{ : }\unicode[STIX]{x1D735}\unicode[STIX]{x1D735}\boldsymbol{B}_{\overline{s}}(\boldsymbol{r},t)+\overline{\unicode[STIX]{x1D70F}}\,\overline{\unicode[STIX]{x1D746}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x2202}_{t}\boldsymbol{B}_{\overline{s}}^{(1)}(\boldsymbol{r},t)+{\textstyle \frac{1}{2}}\overline{\unicode[STIX]{x1D70F}}^{2}\unicode[STIX]{x2202}_{t}^{2}\boldsymbol{B}_{\overline{s}}^{(1)}(\boldsymbol{r},t)+\cdots \,.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

For the terms involving time derivatives, note that it is adequate to calculate $\unicode[STIX]{x2202}_{t}\boldsymbol{B}$ only to first order (i.e. from the Euler equations). (In (3.3), the term involving $\unicode[STIX]{x2202}_{t}^{2}\boldsymbol{B}$ , although formally of the same (second) order in the gradients as the first two terms in the last line, will not be needed.)

The $\text{Q}$  projections are more problematical. For a one-component neutral fluid, Brey et al. argue that for small gradients the $\text{Q}\text{i}\mathscr{L}\text{Q}$ in  $\text{U}$ can be replaced merely by  $\text{i}\mathscr{L}$ . I shall first describe the procedure followed by Brey et al., then indicate a difficulty for the multicomponent case. Brey et al. invoke the identityFootnote 43

(3.4) $$\begin{eqnarray}\text{U}(s)=\text{Q}\text{e}^{-\text{i}\mathscr{L}s}\text{Q}+\displaystyle \int _{0}^{s}\!\text{d}\overline{s}\,\text{Q}\text{e}^{-\text{i}\mathscr{L}(s-\overline{s})}\text{P}\text{i}\mathscr{L}\text{U}(\overline{s}).\end{eqnarray}$$

Because for any phase function  $\widetilde{g}$ one has $\text{i}\mathscr{L}\widetilde{g}=\text{d}\widetilde{g}/\text{d}t$ (see footnote 21 on page 15), when the form $\text{i}\mathscr{L}\boldsymbol{A}s_{+}\prime$ appears to the far right of any expression involving phase-space variables it can be replaced by the right-hand side of (2.27). The electric and magnetic force terms in the momentum equation do not enter the final expressions because they are invariably preceded by a  $\text{Q}$ . In the long-time limit, one is led, after a spatial integration by parts and with the aid of (3.3),Footnote 44 to

(3.5a ) $$\begin{eqnarray}\displaystyle \langle Gs_{+}\prime (\unicode[STIX]{x1D707})\unicode[STIX]{x1D713}_{\unicode[STIX]{x0394}\boldsymbol{B}}(t)\rangle _{0} & = & \displaystyle -\displaystyle \int _{0}^{t}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}\,\boldsymbol{K}_{G}^{\overline{\unicode[STIX]{x1D6FD}}}(\unicode[STIX]{x1D707},\overline{\unicode[STIX]{x1D707}},\overline{\unicode[STIX]{x1D70F}})\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\overline{\boldsymbol{x}}}\unicode[STIX]{x0394}B_{\overline{\unicode[STIX]{x1D6FD}}}(\overline{\unicode[STIX]{x1D707}},t-\overline{\unicode[STIX]{x1D70F}})\end{eqnarray}$$
(3.5b ) $$\begin{eqnarray}\displaystyle & {\approx} & \displaystyle -\displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}\,\boldsymbol{K}_{G}^{\overline{\unicode[STIX]{x1D6FD}}}(\unicode[STIX]{x1D707},\overline{\unicode[STIX]{x1D707}},\overline{\unicode[STIX]{x1D70F}})\boldsymbol{\cdot }\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t)\nonumber\\ \displaystyle & & \displaystyle -\,\displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}\,\boldsymbol{K}_{G}^{\overline{\unicode[STIX]{x1D6FD}}}(\unicode[STIX]{x1D707},\overline{\unicode[STIX]{x1D707}},\overline{\unicode[STIX]{x1D70F}})\overline{\unicode[STIX]{x1D70F}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}[\unicode[STIX]{x2202}_{t}B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t)]^{(1)}\nonumber\\ \displaystyle & & \displaystyle +\,\displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}\,\boldsymbol{K}_{G}^{\overline{\unicode[STIX]{x1D6FD}}}(\unicode[STIX]{x1D707},\overline{\unicode[STIX]{x1D707}},\overline{\unicode[STIX]{x1D70F}})\overline{\unicode[STIX]{x1D746}}\boldsymbol{ : }\unicode[STIX]{x1D735}\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t),\end{eqnarray}$$
where $\boldsymbol{K}=_{\text{a}}\boldsymbol{K}+_{\text{b}}\boldsymbol{K}$ and
(3.6a ) $$\begin{eqnarray}\displaystyle ~_{\text{a}}\boldsymbol{K}_{G}^{\overline{\unicode[STIX]{x1D6FD}}}(\unicode[STIX]{x1D707},\overline{\unicode[STIX]{x1D707}},\overline{\unicode[STIX]{x1D70F}}) & \doteq & \displaystyle \langle \widehat{G}(\unicode[STIX]{x1D707})\text{e}^{-\text{i}\mathscr{L}\overline{\unicode[STIX]{x1D70F}}}\,\widehat{\boldsymbol{J}}\text{}^{\overline{\unicode[STIX]{x1D6FD}}}(\overline{\unicode[STIX]{x1D707}})\rangle _{0},\end{eqnarray}$$
(3.6b ) $$\begin{eqnarray}\displaystyle ~_{\text{b}}\boldsymbol{K}_{G}^{\overline{\unicode[STIX]{x1D6FD}}}(\unicode[STIX]{x1D707},\overline{\unicode[STIX]{x1D707}},\overline{\unicode[STIX]{x1D70F}}) & \doteq & \displaystyle -\displaystyle \int _{0}^{\overline{\unicode[STIX]{x1D70F}}}\!\text{d}\overline{s}\,\langle \widehat{G}(\unicode[STIX]{x1D707})\text{e}^{-\text{i}\mathscr{L}(\overline{\unicode[STIX]{x1D70F}}-\overline{s})}\boldsymbol{A}s_{+}\prime ^{\text{T}}\rangle _{0}\star \unicode[STIX]{x1D614}^{-1}\star \langle (\text{i}\mathscr{L}\boldsymbol{A}s_{+}\prime )U(\overline{s})\widehat{\boldsymbol{J}}\text{}^{\overline{\unicode[STIX]{x1D6FD}}}(\overline{\unicode[STIX]{x1D707}})\rangle _{0}\nonumber\\ \displaystyle & & \displaystyle +\,(\unicode[STIX]{x0394}\text{ correction}).\end{eqnarray}$$

To obtain (3.6b ), the Liouville operator was integrated by parts under the last $F_{0}$  average. The correction terms, not written explicitly, arise from the fact that in general $\text{i}\mathscr{L}F_{0}\neq 0$ ; see the discussion after (2.56).

3.2.1 Two-time $\boldsymbol{x}$ -space correlation functions as velocity integrals over Klimontovich correlations

Before I consider (3.6) in detail, I digress to discuss expectations of the form $M_{0}(\unicode[STIX]{x1D707},\unicode[STIX]{x1D707}s_{+}\prime ,\unicode[STIX]{x1D70F})\doteq \langle \widetilde{a}(\unicode[STIX]{x1D707})\text{e}^{-\text{i}\mathscr{L}\unicode[STIX]{x1D70F}}\widetilde{b}(\unicode[STIX]{x1D707}s_{+}\prime )\rangle _{0}$ , which appear in each of those equations. Here

(3.7) $$\begin{eqnarray}\widetilde{a}(\unicode[STIX]{x1D707})\doteq \mathop{\sum }_{i\in s}a_{s}(\boldsymbol{x}_{i},\boldsymbol{v}_{i})\unicode[STIX]{x1D6FF}(\boldsymbol{r}-\boldsymbol{x}_{i}),\end{eqnarray}$$

where $a_{s}(\boldsymbol{x},\boldsymbol{v})$  is a prescribed function, and similarly for  $\widetilde{b}$ . (At this point, there may also be implicit dependence of  $a$ and  $b$ on the particle coordinates $\boldsymbol{x}_{j\neq i}$ ; that is not indicated explicitly.) A standard manipulation (Klimontovich Reference Klimontovich and ter Harr1967; Krommes & Oberman Reference Krommes and Oberman1976a ) shows that such expectations can be written in terms of correlations of the Klimontovich microdensity  $\widetilde{f}$ . Namely, upon using the definition (2.6) of  $\widetilde{f}$ and the fact that $\text{e}^{-\text{i}\mathscr{L}\unicode[STIX]{x1D70F}}$ moves phase-space coordinates back in time by interval  $\unicode[STIX]{x1D70F}$ , one has

(3.8a ) $$\begin{eqnarray}\displaystyle M_{0}(\unicode[STIX]{x1D707},\unicode[STIX]{x1D707}s_{+}\prime ,\unicode[STIX]{x1D70F}) & = & \displaystyle \left\langle \mathop{\sum }_{i\in s}a_{s}(\boldsymbol{x}_{i},\boldsymbol{v}_{i})\unicode[STIX]{x1D6FF}(\boldsymbol{r}-\boldsymbol{x}_{i})\text{e}^{-\text{i}\mathscr{L}\unicode[STIX]{x1D70F}}\mathop{\sum }_{j\in ss_{+}\prime }b_{ss_{+}\prime }(\boldsymbol{x}_{j},\boldsymbol{v}_{j})\unicode[STIX]{x1D6FF}(\boldsymbol{r}s_{+}\prime -\boldsymbol{x}_{j})\right\rangle _{0}\end{eqnarray}$$
(3.8b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle \overline{n}_{s}\overline{n}_{ss_{+}\prime }\left\langle \frac{1}{\overline{n}_{s}}\mathop{\sum }_{i\in s}a_{s}\boldsymbol{(}\boldsymbol{x}_{i}(0),\boldsymbol{v}_{i}(0)\boldsymbol{)}\unicode[STIX]{x1D6FF}\boldsymbol{(}\boldsymbol{r}-\boldsymbol{x}_{i}(0)\boldsymbol{)}\right.\nonumber\\ \displaystyle & & \displaystyle \times \left.\frac{1}{\overline{n}_{ss_{+}\prime }}\mathop{\sum }_{j\in ss_{+}\prime }b_{ss_{+}\prime }\boldsymbol{(}\boldsymbol{x}_{j}(-\unicode[STIX]{x1D70F}),\boldsymbol{v}_{j}(-\unicode[STIX]{x1D70F})\boldsymbol{)}\unicode[STIX]{x1D6FF}\boldsymbol{(}\boldsymbol{r}s_{+}\prime -\boldsymbol{x}_{j}(-\unicode[STIX]{x1D70F})\boldsymbol{)}\right\rangle _{0}\end{eqnarray}$$
(3.8c ) $$\begin{eqnarray}\displaystyle & = & \displaystyle \overline{n}_{s}\overline{n}_{ss_{+}\prime }\left\langle \displaystyle \int \text{d}\overline{\boldsymbol{v}}\,a_{s}(\boldsymbol{r},\overline{\boldsymbol{v}})\frac{1}{\overline{n}_{s}}\mathop{\sum }_{i\in s}\unicode[STIX]{x1D6FF}\boldsymbol{(}\boldsymbol{r}-\boldsymbol{x}_{i}(0)\boldsymbol{)}\unicode[STIX]{x1D6FF}\boldsymbol{(}\overline{\boldsymbol{v}}-\boldsymbol{v}_{i}(0)\boldsymbol{)}\right.\nonumber\\ \displaystyle & & \displaystyle \times \left.\displaystyle \int \text{d}\overline{\boldsymbol{v}}s_{+}\prime \,b_{ss_{+}\prime }(\boldsymbol{r}s_{+}\prime ,\overline{\boldsymbol{v}}s_{+}\prime )\frac{1}{\overline{n}_{ss_{+}\prime }}\mathop{\sum }_{j\in ss_{+}\prime }\unicode[STIX]{x1D6FF}\boldsymbol{(}\boldsymbol{r}s_{+}\prime -\boldsymbol{x}_{j}(-\unicode[STIX]{x1D70F})\boldsymbol{)}\unicode[STIX]{x1D6FF}\boldsymbol{(}\overline{\boldsymbol{v}}s_{+}\prime -\boldsymbol{v}_{j}(-\unicode[STIX]{x1D70F})\boldsymbol{)}\right\rangle _{0}\end{eqnarray}$$
(3.8d ) $$\begin{eqnarray}\displaystyle & = & \displaystyle \overline{n}_{s}\overline{n}_{ss_{+}\prime }\displaystyle \int \text{d}\overline{\boldsymbol{v}}\,\text{d}\overline{\boldsymbol{v}}s_{+}\prime \,a_{s}(\boldsymbol{r},\overline{\boldsymbol{v}})\langle \widetilde{f}_{s}(\boldsymbol{r},\overline{\boldsymbol{v}},0)\widetilde{f}_{ss_{+}\prime }(\boldsymbol{r}s_{+}\prime ,\overline{\boldsymbol{v}}s_{+}\prime ,-\unicode[STIX]{x1D70F})\rangle _{0}b_{ss_{+}\prime }(\boldsymbol{r}s_{+}\prime ,\overline{\boldsymbol{v}}s_{+}\prime ),\end{eqnarray}$$
the last result holding when neither  $a$ nor  $b$ implicitly depends on any phase-space variable. The random currents on the right-hand side of (2.23) satisfy this property when the potential-energy contributions involving  $\unicode[STIX]{x0394}\unicode[STIX]{x1D749}_{i}$ are neglected; then $a$ and  $b$ depend only on  $\boldsymbol{v}$ and  $s$ . In that case (valid for the classical weakly coupled plasma), the 𝛤-space expectation has been reduced to integrals of the one-point distribution function  $f$ and the two-point Klimontovich correlation function  $C(\unicode[STIX]{x1D70F})\doteq \langle \unicode[STIX]{x1D6FF}f(\unicode[STIX]{x1D70F})\unicode[STIX]{x1D6FF}f(0)\rangle _{0}$ :
(3.9) $$\begin{eqnarray}\displaystyle M_{0}(\unicode[STIX]{x1D707},\unicode[STIX]{x1D707}s_{+}\prime ,\unicode[STIX]{x1D70F}) & = & \displaystyle \left(\overline{n}_{s}\displaystyle \int \text{d}\overline{\boldsymbol{v}}\,a_{s}(\boldsymbol{r},\overline{\boldsymbol{v}})f_{s}(\boldsymbol{r},\overline{\boldsymbol{v}})\right)\left(\overline{n}_{ss_{+}\prime }\displaystyle \int \text{d}\overline{\boldsymbol{v}}s_{+}\prime \,b_{ss_{+}\prime }(\boldsymbol{r}s_{+}\prime ,\overline{\boldsymbol{v}}s_{+}\prime )f_{ss_{+}\prime }(\boldsymbol{r}s_{+}\prime ,\overline{\boldsymbol{v}}s_{+}\prime )\right)\nonumber\\ \displaystyle & & \displaystyle +\,\overline{n}_{s}\overline{n}_{ss_{+}\prime }\displaystyle \int \text{d}\overline{\boldsymbol{v}}\,\text{d}\overline{\boldsymbol{v}}s_{+}\prime \,a_{s}(\boldsymbol{r},\overline{\boldsymbol{v}})C_{sss_{+}\prime }(\boldsymbol{r},\overline{\boldsymbol{v}},\unicode[STIX]{x1D70F},\boldsymbol{r}s_{+}\prime ,\overline{\boldsymbol{v}}s_{+}\prime ,0)b_{ss_{+}\prime }(\boldsymbol{r}s_{+}\prime ,\overline{\boldsymbol{v}}s_{+}\prime ).\end{eqnarray}$$

Here the assumption of stationary statistics was used to shift the $\unicode[STIX]{x1D70F}$  argument.Footnote 45 In the uses I shall make of this formula, the mean-field terms (the first line of (3.9)) will vanish. When potential-energy contributions to  $a$ and  $b$ are included, a generalization of the previous argument shows that the 𝛤-space expectation can be reduced to integrals of correlation functions with no more than four phase-space points.

The theory of the Klimontovich two-time correlation function  $C(\unicode[STIX]{x1D70F})$ will be described in § 5. It is shown there that to lowest order in a weakly coupled plasma the time dependence of the one-sidedFootnote 46 function  $C_{+}(\unicode[STIX]{x1D70F})$ is given for long wavelengths by the linearized Vlasov response functionFootnote 47 modified to include collisional corrections given by the linearized collision operator  $\widehat{\text{C}}$ . This information is sufficient to inform the following discussion, in which I shall consider each of (3.6a ) and (3.6b ) in turn.

3.2.2 Formula (3.6a )

In order to attach some physical significance to formula (3.6a ), consider the limit of homogeneous background statistics. We shall see in § 3.6 and with the aid of footnote 45 on page 36 that the Braginskii (weakly coupled Navier–Stokes) transport coefficients that multiply gradients will arise from the $k\rightarrow 0$ limit of the $\overline{\unicode[STIX]{x1D70F}}$ integrals of expression (3.6a ) with $\widehat{G}$  replaced by  $\widehat{\boldsymbol{J}}^{\unicode[STIX]{x1D6FC}}$ . The first-order gradient-driven transport fluxes are then

(3.10) $$\begin{eqnarray}-\displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\boldsymbol{v}\mathop{\sum }_{\overline{s}}\displaystyle \int \text{d}\overline{\boldsymbol{v}}\,\widehat{\boldsymbol{J}}_{s}\!\text{}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{v})C_{s\overline{s},\boldsymbol{k}=\mathbf{0}}(\boldsymbol{v},\overline{\boldsymbol{v}},\overline{\unicode[STIX]{x1D70F}})\widehat{\boldsymbol{J}}_{\overline{s}}\!\text{}^{\overline{\unicode[STIX]{x1D6FD}}\text{T}}(\overline{\boldsymbol{v}})\boldsymbol{\cdot }\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FD}},\overline{s}},\end{eqnarray}$$

where both  $\unicode[STIX]{x1D6FC}$ and  $\overline{\unicode[STIX]{x1D6FD}}$ may assume the values  $\boldsymbol{p}$ or  $e$ . ( $\widehat{\boldsymbol{J}}\text{}^{n}=\mathbf{0}$ ; see item (i) in appendix D as well as footnote 75 on page 81.) According to the discussion in the last paragraph, the time dependence is given by $\exp (-\widehat{\text{C}}\overline{\unicode[STIX]{x1D70F}})C(\unicode[STIX]{x1D70F}=0)$ , with $C(\unicode[STIX]{x1D70F}=0)$ being given by (1.8). At $\boldsymbol{k}=\mathbf{0}$ the Fourier transform of the pair correlation function  $g$ vanishes due to the normalization constraint, so one has

(3.11) $$\begin{eqnarray}C_{s\overline{s},\boldsymbol{k}=\mathbf{0}}(\boldsymbol{v},\overline{\boldsymbol{v}},\unicode[STIX]{x1D70F}=0)=\overline{n}_{s}^{-1}\unicode[STIX]{x1D6FF}(\boldsymbol{v}-\overline{\boldsymbol{v}})\unicode[STIX]{x1D6FF}_{s\overline{s}}\,f_{0,\overline{s}}(\overline{\boldsymbol{v}});\end{eqnarray}$$

note the appearance of the one-particle reference distribution  $f_{0}$ in this expression. Because $\widehat{\text{C}}$  is isotropic in velocity space, symmetry considerations restrict  $\overline{\unicode[STIX]{x1D6FD}}$ to equal  $\unicode[STIX]{x1D6FC}$ . Upon performing the $\overline{v}$  integration and the  $\overline{s}$ and  $\overline{\unicode[STIX]{x1D6FD}}$ summations in (3.10), one is led to the transport matrices

(3.12) $$\begin{eqnarray}\unicode[STIX]{x1D63F}_{s}^{\unicode[STIX]{x1D6FC}}=\displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\,\langle \,\widehat{\boldsymbol{J}}_{(s)}\!\!\!\text{}^{\unicode[STIX]{x1D6FC}}\,\,|\text{e}^{-\widehat{\text{C}}\overline{\unicode[STIX]{x1D70F}}}|\,\,\widehat{\boldsymbol{J}}_{(s)}\!\!\!\text{}^{\unicode[STIX]{x1D6FC}\text{T}}\rangle _{0},\end{eqnarray}$$

where the presence of the vertical bar indicates the $\unicode[STIX]{x1D707}$ -space scalar product (1:3.15) used in Part 1 (here averaged with the local equilibrium distribution). This reproduces the transport formulas discussed in Part 1 for the irreversible fluxes that are linearly proportional to gradients (now evaluated locally rather than with absolute-equilibrium parameters).

3.2.3 Formula (3.6b )

Brey et al. argue that (3.6b ) is negligible in a gradient expansion. They say,

From the definition of  $\text{P}$ , it follows that the sequence $\text{P}\mathscr{L}$ introduces factors of  $\mathscr{L}\boldsymbol{A}$ and these yield $\ldots$ gradient operators acting on  $\boldsymbol{B}$ . If we keep terms only up to second order in gradients of  $\boldsymbol{B}$ , we see that (3.6b ) $\ldots$ can be neglected since it is proportional to $\langle G(\boldsymbol{r})\text{Q}\exp [-(\overline{\unicode[STIX]{x1D70F}}-\overline{\unicode[STIX]{x1D70F}}s_{+}\prime )\text{i}\mathscr{L}]\boldsymbol{A}s_{+}\prime \rangle _{0}$ evaluated to zeroth order, which vanishes. This may easily be seen, because to this order

(3.13) $$\begin{eqnarray}\langle G(\boldsymbol{r})\text{Q}\exp [-(\overline{\unicode[STIX]{x1D70F}}-\overline{\unicode[STIX]{x1D70F}}s_{+}\prime )\text{i}\mathscr{L}]\boldsymbol{A}s_{+}\prime \rangle _{0}=\langle G(\boldsymbol{r})\text{Q}\boldsymbol{A}s_{+}\prime \rangle _{0}=0.\end{eqnarray}$$

This argument must be revised in the multispecies case because of the exchange terms on the right-hand side of (2.26), which are not proportional to gradients. However, one does expect that the effects of those terms will be small, proportional to  $\unicode[STIX]{x0394}\boldsymbol{u}$ or  $\unicode[STIX]{x0394}T$ , and I have assumed that those are of the same order of smallness as are the gradients. Therefore, the arguments of Brey et al. might appear to suggest that the effect of (3.6b ), which to lowest order multiplies $-\unicode[STIX]{x1D735}\boldsymbol{B}$ , is negligible.

There is, however, a subtlety, which is that the left-hand side of (3.13) appears under an infinite time integral; it is not clear that arguments based on truncated power-series expansion are adequate. To make the mathematics match the discussion in Part 1 as closely as possible, assume that after averaging it is legitimate to replace  $\text{i}\mathscr{L}$ by  $\widehat{\text{C}}$ (this ignores a $\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}$ streaming term) and treat the expectations and projections as the $\unicode[STIX]{x1D707}$ -space ones used in Part 1. The integral of (3.6b ) then becomes

(3.14a ) $$\begin{eqnarray}\displaystyle \lim _{t\rightarrow \infty }\displaystyle \int _{0}^{t}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\;_{\text{b}}\boldsymbol{K}_{G}^{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D707},\overline{\unicode[STIX]{x1D707}},\overline{\unicode[STIX]{x1D70F}}) & = & \displaystyle \lim _{t\rightarrow \infty }\displaystyle \int _{0}^{t}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int _{0}^{\overline{\unicode[STIX]{x1D70F}}}\!\text{d}\overline{s}\,\langle \widehat{G}(\unicode[STIX]{x1D707})|\text{e}^{-\widehat{\text{C}}(\overline{\unicode[STIX]{x1D70F}}-\overline{s})}|\text{P}\widehat{\text{C}}\text{U}(\overline{s})\widehat{\boldsymbol{J}}(\overline{\unicode[STIX]{x1D707}})\rangle _{0}\qquad\end{eqnarray}$$
(3.14b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle \lim _{t\rightarrow \infty }\displaystyle \int _{0}^{t}\!\text{d}\overline{s}\displaystyle \int _{\overline{s}}^{t}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\,\langle \widehat{G}(\unicode[STIX]{x1D707})|\text{e}^{-\widehat{\text{C}}(\overline{\unicode[STIX]{x1D70F}}-\overline{s})}|\text{P}\widehat{\text{C}}\text{U}(\overline{s})\widehat{\boldsymbol{J}}(\overline{\unicode[STIX]{x1D707}})\rangle _{0}\end{eqnarray}$$
(3.14c ) $$\begin{eqnarray}\displaystyle & = & \displaystyle \lim _{t\rightarrow \infty }\displaystyle \int _{0}^{t}\!\text{d}\overline{s}\displaystyle \int _{0}^{t-\overline{s}}\!\text{d}\overline{r}\,\langle \widehat{G}(\unicode[STIX]{x1D707})|\text{e}^{-\widehat{\text{C}}\overline{r}}|\text{P}\widehat{\text{C}}\text{U}(\overline{s})\widehat{\boldsymbol{J}}(\overline{\unicode[STIX]{x1D707}})\rangle _{0}.\end{eqnarray}$$
The modified propagator  $\text{U}(\overline{s})$ constrains  $\overline{s}$ to be less than or of the order of a collision time, whereas $t\rightarrow \infty$ . Thus, $\overline{s}$  is negligible in the upper limit of the second integral, which can then be performed to give
(3.15) $$\begin{eqnarray}\lim _{t\rightarrow \infty }\displaystyle \int _{0}^{t}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\;_{\text{b}}\boldsymbol{K}_{G}^{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D707},\overline{\unicode[STIX]{x1D707}},\overline{\unicode[STIX]{x1D70F}})=\lim _{t\rightarrow \infty }\left\langle \widehat{G}(\unicode[STIX]{x1D707})|\widehat{\text{C}}^{-1}(1-\text{e}^{-\widehat{\text{C}}t})|\text{P}\widehat{\text{C}}\left(\displaystyle \int _{0}^{\infty }\text{d}\overline{s}\,\text{U}(\overline{s})\right)\widehat{\boldsymbol{J}}(\overline{\unicode[STIX]{x1D707}})\right\rangle _{0}.\end{eqnarray}$$

The properties of this expression differ for one-component and multicomponent systems. In a one-component system, the basis functions from which $\text{P}$  is constructed are the null eigenfunctions of  $\widehat{\text{C}}$ ; thus, $\widehat{\text{C}}\text{P}=0$ . One has

(3.16) $$\begin{eqnarray}\text{Q}\widehat{\text{C}}^{-1}(1-\text{e}^{-\widehat{\text{C}}t})\text{P}=\text{Q}(t-{\textstyle \frac{1}{2}}\widehat{\text{C}}\,t^{2}+\cdots \,)\text{P}=\text{Q}\,t\,\text{P}=0.\end{eqnarray}$$

Here ‘zero’ here means that the effect is at least of first order in  $k$ , as can be seen by adding the streaming contribution $\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}$ to  $\widehat{\text{C}}$ . Upon similarly processing the factor $\text{P}\widehat{\text{C}}\int _{0}^{\infty }\,\text{d}\overline{s}\,\text{U}(\overline{s})\widehat{\boldsymbol{J}}(\overline{\unicode[STIX]{x1D707}})$ , one finds that that too is $O(k)$ . Since (3.6b ) multiplies $\unicode[STIX]{x1D735}\boldsymbol{B}$ , one concludes that for a one-component system the net effect of (3.6b ) is of third order in the gradients, thus is negligible to Burnett order. This recovers the conclusion of Brey et al., which was obtained directly from the expression involving the Liouville operator. Brey et al. did not perform the $\overline{\unicode[STIX]{x1D70F}}$  integral as was done above, but rather examined the power-series expansion of $\exp (-\text{i}\mathscr{L}\overline{\unicode[STIX]{x1D70F}})$ , noting that the ultimate effect of the factor $\text{i}\mathscr{L}\boldsymbol{A}s_{+}\prime$ is to produce a term proportional to a gradient. Similar arguments will be used later; the manipulations above justify that procedure for a one-component system.

The situation is different for a multicomponent system. In that case $\widehat{\text{C}}\text{P}\neq 0$ because momentum and energy can be exchanged between unlike species. Then, in the limit $t\rightarrow \infty$ one has $\exp (-\widehat{\text{C}}t)\rightarrow 0$ and one is left with the construction $\widehat{\text{C}}^{-1}\text{P}\widehat{\text{C}}$ . Although I shall not work it out in detail, the ultimate value of expression (3.15) will generate a small exchange contribution that will ultimately multiply $\unicode[STIX]{x1D735}\boldsymbol{B}$ . That second-order term should be retained, in principle; however, I shall neglect all such effects. The $\unicode[STIX]{x0394}$  correction terms are neglected for the same reason.

Mathematically, the various behaviours are an example of an interchange of limits. That is, with $\widehat{\text{C}}\rightarrow \unicode[STIX]{x1D708}$ and $\unicode[STIX]{x1D70F}_{\text{coll}}\doteq \unicode[STIX]{x1D708}^{-1}$ , one has

(3.17) $$\begin{eqnarray}\unicode[STIX]{x1D708}^{-1}(1-\text{e}^{-\unicode[STIX]{x1D708}t})=\unicode[STIX]{x1D70F}_{\text{coll}}(1-\text{e}^{-t/\unicode[STIX]{x1D70F}_{\text{coll}}})\rightarrow \left\{\begin{array}{@{}ll@{}}t\quad & \text{for }t\text{ fixed},\unicode[STIX]{x1D70F}_{\text{coll}}\rightarrow \infty ;\\ \unicode[STIX]{x1D70F}_{\text{coll}}\quad & \text{for }\unicode[STIX]{x1D70F}_{\text{coll}}\text{ fixed},t\rightarrow \infty .\end{array}\right.\end{eqnarray}$$

In future manipulations of second-order terms, I shall follow Brey et al. in counting the powers of gradients as though the system contains just one component. That is, the first limit in (3.17) is used. This can be interpreted as a formal way of ordering out second-order exchange effects – they may happen, but only on a time scale longer than that of interest.

3.3 Term $(\text{i}\text{i}_{\unicode[STIX]{x0394}})$

From (2.60) and (2.55b ), one has

(3.18) $$\begin{eqnarray}\text{term }(\text{ii}_{\unicode[STIX]{x0394}})\doteq \langle Gs_{+}\prime (\unicode[STIX]{x1D707})\unicode[STIX]{x1D6F6}_{\unicode[STIX]{x0394}}(t)\rangle _{0}=-\displaystyle \int _{0}^{t}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}\,\breve{K}_{G}^{\overline{\unicode[STIX]{x1D6FD}}}(\unicode[STIX]{x1D707},\overline{\unicode[STIX]{x1D707}},\overline{\unicode[STIX]{x1D70F}})B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t),\end{eqnarray}$$

where

(3.19) $$\begin{eqnarray}\breve{\boldsymbol{K}}_{G}(\unicode[STIX]{x1D707},\overline{\unicode[STIX]{x1D707}},\overline{\unicode[STIX]{x1D70F}})\doteq \langle Gs_{+}\prime (\unicode[STIX]{x1D707})\text{U}(\overline{\unicode[STIX]{x1D70F}})\dot{\boldsymbol{A}}_{\unicode[STIX]{x0394},\overline{s}}^{\prime }(\overline{\unicode[STIX]{x1D707}})\rangle _{0}.\end{eqnarray}$$

This formula may again be simplified by using the identity (3.4). Thus, one can write $\breve{\boldsymbol{K}}=_{\text{a}}\breve{\boldsymbol{K}}+_{\text{b}}\breve{\boldsymbol{K}}$ , where

(3.20a ) $$\begin{eqnarray}\displaystyle ~_{\text{a}}\breve{\boldsymbol{K}}_{G} & \doteq & \displaystyle \langle Gs_{+}\prime \text{e}^{-\text{i}\mathscr{L}\overline{\unicode[STIX]{x1D70F}}}\dot{\boldsymbol{A}}_{\unicode[STIX]{x0394}}^{\prime }(\overline{\unicode[STIX]{x1D707}})\rangle _{0},\end{eqnarray}$$
(3.20b ) $$\begin{eqnarray}\displaystyle ~_{\text{b}}\breve{\boldsymbol{K}}_{G} & \doteq & \displaystyle \displaystyle \int _{0}^{\overline{\unicode[STIX]{x1D70F}}}\!\text{d}\overline{s}\,\langle Gs_{+}\prime \ldots \rangle _{0}+(\unicode[STIX]{x0394}\text{ correction}).\end{eqnarray}$$

3.3.1 Formula (3.20a )

I show in appendix E that $_{\text{a}}\breve{\boldsymbol{K}}$  gives the hydrodynamic part of the Navier–Stokes exchange effects as calculated in Part 1.

3.3.2 Formula (3.20b )

If one generalizes the above quote that surrounds (3.13) by replacing ‘gradient operators’ with ‘gradient or $\unicode[STIX]{x0394}$ operators’, one might be led to conclude that (3.20b ) is negligible, apparently giving a contribution that is at least of third order. However, the discussion in § 1:3.2 of the multicomponent plasma shows that there is a difficulty with this argument for the multispecies case, for there it was shown that the non-hydrodynamic part of the momentum exchange is not negligible but is rather of order unity relative to the hydrodynamic part. Manipulations identical to those done for (3.6b ) show that

(3.21) $$\begin{eqnarray}\lim _{t\rightarrow \infty }\displaystyle \int _{0}^{t}\!\,\text{d}\overline{\unicode[STIX]{x1D70F}}\,_{\text{b}}\breve{K}_{G}^{\overline{\unicode[STIX]{x1D6FD}}}(\unicode[STIX]{x1D707},\overline{\unicode[STIX]{x1D707}},\overline{\unicode[STIX]{x1D70F}})=\left\langle \widehat{G}\widehat{\text{C}}^{-1}\text{P}\widehat{\text{C}}\displaystyle \int _{0}^{\infty }\text{d}\overline{s}\,U(\overline{s})\widehat{{\dot{A}}}_{\unicode[STIX]{x0394}}\!\!\!\text{}s_{+}\prime \text{}^{\overline{\unicode[STIX]{x1D6FD}}}\right\rangle .\end{eqnarray}$$

Upon comparing this result to the analogous calculations in Part 1, one sees that the last integral is essentially the orthogonal projection $|\text{Q}\unicode[STIX]{x1D712}\!\rangle$ , and the construction $\widehat{\text{C}}^{-1}\text{P}\widehat{\text{C}}|\text{Q}\unicode[STIX]{x1D712}\!\rangle$ is the one that would follow from (1:3.35) by formally multiplying that equation through by  $\widehat{\text{C}}^{-1}$ . Thus, the correct answer for the non-hydrodynamic part of the exchange effect follows from (3.20b ).

3.4 Term $(\text{i}\text{i}\text{i})$

Term ( $\text{iii}_{\text{b}}$ ) is explicitly

(3.22) $$\begin{eqnarray}\displaystyle \frac{1}{2}\langle Gs_{+}\prime \text{Q}\unicode[STIX]{x1D713}_{\unicode[STIX]{x0394}\boldsymbol{B}}^{2}\rangle _{0} & = & \displaystyle \frac{1}{2}\left\langle Gs_{+}\prime \text{Q}\left(-\displaystyle \int _{0}^{t}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\,\text{U}(\overline{\unicode[STIX]{x1D70F}})\text{i}\mathscr{L}\boldsymbol{A}s_{+}\prime \star \unicode[STIX]{x0394}\boldsymbol{B}(t-\overline{\unicode[STIX]{x1D70F}})\right)\right.\nonumber\\ \displaystyle & & \displaystyle \times \left.\left(-\displaystyle \int _{0}^{t}\!\text{d}\overline{\unicode[STIX]{x1D70F}}s_{+}\prime \,\text{U}(\overline{\unicode[STIX]{x1D70F}}s_{+}\prime )\text{i}\mathscr{L}\boldsymbol{A}s_{+}\prime \star \unicode[STIX]{x0394}\boldsymbol{B}(t-\overline{\unicode[STIX]{x1D70F}}s_{+}\prime )\right)\right\rangle _{0}.\end{eqnarray}$$

Because this expression contains two factors of  $\text{i}\mathscr{L}\boldsymbol{A}s_{+}\prime$ , each of which generates either gradients or exchange effects, it is adequate to second order to use the time-local forms  $\unicode[STIX]{x0394}\boldsymbol{B}(t)$ . One then has the construction (for a particular symmetric function  $F$ )

(3.23) $$\begin{eqnarray}\frac{1}{2}\displaystyle \int _{0}^{t}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int _{0}^{t}\!\text{d}\overline{\unicode[STIX]{x1D70F}}s_{+}\prime \,F(\overline{\unicode[STIX]{x1D70F}},\overline{\unicode[STIX]{x1D70F}}s_{+}\prime )=\displaystyle \int _{0}^{t}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int _{0}^{\overline{\unicode[STIX]{x1D70F}}}\!\text{d}\overline{\unicode[STIX]{x1D70F}}s_{+}\prime \,F(\overline{\unicode[STIX]{x1D70F}},\overline{\unicode[STIX]{x1D70F}}s_{+}\prime ).\end{eqnarray}$$

For the $\overline{\unicode[STIX]{x1D70F}}s_{+}\prime$  integral, invoke the identityFootnote 48

(3.24) $$\begin{eqnarray}\displaystyle \int _{0}^{\unicode[STIX]{x1D70F}}\!\,\text{d}\overline{\unicode[STIX]{x1D70F}}\,\text{U}(\overline{\unicode[STIX]{x1D70F}})\text{i}\mathscr{L}\boldsymbol{A}=\boldsymbol{A}-\text{e}^{-\text{Q}\text{i}\mathscr{L}\unicode[STIX]{x1D70F}}\boldsymbol{A},\end{eqnarray}$$

with $\unicode[STIX]{x1D70F}$ being replaced by  $\overline{\unicode[STIX]{x1D70F}}$ . The contribution from the  $\boldsymbol{A}$ term (with $\boldsymbol{A}\rightarrow \boldsymbol{A}s_{+}\prime$ ) cancels term ( $\text{iii}_{\text{a}}$ ). Furthermore, one hasFootnote 49

(3.25) $$\begin{eqnarray}\text{U}(\unicode[STIX]{x1D70F})\doteq \text{Q}\text{e}^{-\text{Q}\text{i}\mathscr{L}\text{Q}\unicode[STIX]{x1D70F}}\text{Q}=\text{e}^{-\text{Q}\text{i}\mathscr{L}\unicode[STIX]{x1D70F}}\text{Q}=\text{R}_{1}(\unicode[STIX]{x1D70F})\text{Q}.\end{eqnarray}$$

In the last expression, I used one instance of the shorthand notation

(3.26a-c ) $$\begin{eqnarray}\text{R}_{0}(\unicode[STIX]{x1D70F})\doteq \text{e}^{-\text{i}\mathscr{L}\unicode[STIX]{x1D70F}},\quad \text{R}_{1}(\unicode[STIX]{x1D70F})\doteq \text{e}^{-\text{Q}\text{i}\mathscr{L}\unicode[STIX]{x1D70F}},\quad \text{R}_{2}(\unicode[STIX]{x1D70F})\doteq \text{e}^{-\text{Q}\text{i}\mathscr{L}\text{Q}\unicode[STIX]{x1D70F}}.\end{eqnarray}$$

Thus,

(3.27) $$\begin{eqnarray}\displaystyle \text{term (iii)} & = & \displaystyle -\displaystyle \int _{0}^{t}\!\,\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}s_{+}\prime \,\langle \widehat{G}(\unicode[STIX]{x1D707})[\text{R}_{1}(\overline{\unicode[STIX]{x1D70F}})\text{Q}\text{i}\mathscr{L}A^{\prime \overline{\unicode[STIX]{x1D6FD}}}(\overline{\unicode[STIX]{x1D707}})]\text{R}_{1}(\overline{\unicode[STIX]{x1D70F}})A^{\prime \overline{\unicode[STIX]{x1D6FE}}}(\overline{\unicode[STIX]{x1D707}}s_{+}\prime )\rangle _{0}\nonumber\\ \displaystyle & & \displaystyle \times \,\unicode[STIX]{x0394}B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r}-\overline{\unicode[STIX]{x1D746}},t)\unicode[STIX]{x0394}B_{\overline{\unicode[STIX]{x1D6FE}}}(\boldsymbol{r}-\overline{\unicode[STIX]{x1D746}}s_{+}\prime ,t).\end{eqnarray}$$

The first  $\unicode[STIX]{x0394}\boldsymbol{B}$ is operated on by  $\text{i}\mathscr{L}\boldsymbol{A}s_{+}\prime$ , so it is already of first order. Because $\text{R}_{1}(\overline{\unicode[STIX]{x1D70F}})=1+O(\text{Q}\text{i}\mathscr{L}\overline{\unicode[STIX]{x1D70F}})$ , use of the expansion (3.3) shows that the second  $\unicode[STIX]{x0394}\boldsymbol{B}$ factor is also at least of first order.

With exchange effects neglected, one may replace $\text{Q}\text{i}\mathscr{L}\boldsymbol{A}s_{+}\prime (\overline{\unicode[STIX]{x1D707}})$ by $\text{Q}\overline{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }\boldsymbol{J}(\overline{\unicode[STIX]{x1D707}})$ . Upon integrating the  $\overline{\unicode[STIX]{x1D735}}$ by parts and using the definition $\overline{\unicode[STIX]{x1D746}}\doteq \boldsymbol{r}-\overline{\boldsymbol{x}}$ and the expansion (3.3), one finds

(3.28) $$\begin{eqnarray}\displaystyle \text{term (iii)} & = & \displaystyle -\displaystyle \int _{0}^{t}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\!\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}s_{+}\prime \,\langle \widehat{G}(\unicode[STIX]{x1D707})[\text{R}_{1}(\overline{\unicode[STIX]{x1D70F}})\widehat{\boldsymbol{J}}\text{}^{\overline{\unicode[STIX]{x1D6FD}}}(\overline{\unicode[STIX]{x1D707}})]\text{R}_{1}(\overline{\unicode[STIX]{x1D70F}})As_{+}\prime ^{\overline{\unicode[STIX]{x1D6FE}}}(\overline{\unicode[STIX]{x1D707}}s_{+}\prime )\rangle _{0}\nonumber\\ \displaystyle & & \displaystyle \boldsymbol{\cdot }\,[\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t)]\unicode[STIX]{x0394}B_{\overline{\unicode[STIX]{x1D6FE}}}(\boldsymbol{r}-\overline{\unicode[STIX]{x1D746}}s_{+}\prime ,t).\end{eqnarray}$$

Although both factors of expression (3.28) involve $\text{R}_{1}(\overline{\unicode[STIX]{x1D70F}})$ , the first  $\text{R}_{1}$ acts on the subtracted flux $\widehat{\boldsymbol{J}}$ (which lives in the orthogonal subspace) while the second one acts on  $\boldsymbol{A}s_{+}\prime$ (which lives in the hydrodynamic subspace). To lowest order in the gradients, one sees from (3.13) that the first  $\text{R}_{1}$ may be replaced by  $\text{R}_{0}$ . Brey et al. show that it is fruitful to manipulate the second  $\text{R}_{1}$ by using the identity

(3.29) $$\begin{eqnarray}\text{e}^{-\text{Q}\text{i}\mathscr{L}\unicode[STIX]{x1D70F}}=\text{e}^{-\text{i}\mathscr{L}\unicode[STIX]{x1D70F}}+\displaystyle \int _{0}^{\unicode[STIX]{x1D70F}}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\,\text{e}^{-\text{i}\mathscr{L}(\unicode[STIX]{x1D70F}-\overline{\unicode[STIX]{x1D70F}})}\text{P}\text{i}\mathscr{L}\text{e}^{-\text{Q}\text{i}\mathscr{L}\overline{\unicode[STIX]{x1D70F}}}.\end{eqnarray}$$

Thus, $\text{term (iii)}=\text{term (iii-c)}+\text{term (iii-d)}$ , where

(3.30a ) $$\begin{eqnarray}\displaystyle \text{term (iii-c)} & \doteq & \displaystyle \displaystyle \int _{0}^{t}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\!\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}s_{+}\prime \,\langle \widehat{G}(\unicode[STIX]{x1D707})[\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}})\widehat{\boldsymbol{J}}\text{}^{\overline{\unicode[STIX]{x1D6FD}}}(\overline{\unicode[STIX]{x1D707}})]\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}})As_{+}\prime ^{\overline{\unicode[STIX]{x1D6FE}}s_{+}\prime }(\overline{\unicode[STIX]{x1D707}}s_{+}\prime )\rangle _{0}\nonumber\\ \displaystyle & & \displaystyle \boldsymbol{\cdot }\,\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t)\unicode[STIX]{x0394}B_{\overline{\unicode[STIX]{x1D6FE}}s_{+}\prime }(\boldsymbol{r}-\overline{\unicode[STIX]{x1D746}}s_{+}\prime ,t),\end{eqnarray}$$
(3.30b ) $$\begin{eqnarray}\displaystyle \text{term (iii-d)} & \doteq & \displaystyle \displaystyle \int _{0}^{t}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\!\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}s_{+}\prime \,\langle \!\widehat{G}(\unicode[STIX]{x1D707})[\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}})\widehat{\boldsymbol{J}}\text{}^{\overline{\unicode[STIX]{x1D6FD}}}(\overline{\unicode[STIX]{x1D707}})]\nonumber\\ \displaystyle & & \displaystyle \boldsymbol{\cdot }\,\displaystyle \int _{0}^{\unicode[STIX]{x1D70F}}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\,\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}}-\overline{\unicode[STIX]{x1D70F}}s_{+}\prime )\text{P}\text{i}\mathscr{L}\text{R}_{1}(\overline{\unicode[STIX]{x1D70F}}s_{+}\prime )A^{\prime \overline{\unicode[STIX]{x1D6FE}}s_{+}\prime }(\overline{\unicode[STIX]{x1D707}}s_{+}\prime )\!\rangle \!_{0}\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t)\unicode[STIX]{x0394}B_{\overline{\unicode[STIX]{x1D6FE}}s_{+}\prime }(\boldsymbol{r}-\overline{\unicode[STIX]{x1D746}}s_{+}\prime ,t).\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

In $\text{term (iii-c)}$ , Taylor expansion gives $\unicode[STIX]{x0394}B_{\overline{\unicode[STIX]{x1D6FE}}s_{+}\prime }(\boldsymbol{r}-\unicode[STIX]{x1D746}s_{+}\prime ,t)\approx \unicode[STIX]{x0394}B_{\overline{\unicode[STIX]{x1D6FE}}s_{+}\prime ,\overline{s}s_{+}\prime s}(\boldsymbol{r})-\overline{\unicode[STIX]{x1D746}}s_{+}\prime \boldsymbol{\cdot }\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FE}}s_{+}\prime }(\boldsymbol{r},t)$ . $\unicode[STIX]{x0394}\boldsymbol{B}_{\overline{s}s_{+}\prime s}$  is again neglected, as it generates an exchange effect. Thus, term (iii-c) reduces to

(3.31) $$\begin{eqnarray}\text{term (iii-c)}\approx -\boldsymbol{\unicode[STIX]{x1D649}}_{2}^{\overline{\unicode[STIX]{x1D6FD}}\overline{\unicode[STIX]{x1D6FE}}}[G](\unicode[STIX]{x1D707},t):\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FD}}}\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FE}}},\end{eqnarray}$$

where

(3.32) $$\begin{eqnarray}\boldsymbol{\unicode[STIX]{x1D649}}_{2}^{\overline{\unicode[STIX]{x1D6FD}}\overline{\unicode[STIX]{x1D6FE}}}[G](\unicode[STIX]{x1D707},t)\doteq -\displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,\boldsymbol{N}_{2}^{\overline{\unicode[STIX]{x1D6FD}}\overline{\unicode[STIX]{x1D6FE}}}[G](\unicode[STIX]{x1D707},\overline{\unicode[STIX]{x1D707}},\overline{\unicode[STIX]{x1D70F}};t)\overline{\unicode[STIX]{x1D746}}\end{eqnarray}$$

and

(3.33) $$\begin{eqnarray}\boldsymbol{N}_{2}^{\overline{\unicode[STIX]{x1D6FD}}\overline{\unicode[STIX]{x1D6FE}}}[G](\unicode[STIX]{x1D707},\overline{\unicode[STIX]{x1D707}},\overline{\unicode[STIX]{x1D70F}};t)\doteq \langle [\text{e}^{\text{i}\mathscr{L}\overline{\unicode[STIX]{x1D70F}}}\widehat{G}(\unicode[STIX]{x1D707})][\widehat{\pmb{\pmb{\mathscr{J}}}}\text{}^{\overline{\unicode[STIX]{x1D6FD}}}As_{+}\prime ^{\overline{\unicode[STIX]{x1D6FE}}}(\overline{\unicode[STIX]{x1D707}})]\rangle _{0}.\end{eqnarray}$$

When only kinetic contributions are used in  $\widehat{G}$ , $\widehat{\pmb{\pmb{\mathscr{J}}}}$ and  $\boldsymbol{A}s_{+}\prime$ , it is easy to see that (3.33) involves the Klimontovich correlation function for three phase-space points.

The further reduction of term (iii-d) is described in appendix B of Brey et al. (Reference Brey, Zwanzig and Dorfman1981), who also use results from their appendix A, in which a representation of $(\unicode[STIX]{x2202}_{t}\boldsymbol{B})^{(1)}$ is derived. The final result, correct to second order in the gradients and in the absence of exchange effects, is

(3.34) $$\begin{eqnarray}\text{term (iii-d)}=\unicode[STIX]{x1D614}_{2}^{\overline{\unicode[STIX]{x1D6FD}}\overline{\unicode[STIX]{x1D6FE}}}[G](\unicode[STIX]{x1D707},t)\boldsymbol{\cdot }\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t)[\unicode[STIX]{x2202}_{t}B_{\overline{\unicode[STIX]{x1D6FE}}}(\boldsymbol{r},t)]^{(1)},\end{eqnarray}$$

where

(3.35) $$\begin{eqnarray}\unicode[STIX]{x1D614}_{2}^{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}[G](\unicode[STIX]{x1D707},t)\doteq \displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,\boldsymbol{N}_{2}^{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}[G](\unicode[STIX]{x1D707},\overline{\unicode[STIX]{x1D707}},\overline{\unicode[STIX]{x1D70F}};t)\overline{\unicode[STIX]{x1D70F}}.\end{eqnarray}$$

3.5 Term $(\text{i}\text{v})$

We have

(3.36) $$\begin{eqnarray}\text{term (iv)}\doteq \langle Gs_{+}\prime \unicode[STIX]{x1D713}_{\unicode[STIX]{x0394}\boldsymbol{B}^{(2)}}\rangle _{0},\end{eqnarray}$$

where $\unicode[STIX]{x0394}\boldsymbol{B}^{(2)}$ is given by (2.74). Although both of the explicit terms in (2.74) are nominally of second order in $\unicode[STIX]{x0394}\boldsymbol{B}$ , it is not difficult to show, using integration by parts, that the second term is actually of third order in the gradients, hence is negligible. The evaluation of the first term is stated by Brey et al. to be

(3.37) $$\begin{eqnarray}\text{term (iv)}\approx -\displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,\langle [\text{e}^{\text{i}\mathscr{L}\overline{\unicode[STIX]{x1D70F}}}\widehat{G}(\unicode[STIX]{x1D707})]\text{P}\widehat{\pmb{\pmb{\mathscr{J}}}}\text{}^{\overline{\unicode[STIX]{x1D6FD}}}As_{+}\prime ^{\overline{\unicode[STIX]{x1D6FE}}}(\overline{\unicode[STIX]{x1D707}})\rangle _{0}\boldsymbol{\cdot }\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t)\overline{\unicode[STIX]{x1D746}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FE}}}(\boldsymbol{r},t).\end{eqnarray}$$

Since they do not give the detailed manipulations, I present a proof in appendix F. Note that this term adds to term (iii-c) to change the $\pmb{\pmb{\mathscr{J}}}\boldsymbol{A}s_{+}\prime$ in (3.33) to $\text{Q}(\pmb{\pmb{\mathscr{J}}}\boldsymbol{A}s_{+}\prime )$ .

3.6 Summary of the gradient expansion

One can now collect all of the terms. Brey et al. show that the terms involving $(\unicode[STIX]{x2202}_{t}\boldsymbol{B})^{(1)}$ can be combined. One ultimately finds

(3.38) $$\begin{eqnarray}\displaystyle \langle G\rangle & = & \displaystyle \langle G\rangle _{\text{Euler}}-\boldsymbol{k}_{1\unicode[STIX]{x1D735}}^{\overline{\unicode[STIX]{x1D6FD}}}[G](\unicode[STIX]{x1D707},t)\boldsymbol{\cdot }\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t)-k_{1\unicode[STIX]{x0394}}^{\overline{\unicode[STIX]{x1D6FD}}}[G](\unicode[STIX]{x1D707},t)B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t)\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D65C}_{2}^{\overline{\unicode[STIX]{x1D6FD}}}[G](\unicode[STIX]{x1D707},t)\boldsymbol{ : }\unicode[STIX]{x1D735}\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t)-\unicode[STIX]{x1D65D}_{2}^{\overline{\unicode[STIX]{x1D6FD}}\overline{\unicode[STIX]{x1D6FE}}}[G](\unicode[STIX]{x1D707},t)\boldsymbol{ : }\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t)\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FE}}}(\boldsymbol{r},t)\nonumber\\ \displaystyle & & \displaystyle +\,\left[\unicode[STIX]{x2202}_{t}\left(\boldsymbol{k}_{2}^{\overline{\unicode[STIX]{x1D6FD}}}[G](\unicode[STIX]{x1D707},t)\boldsymbol{\cdot }\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t)\right)\right]^{(1)},\end{eqnarray}$$

where

(3.39a ) $$\begin{eqnarray}\displaystyle \boldsymbol{k}_{1\unicode[STIX]{x1D735}}^{\overline{\unicode[STIX]{x1D6FD}}}[G](\unicode[STIX]{x1D707},t) & \doteq & \displaystyle \displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}\,\langle \widehat{G}(\unicode[STIX]{x1D707})\text{e}^{-\text{i}\mathscr{L}\overline{\unicode[STIX]{x1D70F}}}\widehat{\boldsymbol{J}}\text{}^{\overline{\unicode[STIX]{x1D6FD}}}(\overline{\unicode[STIX]{x1D707}})\rangle _{0},\end{eqnarray}$$
(3.39b ) $$\begin{eqnarray}\displaystyle k_{1\unicode[STIX]{x0394}}^{\overline{\unicode[STIX]{x1D6FD}}}[G](\unicode[STIX]{x1D707},t) & \doteq & \displaystyle \displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}\,\langle \widehat{G}(\unicode[STIX]{x1D707})\text{e}^{-\text{i}\text{Q}\mathscr{L}\overline{\unicode[STIX]{x1D70F}}}{\dot{A}}_{\unicode[STIX]{x0394},\overline{s}}^{\prime \overline{\unicode[STIX]{x1D6FD}}}(\overline{\unicode[STIX]{x1D707}})\rangle _{0},\end{eqnarray}$$
(3.39c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D65C}_{2}^{\overline{\unicode[STIX]{x1D6FD}}}[G](\unicode[STIX]{x1D707},t) & \doteq & \displaystyle -\displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}\,\langle \widehat{G}(\unicode[STIX]{x1D707})\text{e}^{-\text{i}\mathscr{L}\overline{\unicode[STIX]{x1D70F}}}\widehat{\boldsymbol{J}}\text{}^{\overline{\unicode[STIX]{x1D6FD}}}(\overline{\unicode[STIX]{x1D707}})\rangle _{0}\overline{\unicode[STIX]{x1D746}},\end{eqnarray}$$
(3.39d ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D65D}_{2}^{\overline{\unicode[STIX]{x1D6FD}}\overline{\unicode[STIX]{x1D6FE}}}[G](\unicode[STIX]{x1D707},t) & \doteq & \displaystyle -\displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}\,\langle \widehat{G}(\unicode[STIX]{x1D707})\text{e}^{-\text{i}\mathscr{L}\overline{\unicode[STIX]{x1D70F}}}\text{Q}[\widehat{\pmb{\pmb{\mathscr{J}}}}\text{}^{\overline{\unicode[STIX]{x1D6FD}}}As_{+}\prime ^{\overline{\unicode[STIX]{x1D6FE}}}(\overline{\unicode[STIX]{x1D707}})]\rangle _{0}\overline{\unicode[STIX]{x1D746}},\end{eqnarray}$$
(3.39e ) $$\begin{eqnarray}\displaystyle \boldsymbol{k}_{2}^{\unicode[STIX]{x1D6FD}}[G](\unicode[STIX]{x1D707},t) & \doteq & \displaystyle \displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}\,\langle \widehat{G}(\unicode[STIX]{x1D707})\text{e}^{-\text{i}\mathscr{L}\overline{\unicode[STIX]{x1D70F}}}\widehat{\pmb{\pmb{\mathscr{J}}}}\text{}^{\unicode[STIX]{x1D6FD}}\rangle _{0}\overline{\unicode[STIX]{x1D70F}}.\end{eqnarray}$$

We have now found a formula for the non-equilibrium average of any quantity  $G$ , correct to first order in exchange terms and second order in gradients. That formula can be used to evaluate the right-hand side of the equation for $\unicode[STIX]{x2202}_{t}\boldsymbol{a}_{s}$ , namely, the averages of the right-hand sides of equations (2.26). The averages of the conserved fluxes are

(3.40) $$\begin{eqnarray}\displaystyle \langle \boldsymbol{J}_{s}^{\unicode[STIX]{x1D6FC}}\rangle & = & \displaystyle \langle \boldsymbol{J}_{s}^{\unicode[STIX]{x1D6FC}}\rangle _{\text{Euler}}-\underbrace{\boldsymbol{k}_{1}^{\overline{\unicode[STIX]{x1D6FD}}}[\boldsymbol{J}_{s}^{\unicode[STIX]{x1D6FC}}](\unicode[STIX]{x1D707},t)\boldsymbol{\cdot }\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t)}_{\text{}_{\unicode[STIX]{x1D735}}\text{NS}_{\overline{\unicode[STIX]{x1D6FD}}}^{\unicode[STIX]{x1D6FC}}}-\underbrace{\unicode[STIX]{x1D65C}_{2}^{\overline{\unicode[STIX]{x1D6FD}}}[\boldsymbol{J}_{s}^{\unicode[STIX]{x1D6FC}}](\unicode[STIX]{x1D707},t)\boldsymbol{ : }\unicode[STIX]{x1D735}\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t)}_{\text{B}_{\overline{\unicode[STIX]{x1D6FD}}}^{\unicode[STIX]{x1D6FC}}}\nonumber\\ \displaystyle & & \displaystyle -\,\underbrace{\unicode[STIX]{x1D65D}_{2}^{\overline{\unicode[STIX]{x1D6FD}}\overline{\unicode[STIX]{x1D6FE}}}[\boldsymbol{J}_{s}^{\unicode[STIX]{x1D6FC}}](\unicode[STIX]{x1D707},t)\boldsymbol{ : }\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t)\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FE}}}(\boldsymbol{r},t)}_{\text{B}_{\overline{\unicode[STIX]{x1D6FD}}\overline{\unicode[STIX]{x1D6FE}}}^{\unicode[STIX]{x1D6FC}}}+\underbrace{\left[\unicode[STIX]{x2202}_{t}\left(\boldsymbol{k}_{2}^{\overline{\unicode[STIX]{x1D6FD}}}[\boldsymbol{J}_{s}^{\unicode[STIX]{x1D6FC}}](\unicode[STIX]{x1D707},t)\boldsymbol{\cdot }\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t)\right)\right]^{(1)}}_{\text{B}^{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x2202}_{t}\boldsymbol{k}_{\overline{\unicode[STIX]{x1D6FD}}}+\text{B}^{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x2202}_{t}B_{\overline{\unicode[STIX]{x1D6FD}}}},\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

and the averages of the first-order exchange terms are

(3.41) $$\begin{eqnarray}X_{s}^{\unicode[STIX]{x1D6FC}}\equiv \langle {\dot{A}}_{\unicode[STIX]{x0394},s}^{\prime \unicode[STIX]{x1D6FC}}\rangle =-\underbrace{k_{1\unicode[STIX]{x1D6E5}}^{\overline{\unicode[STIX]{x1D6FD}}}[{\dot{A}}_{\unicode[STIX]{x0394},s}^{\prime \unicode[STIX]{x1D6FC}}](\unicode[STIX]{x1D707},t)B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t)}_{\text{}_{\unicode[STIX]{x0394}}\text{NS}_{\overline{\unicode[STIX]{x1D6FD}}}^{\unicode[STIX]{x1D6FC}}}.\end{eqnarray}$$

In the above, the various terms have been concisely identified for future reference. NS and B stand for Navier–Stokes and Burnett, respectively. The field indices can assume the values  $n$ , $\boldsymbol{p}$ or  $e$ . For example, in the momentum equation the linear Burnett term includes the contributions $\text{B}_{\boldsymbol{p}}^{\boldsymbol{p}}$ and $\text{B}_{e}^{\boldsymbol{p}}$ .

4 Some specific formulas for transport coefficients

In § 4.1 I follow Brey (Reference Brey1983) and write out the general structure of the dissipative fluid equations for the unmagnetized, one-component fluid. Integrals that define the transport coefficients are recorded in § 4.2. Then in § 4.3 I comment on the first-order exchange effects.

4.1 The Burnett equations for a one-component fluid

An important reference case is the one-component fluid. Those results were recorded for $\boldsymbol{B}^{\text{ext}}=\mathbf{0}$ in appendix A of Brey (Reference Brey1983), and I shall transcribe them here, following Brey’s numbering conventions for the various dissipative coefficients. The results will be used in § 6 to demonstrate the consistency of the calculations of Catto & Simakov of parallel viscosity with the two-time formalism. Note that in the following formulas the effect of wave-induced transport is omitted.

Instead of Brey’s  $\unicode[STIX]{x1D702}$ and  $\unicode[STIX]{x1D706}$ , I shall use $\unicode[STIX]{x1D707}\doteq \unicode[STIX]{x1D702}/nm$ and $\unicode[STIX]{x1D705}\doteq \unicode[STIX]{x1D706}/n$ , which have the dimensions of a diffusion coefficient. Subscripted $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D705}$  quantities relate to the Burnett corrections and have various dimensions. Some supporting algebra is given by Krommes (Reference Krommes2018c , §§ 2 and 3).

In the following definitions, $d$  denotes the dimension of space, $e$  denotes the energy density (in the local frame with $\boldsymbol{u}=\mathbf{0}$ ), $h$  denotes the enthalpy, and $s$  denotes the entropy density. In the limit of an ideal gas, one has

(4.1a-d ) $$\begin{eqnarray}p\rightarrow nT,\quad e\rightarrow \frac{3}{2}nT,\quad h\rightarrow \frac{5}{2}nT,\quad s\rightarrow \ln \left[\vphantom{\left(\frac{1}{2}\right)}\right.\!\!\frac{1}{n}\left(\frac{4\unicode[STIX]{x03C0}me}{3h_{\text{P}}^{2}}\right)^{3/2}\!\!\left.\vphantom{\left(\frac{1}{2}\right)}\right]+\frac{5}{2}.\end{eqnarray}$$

(In the last expression, $h_{\text{P}}$  is Planck’s constant.) The expansion coefficient  $\unicode[STIX]{x1D6FC}$ and isothermal compressibility  $\unicode[STIX]{x1D705}_{T}$ are

(4.2a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}\doteq -\frac{1}{n}\left(\frac{\unicode[STIX]{x2202}n}{\unicode[STIX]{x2202}T}\right)_{p}\rightarrow \frac{1}{T},\quad \unicode[STIX]{x1D705}_{T}\doteq \frac{1}{n}\left(\frac{\unicode[STIX]{x2202}n}{\unicode[STIX]{x2202}p}\right)_{T}\rightarrow \frac{1}{p}.\end{eqnarray}$$

The strain rate and vorticity tensors are

(4.3a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D64E}\doteq {\textstyle \frac{1}{2}}[(\unicode[STIX]{x1D735}\boldsymbol{u})+(\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}],\quad \unicode[STIX]{x1D734}\doteq {\textstyle \frac{1}{2}}[(\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}-(\unicode[STIX]{x1D735}\boldsymbol{u})];\end{eqnarray}$$

the traceless tensor

(4.4) $$\begin{eqnarray}\unicode[STIX]{x1D652}\doteq (\unicode[STIX]{x1D735}\boldsymbol{u})+(\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}-\frac{2}{d}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u})\unicode[STIX]{x1D644}\end{eqnarray}$$

is also useful.

4.1.1 The one-component Euler equations

Let us write $\unicode[STIX]{x1D749}=p\unicode[STIX]{x1D644}+\unicode[STIX]{x1D745}$ . Then the one-component Euler equations are

(4.5a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}n & = & \displaystyle -\unicode[STIX]{x1D735}\boldsymbol{\cdot }(n\boldsymbol{u}),\end{eqnarray}$$
(4.5b ) $$\begin{eqnarray}\displaystyle mn(\unicode[STIX]{x2202}_{t}\boldsymbol{u}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}) & = & \displaystyle -nq(\boldsymbol{E}+c^{-1}\boldsymbol{u}\boldsymbol{\times }\boldsymbol{B}^{\text{ext}})-\unicode[STIX]{x1D735}p,\end{eqnarray}$$
(4.5c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}T+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}T & = & \displaystyle -T\left(\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}e}\right)\bigg|_{n}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}.\end{eqnarray}$$
For the derivation of (4.5c ), see the discussion of (2-S:3.126).

4.1.2 Dissipative momentum flux for a one-component fluid

The dissipative momentum flux to Burnett order is

(4.6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D745}/nm & = & \displaystyle -\unicode[STIX]{x1D707}\unicode[STIX]{x1D652}-\unicode[STIX]{x1D701}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u})\unicode[STIX]{x1D644}\nonumber\\ \displaystyle & & \displaystyle -\,2\unicode[STIX]{x1D707}_{1}(\unicode[STIX]{x1D735}\unicode[STIX]{x1D735}p-\unicode[STIX]{x1D705}_{T}\unicode[STIX]{x1D735}p\,\unicode[STIX]{x1D735}p)+(\unicode[STIX]{x1D707}_{7}-\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D707}_{1})(\unicode[STIX]{x1D735}p\,\unicode[STIX]{x1D735}T+\unicode[STIX]{x1D735}T\,\unicode[STIX]{x1D735}p)\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D707}_{3}\unicode[STIX]{x1D735}\unicode[STIX]{x1D735}T+\unicode[STIX]{x1D707}_{5}\unicode[STIX]{x1D735}T\,\unicode[STIX]{x1D735}T\nonumber\\ \displaystyle & & \displaystyle +\,\left[\unicode[STIX]{x1D707}_{11}-2\left(\frac{\unicode[STIX]{x2202}(n\unicode[STIX]{x1D707}_{1})}{\unicode[STIX]{x2202}n}\right)_{s}\right](\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u})\unicode[STIX]{x1D64E}+(\unicode[STIX]{x1D707}_{12}-2\unicode[STIX]{x1D707}_{1})\unicode[STIX]{x1D64E}\boldsymbol{\cdot }\unicode[STIX]{x1D64E}^{\text{T}}+2\unicode[STIX]{x1D707}_{1}\unicode[STIX]{x1D734}\boldsymbol{\cdot }\unicode[STIX]{x1D734}^{\text{T}}\nonumber\\ \displaystyle & & \displaystyle +\,~\unicode[STIX]{x1D707}_{13}(\unicode[STIX]{x1D64E}^{\text{T}}\boldsymbol{\cdot }\unicode[STIX]{x1D734}+\unicode[STIX]{x1D734}^{\text{T}}\boldsymbol{\cdot }\unicode[STIX]{x1D64E})~\nonumber\\ \displaystyle & & \displaystyle +\,\left\{\vphantom{\left[\unicode[STIX]{x1D707}_{9}-\left(\frac{\unicode[STIX]{x2202}n\unicode[STIX]{x1D707}_{2}}{\unicode[STIX]{x2202}n}\right)_{s}\right]}-\unicode[STIX]{x1D707}_{2}[\unicode[STIX]{x1D6FB}^{2}p-\unicode[STIX]{x1D705}_{T}|\unicode[STIX]{x1D735}p|^{2}]+\unicode[STIX]{x1D707}_{4}\unicode[STIX]{x1D6FB}^{2}T+\unicode[STIX]{x1D707}_{6}|\unicode[STIX]{x1D735}T|^{2}+(\unicode[STIX]{x1D707}_{8}-\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D707}_{2})\unicode[STIX]{x1D735}p\boldsymbol{\cdot }\unicode[STIX]{x1D735}T\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\left[\unicode[STIX]{x1D707}_{9}-\left(\frac{\unicode[STIX]{x2202}(n\unicode[STIX]{x1D707}_{2})}{\unicode[STIX]{x2202}n}\right)_{s}\right](\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u})^{2}+(\unicode[STIX]{x1D707}_{10}-\unicode[STIX]{x1D707}_{2})\text{Tr}(\unicode[STIX]{x1D64E}\boldsymbol{\cdot }\unicode[STIX]{x1D64E}^{\text{T}})+\unicode[STIX]{x1D707}_{2}\text{Tr}(\unicode[STIX]{x1D734}\boldsymbol{\cdot }\unicode[STIX]{x1D734}^{\text{T}})\right\}\unicode[STIX]{x1D644}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Here the first line gives the Navier–Stokes result; the subsequent lines are the Burnett corrections. The viscosities are defined in the next several paragraphs in terms of certain $K$  quantities that are the integrals of two-time correlation functions and are defined in § 4.2. In the following definitions, underbracing indicates the value of an expression for an ideal gas. The origins of the various terms can be traced by noting which $K$  quantities enter the expression for a particular viscosity, then referring to § 4.2, where each formula is linked to one or moreFootnote 50 of the terms in the general result (3.38). $K$  integrals with wavy underlining, such as , are determined by correlation functions with three phase-space points; the others follow from two-point correlations.

The Navier–Stokes viscosity coefficients are

(4.7a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707} & \doteq & \displaystyle (nmT)^{-1}K^{\text{I}},\end{eqnarray}$$
(4.7b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D701} & \doteq & \displaystyle (nmT)^{-1}\left(K^{\text{II}}+\frac{2}{d}K^{\text{I}}\right),\end{eqnarray}$$
where $\unicode[STIX]{x1D707}$  is the kinematic viscosity and $\unicode[STIX]{x1D701}$  is the bulk viscosity.

The Burnett momentum coefficients are

(4.8a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{1} & \doteq & \displaystyle (nmT)^{-1}K^{\text{IV}},\end{eqnarray}$$
(4.8b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{2} & \doteq & \displaystyle (nmT)^{-1}K^{\text{V}},\end{eqnarray}$$
(4.8c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{3} & \doteq & \displaystyle -2(nmT^{2})^{-1}K_{1},\end{eqnarray}$$
(4.8d ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{4} & \doteq & \displaystyle -(nmT^{2})^{-1}K_{2},\end{eqnarray}$$
(4.8e ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{5} & \doteq & \displaystyle 4(nmT^{3})^{-1}K_{1}+2\underbrace{(h/nT)}_{5/2}(nmT^{3})^{-1}K_{3}-2(nmT^{4})^{-1}K_{5}\nonumber\\ \displaystyle & & \displaystyle +\,2n^{-1}(mT)^{-2}\underbrace{\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}T}\left(\frac{h}{n}\right)_{p}}_{5/2}K_{20},\end{eqnarray}$$
(4.8f ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{6} & \doteq & \displaystyle 2(nmT^{3})^{-1}K_{2}+(nmT^{3})^{-1}\underbrace{(h/nT)}_{5/2}K_{4}-(nmT^{4})^{-1}K_{6}\nonumber\\ \displaystyle & & \displaystyle +\,n^{-1}(mT)^{-2}\underbrace{\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}T}\left(\frac{h}{n}\right)_{p}}_{5/2}K_{21},\end{eqnarray}$$
(4.8g ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{7} & \doteq & \displaystyle -(n^{2}mT^{3})^{-1}K_{3}+n^{-1}(mT)^{-2}\underbrace{\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}p}\left(\frac{h}{n}\right)_{T}}_{0}K_{20},\end{eqnarray}$$
(4.8h ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{8} & \doteq & \displaystyle -(n^{2}mT^{3})^{-1}K_{4}+n^{-1}(mT)^{-2}\underbrace{\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}p}\left(\frac{h}{n}\right)_{T}}_{0}K_{21},\end{eqnarray}$$
(4.8i ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{9} & \doteq & \displaystyle -(nmT^{2})^{-1}K_{7}-(nmT)^{-1}\underbrace{\left(\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}e}\right)_{n}}_{2/3}K_{21},\end{eqnarray}$$
(4.8j ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{10} & \doteq & \displaystyle -2(nmT^{2})^{-1}K_{8}+2(nmT)^{-1}K_{21},\end{eqnarray}$$
(4.8k ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{11} & \doteq & \displaystyle -2(nmT^{2})^{-1}(K_{9}+K_{10})-2(nmT)^{-1}\underbrace{\left(\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}e}\right)_{n}}_{2/3}K_{20},\end{eqnarray}$$
(4.8l ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{12} & \doteq & \displaystyle -4(nmT^{2})^{-1}(K_{11}+K_{12})+4(nmT)^{-1}K_{20},\end{eqnarray}$$
(4.8m ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{13} & \doteq & \displaystyle -2(nmT^{2})^{-1}(K_{11}-K_{12})+2(nmT)^{-1}K_{20}.\end{eqnarray}$$

For a weakly coupled gas, these coefficients can be evaluated in the ideal-gas limit. In that limit, the fact that $\text{Tr}\,\widehat{\unicode[STIX]{x1D749}}=0$ provides the constraints

(4.9a-f ) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}K^{\text{II}}=-{\displaystyle \frac{2}{d}}K^{\text{I}},\quad K^{\text{V}}=-{\displaystyle \frac{2}{d}}K^{\text{IV}},\quad K_{2}=-{\displaystyle \frac{2}{d}}K_{1},\quad K_{4}=-{\displaystyle \frac{2}{d}}K_{3},\\ K_{6}=-{\displaystyle \frac{2}{d}}K_{5},\quad K_{21}=-{\displaystyle \frac{2}{d}}K_{20}.\end{array}\right\}\end{eqnarray}$$

Thus, for a weakly coupled gas the bulk viscosity  $\unicode[STIX]{x1D701}$ vanishes (see (4.7b )); one has

(4.10a-f ) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D707}_{2}=-{\displaystyle \frac{2}{d}}\unicode[STIX]{x1D707}_{1},\quad \unicode[STIX]{x1D707}_{4}=-{\displaystyle \frac{1}{d}}\unicode[STIX]{x1D707}_{3},\quad \unicode[STIX]{x1D707}_{6}=-{\displaystyle \frac{1}{d}}\unicode[STIX]{x1D707}_{5},\quad \unicode[STIX]{x1D707}_{8}=-{\displaystyle \frac{2}{d}}\unicode[STIX]{x1D707}_{7},\quad \unicode[STIX]{x1D707}_{9}=-{\displaystyle \frac{1}{d}}\unicode[STIX]{x1D707}_{11},\\ \unicode[STIX]{x1D707}_{10}=-{\displaystyle \frac{1}{d}}\unicode[STIX]{x1D707}_{12};\end{array}\right\} & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

and the dissipative momentum flux reduces to

(4.11) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D745}^{\text{wc}}/nm & = & \displaystyle -\unicode[STIX]{x1D707}\unicode[STIX]{x1D652}\nonumber\\ \displaystyle & & \displaystyle -\,2\unicode[STIX]{x1D707}_{1}\left[\left(\unicode[STIX]{x1D735}\unicode[STIX]{x1D735}p-\frac{1}{p}\unicode[STIX]{x1D735}p\,\unicode[STIX]{x1D735}p\right)-\frac{1}{d}\left(\unicode[STIX]{x1D6FB}^{2}p-\frac{1}{p}|\unicode[STIX]{x1D735}p|^{2}\right)\unicode[STIX]{x1D644}\right]\nonumber\\ \displaystyle & & \displaystyle +\,(\unicode[STIX]{x1D707}_{7}-T^{-1}\unicode[STIX]{x1D707}_{1})\left(\unicode[STIX]{x1D735}p\,\unicode[STIX]{x1D735}T+\unicode[STIX]{x1D735}T\,\unicode[STIX]{x1D735}p-\frac{2}{d}\unicode[STIX]{x1D735}p\boldsymbol{\cdot }\unicode[STIX]{x1D735}T\,\unicode[STIX]{x1D644}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D707}_{3}\left(\unicode[STIX]{x1D735}\unicode[STIX]{x1D735}T-\frac{1}{d}\unicode[STIX]{x1D6FB}^{2}T\,\unicode[STIX]{x1D644}\right)+\unicode[STIX]{x1D707}_{5}\left(\unicode[STIX]{x1D735}T\,\unicode[STIX]{x1D735}T-\frac{1}{d}|\unicode[STIX]{x1D735}T|^{2}\unicode[STIX]{x1D644}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\left[\unicode[STIX]{x1D707}_{11}-2\left(\frac{\unicode[STIX]{x2202}(n\unicode[STIX]{x1D707}_{1})}{\unicode[STIX]{x2202}n}\right)_{s}\right]\left((\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u})\unicode[STIX]{x1D64E}-\frac{1}{d}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u})^{2}\unicode[STIX]{x1D644}\right)\nonumber\\ \displaystyle & & \displaystyle +\,(\unicode[STIX]{x1D707}_{12}-2\unicode[STIX]{x1D707}_{1})\left((\unicode[STIX]{x1D64E}\boldsymbol{\cdot }\unicode[STIX]{x1D64E}^{\text{T}}-\unicode[STIX]{x1D734}\boldsymbol{\cdot }\unicode[STIX]{x1D734}^{\text{T}})-\frac{1}{d}\text{Tr}(\unicode[STIX]{x1D64E}\boldsymbol{\cdot }\unicode[STIX]{x1D64E}^{\text{T}}-\unicode[STIX]{x1D734}\boldsymbol{\cdot }\unicode[STIX]{x1D734}^{\text{T}})\unicode[STIX]{x1D644}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D707}_{13}(\unicode[STIX]{x1D64E}^{\text{T}}\boldsymbol{\cdot }\unicode[STIX]{x1D734}+\unicode[STIX]{x1D734}^{\text{T}}\boldsymbol{\cdot }\unicode[STIX]{x1D64E}).\end{eqnarray}$$

Each line is separately traceless.Footnote 51 The integral expression for the kinematic viscosity  $\unicode[STIX]{x1D707}$ will be shown to agree with the one derived in Part 1.

4.1.3 Dissipative heat flux for a one-component fluid

The dissipative heat flux $\boldsymbol{j}_{_{\text{diss}}}^{e}\equiv \boldsymbol{q}$ to Burnett order is

(4.12) $$\begin{eqnarray}\displaystyle \boldsymbol{q}/n & = & \displaystyle n^{-1}\unicode[STIX]{x1D745}\boldsymbol{\cdot }\boldsymbol{u}-\unicode[STIX]{x1D705}_{}\unicode[STIX]{x1D735}T\nonumber\\ \displaystyle & & \displaystyle +\,\left[\vphantom{\left(\frac{1}{2}\right)}\right.\!\unicode[STIX]{x1D705}_{2}-\unicode[STIX]{x1D705}_{1}T\underbrace{\left(\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}e}\right)_{n}}_{3/2}\!\left.\vphantom{\left(\frac{1}{2}\right)}\right]\unicode[STIX]{x1D735}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u})+\unicode[STIX]{x1D705}_{3}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}\nonumber\\ \displaystyle & & \displaystyle +\,\left\{\vphantom{\left(\frac{1}{2}\right)}\right.\!\unicode[STIX]{x1D705}_{4}-\unicode[STIX]{x1D705}_{1}\underbrace{\left(\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}e}\right)_{n}}_{3/2}-\unicode[STIX]{x1D705}_{1}T\left[\vphantom{\left(\frac{1}{2}\right)}\right.\!\underbrace{\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}T}\left(\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}e}\right)_{n}}_{0}\!\left.\vphantom{\left(\frac{1}{2}\right)}\right]_{p}-\left(\frac{\unicode[STIX]{x2202}(n\unicode[STIX]{x1D705}_{1})}{\unicode[STIX]{x2202}n}\right)_{s}\!\!\left.\vphantom{\left(\frac{1}{2}\right)}\right\}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u})\unicode[STIX]{x1D735}T\nonumber\\ \displaystyle & & \displaystyle +\,[(\unicode[STIX]{x1D705}_{5}-\unicode[STIX]{x1D705}_{1})\unicode[STIX]{x1D64E}+(\unicode[STIX]{x1D705}_{6}+\unicode[STIX]{x1D705}_{1})\unicode[STIX]{x1D734}]\boldsymbol{\cdot }\unicode[STIX]{x1D735}T\nonumber\\ \displaystyle & & \displaystyle +\,\left(\vphantom{\left(\frac{1}{2}\right)}\right.\!\unicode[STIX]{x1D705}_{7}\unicode[STIX]{x1D64E}+\left\{\vphantom{\left(\frac{1}{2}\right)}\right.\!\!\unicode[STIX]{x1D705}_{8}-\unicode[STIX]{x1D705}_{1}T\left[\vphantom{\left(\frac{1}{2}\right)}\right.\!\underbrace{\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}p}\left(\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}e}\right)_{n}}_{0}\!\left.\vphantom{\left(\frac{1}{2}\right)}\right]_{T}\!\!\left.\vphantom{\left(\frac{1}{2}\right)}\right\}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u})\unicode[STIX]{x1D644}\!\!\left.\vphantom{\left(\frac{1}{2}\right)}\right)\boldsymbol{\cdot }\unicode[STIX]{x1D735}p,\end{eqnarray}$$

where the Navier–Stokes thermal diffusivity is

(4.13) $$\begin{eqnarray}\unicode[STIX]{x1D705}_{}\doteq n^{-1}T^{-2}K^{\text{III}}\end{eqnarray}$$

and the Burnett energy coefficients are

(4.14a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{1} & \doteq & \displaystyle (nT^{2})^{-1}K^{\text{VI}},\end{eqnarray}$$
(4.14b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{2} & \doteq & \displaystyle -(nT)^{-1}(K_{1}+K_{2}),\end{eqnarray}$$
(4.14c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{3} & \doteq & \displaystyle -(nT)^{-1}K_{1},\end{eqnarray}$$
(4.14d ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{4} & \doteq & \displaystyle (nT^{2})^{-1}\left\{K_{1}+K_{2}-T^{-1}(K_{13}+K_{19})+(nT)^{-1}h\,K_{17}\vphantom{\left(\frac{1}{2}\right)}\right.\!\!\nonumber\\ \displaystyle & & \displaystyle +T\left[\vphantom{\left(\frac{1}{2}\right)}\right.\!\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}T}\underbrace{\left(\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}n}\right)_{e}}_{0}\!\left.\vphantom{\left(\frac{1}{2}\right)}\right]_{p}K_{22}+T\left[\vphantom{\left(\frac{1}{2}\right)}\right.\!\underbrace{\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}T}\left(\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}e}\right)_{n}}_{0}\!\left.\vphantom{\left(\frac{1}{2}\right)}\right]_{p}K_{23}\!\!\left.\vphantom{\left(\frac{1}{2}\right)}\right\},\end{eqnarray}$$
(4.14e ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{5} & \doteq & \displaystyle (nT^{2})^{-1}\left\{\vphantom{\left(\frac{1}{2}\right)}\right.\!\!3K_{1}+K_{2}+2(nT)^{-1}hK_{16}-T^{-1}(2K_{18}+K_{14}+K_{15})\nonumber\\ \displaystyle & & \displaystyle +\left[\vphantom{\left(\frac{1}{2}\right)}\right.\!1+\underbrace{\left(\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}e}\right)_{n}}_{2/3}\!\left.\vphantom{\left(\frac{1}{2}\right)}\right]K_{23}+\left[\vphantom{\left(\frac{1}{2}\right)}\right.\!\underbrace{\left(\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}n}\right)_{e}}_{0}-\underbrace{\frac{h}{n}}_{(5/2)T}\!\left.\vphantom{\left(\frac{1}{2}\right)}\right]K_{22}\!\!\left.\vphantom{\left(\frac{1}{2}\right)}\right\},\end{eqnarray}$$
(4.14f ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{6} & \doteq & \displaystyle (nT^{2})^{-1}\left\{\vphantom{\left(\frac{1}{2}\right)}\right.\!\!K_{1}-K_{2}-T^{-1}(K_{14}-K_{15})\nonumber\\ \displaystyle & & \displaystyle -\left[1+\left(\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}e}\right)_{n}\right]K_{23}-\left[\vphantom{\left(\frac{1}{2}\right)}\right.\!\underbrace{\left(\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}n}\right)_{e}}_{0}-\underbrace{\frac{h}{n}}_{(5/2)T}\!\left.\vphantom{\left(\frac{1}{2}\right)}\right]K_{22}\!\!\left.\vphantom{\left(\frac{1}{2}\right)}\right\},\end{eqnarray}$$
(4.14g ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{7} & \doteq & \displaystyle -2(nT)^{-2}K_{16}-2\unicode[STIX]{x1D707}_{1},\end{eqnarray}$$
(4.14h ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{8} & \doteq & \displaystyle -(nT)^{-2}K_{17}-\unicode[STIX]{x1D707}_{2}+\frac{1}{nT}\left[\vphantom{\left(\frac{1}{2}\right)}\right.\!\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}p}\underbrace{\left(\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}n}\right)_{e}}_{0}\!\left.\vphantom{\left(\frac{1}{2}\right)}\right]_{T}K_{22}+\frac{1}{nT}\underbrace{\left[\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}p}\left(\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}e}\right)_{n}\right]_{T}}_{0}K_{23}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
Note that all of the Burnett energy corrections are negligible when $\boldsymbol{u}$  is ordered small.

4.2 Integrals of correlation functions

The $\unicode[STIX]{x1D707}$ and  $\unicode[STIX]{x1D705}_{}$ coefficients that appear in the previous subsections are defined in terms of various integrals of correlation tensors. Symmetry considerations reduce those tensors to a collection of scalar quantities  $K^{i}$ and  $K_{i}$ defined as follows, using Brey’s numbering conventions. The origin of each term is indicated by the notation such as $\text{NS}_{\boldsymbol{p}}^{\boldsymbol{p}}$ that was introduced in conjunction with (3.38). The one-component results tabulated here are correct for the unmagnetized case. When $\boldsymbol{B}^{\text{ext}}\neq \mathbf{0}$ , additional transport coefficients must be introduced. That was done for linear response in § 1:3; I shall eschew that exercise for the Burnett coefficients. For multispecies plasmas, modified propagators must be used instead of  $\text{R}_{0}$ .

(4.15a ) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{NS}_{\boldsymbol{p}}^{\boldsymbol{p}}:\quad \displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\,\langle \widehat{\unicode[STIX]{x1D70F}}_{ij}(\mathbf{0})\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}})\widehat{\mathscr{T}}_{jk}\rangle =K^{\text{I}}(\unicode[STIX]{x1D6FF}_{ik}\unicode[STIX]{x1D6FF}_{jl}+\unicode[STIX]{x1D6FF}_{il}\unicode[STIX]{x1D6FF}_{jk})+K^{\text{II}}\unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x1D6FF}_{kl},\end{eqnarray}$$
(4.15b ) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{NS}_{e}^{e}:\quad \displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\,\langle \widehat{J}_{i}^{E}(\mathbf{0})\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}})\widehat{\mathscr{J}}_{j}^{E}\rangle =K^{\text{III}}\unicode[STIX]{x1D6FF}_{ij},\end{eqnarray}$$
(4.15c ) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{B}_{\boldsymbol{p}}^{\boldsymbol{p}}:\quad \displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\,\langle \widehat{\unicode[STIX]{x1D70F}}_{ij}(\mathbf{0})\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}})\widehat{\mathscr{T}}_{kl}\rangle \overline{\unicode[STIX]{x1D70F}}=K^{\text{IV}}(\unicode[STIX]{x1D6FF}_{ik}\unicode[STIX]{x1D6FF}_{jl}+\unicode[STIX]{x1D6FF}_{il}\unicode[STIX]{x1D6FF}_{jk})+K^{\text{V}}\unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x1D6FF}_{kl},\end{eqnarray}$$
(4.15d ) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{B}_{e}^{e}:\quad \displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\,\langle \widehat{J}_{i}^{E}(\mathbf{0})\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}})\widehat{\mathscr{J}}_{j}^{E}\rangle \overline{\unicode[STIX]{x1D70F}}=K^{\text{VI}}\unicode[STIX]{x1D6FF}_{ij},\end{eqnarray}$$
(4.15e ) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{B}_{e}^{\boldsymbol{p}}:\quad \displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\!\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,\langle \widehat{\unicode[STIX]{x1D70F}}_{ij}(\mathbf{0})\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}})\widehat{J}_{k}^{E}(\overline{\boldsymbol{x}})\rangle \overline{x}_{l}=K_{1}(\unicode[STIX]{x1D6FF}_{ik}\unicode[STIX]{x1D6FF}_{jl}+\unicode[STIX]{x1D6FF}_{il}\unicode[STIX]{x1D6FF}_{jk})+K_{2}\unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x1D6FF}_{kl},\end{eqnarray}$$
(4.15f ) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{B}_{en}^{\boldsymbol{p}}:\quad \displaystyle \int \text{d}\overline{\unicode[STIX]{x1D70F}}\!\displaystyle \int _{0}^{\infty }\text{d}\overline{\boldsymbol{x}}\,\langle \widehat{\unicode[STIX]{x1D70F}}_{ij}(\mathbf{0})\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}})\widehat{\mathscr{J}}_{k}^{E}Ns_{+}\prime (\overline{\boldsymbol{x}})\rangle \overline{x}_{l}\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad \quad =K_{3}(\unicode[STIX]{x1D6FF}_{ik}\unicode[STIX]{x1D6FF}_{jl}+\unicode[STIX]{x1D6FF}_{il}\unicode[STIX]{x1D6FF}_{jk})+K_{4}\unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x1D6FF}_{kl},\end{eqnarray}$$
(4.15g ) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{B}_{ee}^{\boldsymbol{p}}:\quad \displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\!\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,\langle \widehat{\unicode[STIX]{x1D70F}}_{ij}(\mathbf{0})\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}})\widehat{\mathscr{J}}_{k}^{E}Es_{+}\prime (\overline{\boldsymbol{x}})\rangle \overline{x}_{l}=K_{5}(\unicode[STIX]{x1D6FF}_{ik}\unicode[STIX]{x1D6FF}_{jl}+\unicode[STIX]{x1D6FF}_{il}\unicode[STIX]{x1D6FF}_{jk})+K_{6}\unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x1D6FF}_{kl},\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
(4.15h ) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{B}_{\boldsymbol{p}\boldsymbol{p}}^{\boldsymbol{p}}:\quad \displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\!\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,\langle \widehat{\unicode[STIX]{x1D70F}}_{ij}(\mathbf{0})\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}})\widehat{\mathscr{T}}_{kl}Gs_{+}\prime _{m}(\overline{\boldsymbol{x}})\rangle \overline{x}_{n}\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad \quad =K_{7}\unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x1D6FF}_{kl}\unicode[STIX]{x1D6FF}_{mn}+K_{8}\unicode[STIX]{x1D6FF}_{ij}(\unicode[STIX]{x1D6FF}_{km}\unicode[STIX]{x1D6FF}_{ln}+\unicode[STIX]{x1D6FF}_{kn}\unicode[STIX]{x1D6FF}_{lm})\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad \qquad +\,K_{9}\unicode[STIX]{x1D6FF}_{kl}(\unicode[STIX]{x1D6FF}_{im}\unicode[STIX]{x1D6FF}_{jn}+\unicode[STIX]{x1D6FF}_{in}\unicode[STIX]{x1D6FF}_{jm})+K_{10}\unicode[STIX]{x1D6FF}_{mn}(\unicode[STIX]{x1D6FF}_{ik}\unicode[STIX]{x1D6FF}_{jl}+\unicode[STIX]{x1D6FF}_{il}\unicode[STIX]{x1D6FF}_{jk})\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad \qquad +\,K_{11}[\unicode[STIX]{x1D6FF}_{km}(\unicode[STIX]{x1D6FF}_{il}\unicode[STIX]{x1D6FF}_{jn}+\unicode[STIX]{x1D6FF}_{jl}\unicode[STIX]{x1D6FF}_{in})+\unicode[STIX]{x1D6FF}_{lm}(\unicode[STIX]{x1D6FF}_{ik}\unicode[STIX]{x1D6FF}_{jn}+\unicode[STIX]{x1D6FF}_{jk}\unicode[STIX]{x1D6FF}_{in})]\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad \qquad +\,K_{12}[\unicode[STIX]{x1D6FF}_{kn}(\unicode[STIX]{x1D6FF}_{il}\unicode[STIX]{x1D6FF}_{jm}+\unicode[STIX]{x1D6FF}_{jl}\unicode[STIX]{x1D6FF}_{im})+\unicode[STIX]{x1D6FF}_{ln}(\unicode[STIX]{x1D6FF}_{ik}\unicode[STIX]{x1D6FF}_{jm}+\unicode[STIX]{x1D6FF}_{jk}\unicode[STIX]{x1D6FF}_{im})],\end{eqnarray}$$
(4.15i ) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{B}_{e\boldsymbol{p}}^{e}:\quad \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\!\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,\langle \widehat{J}_{i}^{E}(\mathbf{0})\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}})\widehat{\mathscr{J}}_{j}^{E}Gs_{+}\prime _{k}(\overline{\boldsymbol{x}})\rangle \overline{x}_{l}\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad \quad =K_{13}\unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x1D6FF}_{kl}+K_{14}\unicode[STIX]{x1D6FF}_{ik}\unicode[STIX]{x1D6FF}_{jl}+K_{15}\unicode[STIX]{x1D6FF}_{il}\unicode[STIX]{x1D6FF}_{jk},\end{eqnarray}$$
(4.15j ) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{B}_{\boldsymbol{p}n}^{e}:\quad \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\!\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,\langle \widehat{J}_{i}^{E}(\mathbf{0})\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}})\widehat{\mathscr{T}}_{jk}Ns_{+}\prime (\overline{\boldsymbol{x}})\rangle \overline{x}_{l}\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad \quad =K_{16}(\unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x1D6FF}_{kl}+\unicode[STIX]{x1D6FF}_{ik}\unicode[STIX]{x1D6FF}_{jl})+K_{17}\unicode[STIX]{x1D6FF}_{il}\unicode[STIX]{x1D6FF}_{jk},\end{eqnarray}$$
(4.15k ) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{B}_{\boldsymbol{p}e}^{e}:\quad \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\!\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,\langle \widehat{J}_{i}^{E}(\mathbf{0})\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}})\widehat{\mathscr{T}}_{jk}Es_{+}\prime (\overline{\boldsymbol{x}})\rangle \overline{x}_{l}\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad \quad =K_{18}(\unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x1D6FF}_{kl}+\unicode[STIX]{x1D6FF}_{ik}\unicode[STIX]{x1D6FF}_{jl})+K_{19}\unicode[STIX]{x1D6FF}_{il}\unicode[STIX]{x1D6FF}_{jk},\end{eqnarray}$$
(4.15l ) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{B}_{-\text{P}(\boldsymbol{p}\boldsymbol{p}+en+ee)}^{\boldsymbol{p}}:\quad \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\!\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,\langle \widehat{\unicode[STIX]{x1D70F}}_{ij}(\mathbf{0})\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}})P_{k}(\overline{\boldsymbol{x}})\rangle \overline{x}_{l}\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad \quad =K_{20}(\unicode[STIX]{x1D6FF}_{ik}\unicode[STIX]{x1D6FF}_{jl}+\unicode[STIX]{x1D6FF}_{il}\unicode[STIX]{x1D6FF}_{jk})+K_{21}\unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x1D6FF}_{kl},\end{eqnarray}$$
(4.15m ) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{B}_{-\text{P}(\boldsymbol{p}n+\boldsymbol{p}e+e\boldsymbol{p})}^{e}:\quad \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\!\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,\langle \widehat{J}_{i}^{E}(\mathbf{0})\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}})N(\overline{\boldsymbol{x}})\rangle \overline{x}_{j}=K_{22}\unicode[STIX]{x1D6FF}_{ij},\end{eqnarray}$$
(4.15n ) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{B}_{-\text{P}(\boldsymbol{p}n+\boldsymbol{p}e+e\boldsymbol{p})}^{e}:\quad \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\!\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,\langle \widehat{J}_{i}^{E}(\mathbf{0})\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}})E(\overline{\boldsymbol{x}})\rangle \overline{x}_{j}=K_{23}\unicode[STIX]{x1D6FF}_{ij}.\end{eqnarray}$$

These expressions involve two-time correlation functions $C(t,ts_{+}\prime )$ with $m$ phase-space arguments associated with time  $t$ and $n$  phase-space arguments associated with time  $ts_{+}\prime$ ; I shall use the notation $C^{(m,n)}(t,ts_{+}\prime )$ . For weakly coupled systems, where the potential parts of the microscopic fluxes may be neglected, one requires only $m=1$ (in later discussion, $m>1$ will be required) and $n=1$ or $n=2$ . I shall assume stationary statistics, which implies that $C^{(m,n)}(t,ts_{+}\prime )$  depends only on the time difference $\unicode[STIX]{x1D70F}\doteq t-ts_{+}\prime$ ; only the one-sided functions $C_{+}^{(m;n)}(\unicode[STIX]{x1D70F})\doteq H(\unicode[STIX]{x1D70F})C^{(m,n)}(\unicode[STIX]{x1D70F})$ are required. What to do about spatial dependence is more complicated. As I remarked in § 1, if one is to take account of the convective amplification of fluctuations (Kent & Taylor Reference Kent and Taylor1969), it is not permissible to assume homogeneous statistics. However, even if one ignores that possibility, the implications of spatial homogeneity are non-trivial (and useful for later comparison with the work of Catto & Simakov). Homogeneity implies that $C^{(1,n)}$  is a function of $n$ spatial differences, which by convention I shall refer to $\boldsymbol{x}$ ( $\boldsymbol{x}=\mathbf{0}$ in the above formulas). This leads one to introduce $\unicode[STIX]{x1D746}s_{+}\prime \doteq \boldsymbol{x}-\boldsymbol{x}s_{+}\prime$ and $\unicode[STIX]{x1D746}s_{+}\prime \prime \doteq \boldsymbol{x}-\boldsymbol{x}s_{+}\prime \prime$ . Thus, for example,

(4.16a ) $$\begin{eqnarray}\displaystyle C_{+}^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ) & = & \displaystyle C_{+}^{(1;1)}(\boldsymbol{v},\unicode[STIX]{x1D70F};\unicode[STIX]{x1D746}s_{+}\prime ,\boldsymbol{v}s_{+}\prime )=\displaystyle \int \frac{\text{d}\boldsymbol{k}s_{+}\prime }{(2\unicode[STIX]{x03C0})^{3}}\,\text{e}^{\text{i}\boldsymbol{k}s_{+}\prime \boldsymbol{\cdot }\unicode[STIX]{x1D746}s_{+}\prime }\widehat{\text{C}}_{+;\boldsymbol{k}s_{+}\prime }^{(1;1)}(\boldsymbol{v},\unicode[STIX]{x1D70F};\boldsymbol{v}s_{+}\prime ),\end{eqnarray}$$
(4.16b ) $$\begin{eqnarray}\displaystyle C_{+}^{(1;2)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime ) & = & \displaystyle C_{+}^{(1;2)}(\boldsymbol{v}_{1},\unicode[STIX]{x1D70F};\unicode[STIX]{x1D746}s_{+}\prime ,\boldsymbol{v}s_{+}\prime ,\unicode[STIX]{x1D746}s_{+}\prime \prime ,\boldsymbol{v}s_{+}\prime \prime )\nonumber\\ \displaystyle & = & \displaystyle \displaystyle \int \frac{\text{d}\boldsymbol{k}s_{+}\prime }{(2\unicode[STIX]{x03C0})^{3}}\displaystyle \int \frac{\text{d}\boldsymbol{k}s_{+}\prime \prime }{(2\unicode[STIX]{x03C0})^{3}}\,\text{e}^{\text{i}\boldsymbol{k}s_{+}\prime \boldsymbol{\cdot }\unicode[STIX]{x1D746}s_{+}\prime +\text{i}\boldsymbol{k}s_{+}\prime \prime \boldsymbol{\cdot }\unicode[STIX]{x1D746}s_{+}\prime \prime }\widehat{\text{C}}_{+;\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\boldsymbol{v}_{1},\unicode[STIX]{x1D70F};\boldsymbol{v}s_{+}\prime ,\boldsymbol{v}s_{+}\prime \prime ),\qquad\end{eqnarray}$$
(4.16c ) $$\begin{eqnarray}\displaystyle C_{+}^{(2;1)}(\text{}\underline{1},\text{}\underline{2};\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ) & = & \displaystyle C_{+}^{(2;1)}(\unicode[STIX]{x1D746},\boldsymbol{v}_{1},\boldsymbol{v}_{2},\unicode[STIX]{x1D70F};\unicode[STIX]{x1D746}s_{+}\prime ,\boldsymbol{v}s_{+}\prime )\nonumber\\ \displaystyle & = & \displaystyle \displaystyle \int \frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\displaystyle \int \frac{\text{d}\boldsymbol{k}s_{+}\prime }{(2\unicode[STIX]{x03C0})^{3}}\,\text{e}^{\text{i}\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x1D746}+\text{i}\boldsymbol{k}s_{+}\prime \boldsymbol{\cdot }\unicode[STIX]{x1D746}s_{+}\prime }\widehat{\text{C}}_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime }^{(2;1)}(\boldsymbol{v}_{1},\boldsymbol{v}_{2},\unicode[STIX]{x1D70F};\boldsymbol{v}s_{+}\prime ).\end{eqnarray}$$
For weak coupling, formulas that involve the total amount of a flux (e.g. $\widehat{\pmb{\pmb{\mathscr{J}}}}^{E}$ ) involve the spatial integral over $\boldsymbol{x}-\boldsymbol{x}s_{+}\prime$ and thus require $\widehat{\text{C}}_{\boldsymbol{k}s_{+}\prime =\mathbf{0}}$ . Formulas that involve an $\overline{\boldsymbol{x}}$  weighting can be expressed in terms of a wavenumber derivative. Specifically, upon replacing  $\overline{\boldsymbol{x}}$ by  $\boldsymbol{x}s_{+}\prime \prime$ ,
(4.17a ) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \int \text{d}\boldsymbol{x}s_{+}\prime \,\text{d}\boldsymbol{x}s_{+}\prime \prime \,C_{+}^{(1;2)}(\boldsymbol{x}=\mathbf{0},\boldsymbol{v},\unicode[STIX]{x1D70F};\boldsymbol{x}s_{+}\prime ,\boldsymbol{v}s_{+}\prime ,\boldsymbol{x}s_{+}\prime \prime ,\boldsymbol{v}s_{+}\prime \prime )\boldsymbol{x}s_{+}\prime \prime \nonumber\\ \displaystyle & & \displaystyle \quad =-\displaystyle \int \text{d}\unicode[STIX]{x1D746}s_{+}\prime \,\text{d}\unicode[STIX]{x1D746}s_{+}\prime \prime \,C_{+}^{(1;2)}(\boldsymbol{v},\unicode[STIX]{x1D70F};\unicode[STIX]{x1D746}s_{+}\prime ,\boldsymbol{v}s_{+}\prime ,\unicode[STIX]{x1D746}s_{+}\prime \prime ,\boldsymbol{v}s_{+}\prime \prime )\unicode[STIX]{x1D746}s_{+}\prime \prime\end{eqnarray}$$
(4.17b ) $$\begin{eqnarray}\displaystyle & & \displaystyle \quad =\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}(-\text{i}\boldsymbol{k}s_{+}\prime \prime )}\widehat{\text{C}}_{+;\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\boldsymbol{v},\unicode[STIX]{x1D70F};\boldsymbol{v}s_{+}\prime ,\boldsymbol{v}s_{+}\prime \prime )|_{\boldsymbol{k}s_{+}\prime =\mathbf{0},\,\boldsymbol{k}s_{+}\prime \prime =\mathbf{0}}.\end{eqnarray}$$
A discussion of the evaluation of such correlation functions and their time integrals is given in § 5.

4.3 Exchange terms

Equation (3.41), together with the definition (3.39b ), provides a representation of first-order interspecies momentum and energy exchange in terms of two-time correlations. It is not immediately obvious that those formulas are consistent with the results already known to Braginskii. Therefore, as an example I work out in appendix E the hydrodynamic contribution to momentum exchange and demonstrate complete agreement with the analogous calculation in Part 1.

5 Theory of two-time correlation functions for weakly coupled plasmas

The formulas in § 4.2 express the transport coefficients in terms of various two-time correlation functions. Equations (4.15f )–(4.15k ) require three-point correlations in phase space, while the rest of equations (4.15) require merely two-point correlations. The exchange terms discussed in appendix  E require four-point correlations. Calculating such multipoint correlation functions for the general case of strong coupling is a formidable challenge even though only the low-frequency, long-wavelength behaviour is of interest; however, the task is relatively simple for weak coupling provided that one ignores issues with long-ranged collisional correlations and convective amplification of fluctuations (i.e. does calculations that only retain effects that lead to the Balescu–Lenard or Landau collision operators).

5.1 Heuristic physics of two-time correlations

Before proceeding to the details, I shall give a qualitative introduction that relies on the fact that cumulants are related to functional derivatives. I have already introduced this topic in § 1.3, where I pointed out that the two-time Green’s function of a linear equation is the functional derivative of the basic one-time field with respect to an external source  $\widehat{\unicode[STIX]{x1D702}}$ . The generalization to statistical theory is well known. For example, for continuous classical fields (no particle discreteness effects), Martin et al. (Reference Martin, Siggia and Rose1973) have shown that the $n$ -time cumulant $C_{n}$ of the random field  $\widetilde{\unicode[STIX]{x1D713}}(t)$ is the functional derivative of $C_{n-1}(t_{1},\ldots ,t_{n-1})$ with respect to a source field $\unicode[STIX]{x1D702}(t_{n})$ .Footnote 52 That is, a cumulant generating functional isFootnote 53

(5.1) $$\begin{eqnarray}Z[\unicode[STIX]{x1D702}]\doteq \ln \left\langle \exp \left(\int _{-\infty }^{\infty }\!\,\text{d}\overline{t}\,\widetilde{\unicode[STIX]{x1D713}}(\overline{t})\unicode[STIX]{x1D702}(\overline{t})\right)\right\rangle ,\end{eqnarray}$$

and one has

(5.2a,b ) $$\begin{eqnarray}C_{1}(t)=\frac{\unicode[STIX]{x1D6FF}Z[\unicode[STIX]{x1D702}]}{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D702}(t)},\quad C_{2}(t,ts_{+}\prime )=\frac{\unicode[STIX]{x1D6FF}^{2}Z[\unicode[STIX]{x1D702}]}{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D702}(t)\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D702}(ts_{+}\prime )}=\frac{\unicode[STIX]{x1D6FF}C_{1}(t)}{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D702}(ts_{+}\prime )},\quad \ldots ,\end{eqnarray}$$

with the physical cumulants following in the limit $\unicode[STIX]{x1D702}\rightarrow 0$ . For situations in which particle discreteness effects are important, a one-time cumulant generating functional was discussed by Dawson & Nakayama (Reference Dawson and Nakayama1967), and the generalization to two-time cumulants was given by Krommes (Reference Krommes1975) and Krommes & Oberman (Reference Krommes and Oberman1976a ). (These topics are discussed in detail in § 5.2.) Now suppose that the nonlinear kinetic equation holds (I set $\boldsymbol{B}^{\text{ext}}=\mathbf{0}$ for simplicity and assume a bilinear collision operator such as the Landau operator):

(5.3) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\,f+\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}f+(\pmb{\mathbb{E}}f)\boldsymbol{\cdot }\boldsymbol{\unicode[STIX]{x2202}}f+\text{C}[\,f,\overline{f}]=0.\end{eqnarray}$$

Here $\pmb{\mathbb{E}}$  is the electric-field operator defined for the case of homogeneous statistics by (1:2.38). Without worrying about details, which will be discussed later, functionally differentiate (5.3) to obtain (Krommes & Oberman Reference Krommes and Oberman1976a )

(5.4) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}C_{2}(t,ts_{+}\prime )+\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}C_{2}+(\pmb{\mathbb{E}}f)\boldsymbol{\cdot }\boldsymbol{\unicode[STIX]{x2202}}C+(\boldsymbol{\unicode[STIX]{x2202}}f)\boldsymbol{\cdot }\pmb{\mathbb{E}}C_{2}+\widehat{\text{C}}[\,f]C_{2}=0,\end{eqnarray}$$

where $\widehat{\text{C}}$  is the linearized collision operator. This shows that at long wavelengths the two-time function $C_{2}(\unicode[STIX]{x1D70F})$ decays on the collisional time scale. If one assumes homogeneous statistics, it is only a matter of filling in the details to integrate  $C_{2}$ according to formulas like (4.15a ) and obtain the same Navier–Stokes transport coefficients that were discussed in Part 1. More generally, solution of (5.4) for weakly inhomogeneous statistics will ultimately lead to first-order transport coefficients defined in terms of convectively amplified fluctuation spectra (Kent & Taylor Reference Kent and Taylor1969).

Upon differentiating (5.4), one obtains schematically

(5.5) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}C_{3}(t,ts_{+}\prime ,ts_{+}\prime \prime )}{\unicode[STIX]{x2202}t}+\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}C_{3}+\boldsymbol{E}\boldsymbol{\cdot }\boldsymbol{\unicode[STIX]{x2202}}C_{3}+\widehat{\text{C}}[\,f]C_{3}=-\frac{\unicode[STIX]{x1D6FF}\displaystyle \widehat{\text{C}}[\,f]}{\unicode[STIX]{x1D6FF}f}C_{2}C_{2}-2(\pmb{\mathbb{E}}C_{2})\boldsymbol{\cdot }\boldsymbol{\unicode[STIX]{x2202}}C_{2},\end{eqnarray}$$

after which one can set $t^{\prime \prime }=t^{\prime }$ . As we shall see later in more detail, the first term on the right-hand side is related to the action of the nonlinear collision operator acting on  $C_{2}$ , which then drives the triplet correlation function that determines the non-Gaussian Burnett transport effects. The significance of the last term is discussed in the paragraph before § G.3.1 on page 98.

5.2 The two-time cumulant hierarchy and correlation functions

I shall now discuss the formal derivation of the previous results. For conciseness, I shall continue to set $\boldsymbol{B}^{\text{ext}}=\mathbf{0}$ ; the way to add the Lorentz force to the final formulas will be clear. Most powerfully, one has available the renormalized theory of Rose (Reference Rose1979), which generalizes continuum statistical dynamics – for example, the formalism of Martin et al. (Reference Martin, Siggia and Rose1973) – to include particle discreteness. However, for a weakly coupled, near-equilibrium system it is more expeditious to proceed via a two-time generalization of the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy. The one- and two-time hierarchies were discussed in appendix C of Krommes (Reference Krommes1975) via a generating-functional approach; the following material is taken directly from that appendix; see also appendix A of Krommes & Oberman (Reference Krommes and Oberman1976a ). For one-time physics, define the generating function

(5.6) $$\begin{eqnarray}S[\unicode[STIX]{x1D702}]\doteq \langle \text{e}^{\widetilde{\unicode[STIX]{x1D6F7}}_{\unicode[STIX]{x1D702}}(t)}\rangle ,\end{eqnarray}$$

where

(5.7) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D6F7}}_{\unicode[STIX]{x1D702}}(t)\doteq \displaystyle \int \text{d}\overline{\boldsymbol{q}}\,\widetilde{f}(\overline{\boldsymbol{q}},t)\unicode[STIX]{x1D702}(\overline{\boldsymbol{q}})\end{eqnarray}$$

is a random functional of  $\unicode[STIX]{x1D702}(\boldsymbol{q})$ . Note that there is no time integral in this formula. The one-time distribution functions are defined in the thermodynamic limit as (Dawson & Nakayama Reference Dawson and Nakayama1967),

(5.8) $$\begin{eqnarray}f_{\unicode[STIX]{x1D702}}^{(s)}(\text{}\underline{1},\ldots ,\text{}\underline{s},t)=\text{D}_{s}\text{D}_{s-1}\ldots \text{D}_{1}\langle S\rangle ,\end{eqnarray}$$

where

(5.9) $$\begin{eqnarray}\text{D}_{s}\doteq \frac{\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D702}(\text{}\underline{s})}-\mathop{\sum }_{i<s}\overline{n}_{s}^{-1}\unicode[STIX]{x1D6FF}(\text{}\underline{i}-\text{}\underline{s})\end{eqnarray}$$

and an underline signifies that time should be omitted from the implied set of variables. The usual $s$ -body distributions are the $\unicode[STIX]{x1D702}\rightarrow 0$ limit of $f_{\unicode[STIX]{x1D702}}^{(s)}$ . Similarly, one-time cumulants (denoted here by an overline) follow by differentiating $\ln \langle S\rangle$ :

(5.10) $$\begin{eqnarray}\overline{f}_{\unicode[STIX]{x1D702}}^{(s)}=\text{D}_{s}\ldots \text{D}_{1}\ln \langle S\rangle .\end{eqnarray}$$

The properties of the logarithm lead to the well-known cluster expansion

(5.11) $$\begin{eqnarray}f_{\unicode[STIX]{x1D702}}^{(s)}(\text{}\underline{1},\ldots ,\text{}\underline{s})=\mathop{\sum }_{n=1}^{s}\mathop{\sum }_{P}\overline{f}_{\unicode[STIX]{x1D702}}\boldsymbol{(}N(P_{1})\boldsymbol{)}(P_{1})\overline{f}_{\unicode[STIX]{x1D702}}\boldsymbol{(}N(P_{2})\boldsymbol{)}(P_{2})\ldots \overline{f}_{\unicode[STIX]{x1D702}}\boldsymbol{(}N(P_{n})\boldsymbol{)}(P_{n}),\end{eqnarray}$$

where $P_{i}$  is a subset of $\{\text{}\underline{1},\ldots ,\text{}\underline{s}\}$ , $N(P_{i})$  is the number of members of  $P_{i}$ , $\sum _{i=1}^{n}\!N(P_{i})=s$ and $\sum _{P}$ is the sum over all distinct and disjoint subsets of $\{\text{}\underline{1},\,\text{}\underline{2},\ldots ,\text{}\underline{s}\}$ :

(5.12) $$\begin{eqnarray}\mathop{\bigcup }_{i=1}^{n}P_{i}=\{\text{}\underline{1},\,\text{}\underline{2},\ldots ,\text{}\underline{s}\}\equiv \{\text{}\underline{s}\}.\end{eqnarray}$$

To obtain time-evolution equations, consider the time derivative of the generating functional:

(5.13a ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}S}{\unicode[STIX]{x2202}t} & = & \displaystyle \left\langle \frac{\text{d}\widetilde{\unicode[STIX]{x1D6F7}}_{\unicode[STIX]{x1D702}}}{\text{d}t}\,\text{e}^{\widetilde{\unicode[STIX]{x1D6F7}}_{\unicode[STIX]{x1D702}}}\right\rangle\end{eqnarray}$$
(5.13b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle \left\langle \left(\displaystyle \int \text{d}\widehat{\boldsymbol{q}}\,\frac{\unicode[STIX]{x2202}\widetilde{f}(\widehat{\boldsymbol{q}},t)}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D702}(\widehat{\boldsymbol{q}})\right)\text{e}^{\widetilde{\unicode[STIX]{x1D6F7}}_{\unicode[STIX]{x1D702}}}\right\rangle\end{eqnarray}$$
(5.13c ) $$\begin{eqnarray}\displaystyle & = & \displaystyle -\displaystyle \int \text{d}\widehat{\boldsymbol{q}}\,\left\langle \left[\widehat{\boldsymbol{v}}\boldsymbol{\cdot }\widehat{\unicode[STIX]{x1D735}}\widetilde{f}(\widehat{\boldsymbol{q}},t)+\left(\frac{q}{m}\right)_{\widehat{s}}\pmb{\mathbb{E}}\widetilde{f}(\widehat{\boldsymbol{q}},t)\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}\widetilde{f}}{\unicode[STIX]{x2202}\widehat{\boldsymbol{v}}}\right]\text{e}^{\widetilde{\unicode[STIX]{x1D6F7}}_{\unicode[STIX]{x1D702}}}\right\rangle \unicode[STIX]{x1D702}(\widehat{\boldsymbol{q}}).\end{eqnarray}$$
Upon functionally differentiating this equation, one arrives at the usual BBGKY hierarchy
(5.14) $$\begin{eqnarray}\displaystyle 0 & = & \displaystyle \unicode[STIX]{x2202}_{t}f_{\unicode[STIX]{x1D702}}^{(s)}+\mathop{\sum }_{i\in \{s\}}\boldsymbol{v}_{i}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{i}f_{\unicode[STIX]{x1D702}}^{(s)}+\mathop{\sum }_{i\neq j\in \{s\}}\unicode[STIX]{x1D750}_{ij}\boldsymbol{\cdot }(q_{j}\boldsymbol{\unicode[STIX]{x2202}}_{i})f_{\unicode[STIX]{x1D702}}^{(s)}+\mathop{\sum }_{i\in \{s\}}\boldsymbol{\unicode[STIX]{x2202}}_{i}\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{i},\text{}\underline{s+1})f_{\unicode[STIX]{x1D702}}^{(s+1)}\nonumber\\ \displaystyle & & \displaystyle +\,O(\unicode[STIX]{x1D702}).\end{eqnarray}$$

A similar result holds for the cumulant hierarchy. Upon indicating one-time cumulants with overlines, one has

(5.15) $$\begin{eqnarray}\displaystyle 0 & = & \displaystyle \frac{\text{d}^{(s)}\overline{f}_{\unicode[STIX]{x1D702}}^{(s)}}{\text{d}t}+\mathop{\sum }_{i\neq j\in \{s\}}\unicode[STIX]{x1D750}_{ij}\boldsymbol{\cdot }(q_{j}\boldsymbol{\unicode[STIX]{x2202}}_{i})\left(\mathop{\sum }_{n=1}^{s-1}\mathop{\sum }_{P}\overline{f}_{\unicode[STIX]{x1D702}}^{(s-n)}(\ldots ,\text{}\underline{i},\ldots )\overline{f}_{\unicode[STIX]{x1D702}}^{(n)}(\ldots \,\text{}\underline{j}\,\ldots )\right.\nonumber\\ \displaystyle & & \displaystyle \quad \qquad \qquad \qquad \qquad \qquad +\left.\overline{f}_{\unicode[STIX]{x1D702}}^{(s)}(\text{}\underline{1},\ldots ,\text{}\underline{s})\vphantom{\mathop{\sum }_{n=1}^{s-1}}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{i\in \{s\}}\boldsymbol{\unicode[STIX]{x2202}}_{i}\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{i},\text{}\underline{s+1})\left(\mathop{\sum }_{n=1}^{s}\mathop{\sum }_{P}\overline{f}_{\unicode[STIX]{x1D702}}^{(s+1-n)}(\ldots ,\text{}\underline{i},\ldots )\overline{f}_{\unicode[STIX]{x1D702}}^{(n)}(\ldots ,\text{}\underline{s+1},\ldots )\right.\nonumber\\ \displaystyle & & \displaystyle \quad \qquad \qquad \qquad \qquad +\left.\overline{f}_{\unicode[STIX]{x1D702}}^{(s+1)}(\text{}\underline{1},\ldots ,\text{}\underline{s+1})\vphantom{\mathop{\sum }_{n=1}^{s-1}}\right)\nonumber\\ \displaystyle & & \displaystyle +\,O(\unicode[STIX]{x1D702}).\end{eqnarray}$$

Again, $\sum _{P}$ means to sum over all distinct permutations of disjoint subsets.

I shall use the standard notation $\overline{f}^{(1)}\equiv f$ , $\overline{f}^{(2)}\equiv g$ , $\overline{f}^{(3)}\equiv h$ and $\overline{f}^{(4)}\equiv k$ . Upon noting that $\pmb{\mathbb{E}}(\text{}\underline{1},\overline{\text{}\underline{1}})f(\overline{\text{}\underline{1}},t)=\boldsymbol{E}(1)$ and defining the Landau operator Footnote 54 for particle 1 as

(5.16) $$\begin{eqnarray}\text{i}\text{L}_{1}\equiv \text{i}\text{L}(1,\overline{1})\doteq \boldsymbol{v}_{1}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{1}\unicode[STIX]{x1D6FF}(\text{}\underline{1}-\overline{\text{}\underline{1}})+\boldsymbol{E}(1)\boldsymbol{\cdot }\boldsymbol{\unicode[STIX]{x2202}}_{1}\unicode[STIX]{x1D6FF}(\text{}\underline{1}-\overline{\text{}\underline{1}})+\boldsymbol{\unicode[STIX]{x2202}}_{1}f\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{1},\text{}\underline{\overline{1}})\end{eqnarray}$$

(the last term is responsible for crucial polarization or self-consistent response effects), one can write the first three members of the cumulant hierarchy (now dropping the $\unicode[STIX]{x1D702}$  subscript) as

(5.17a ) $$\begin{eqnarray}\displaystyle 0 & = & \displaystyle \unicode[STIX]{x2202}_{t}\,f(\text{}\underline{1},t)+\boldsymbol{v}_{1}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{1}+\boldsymbol{E}\boldsymbol{\cdot }\boldsymbol{\unicode[STIX]{x2202}}_{1}f+\boldsymbol{\unicode[STIX]{x2202}}_{1}\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{1},\overline{\text{}\underline{2}})g(\text{}\underline{1},\overline{\text{}\underline{2}},t),\end{eqnarray}$$
(5.17b ) $$\begin{eqnarray}\displaystyle 0 & = & \displaystyle (\unicode[STIX]{x2202}_{t}+\text{i}\text{L}_{1}+\text{i}\text{L}_{2})g(\text{}\underline{1},\text{}\underline{2},t)+\unicode[STIX]{x1D750}_{12}\boldsymbol{\cdot }(q_{2}\boldsymbol{\unicode[STIX]{x2202}}_{1}-q_{1}\boldsymbol{\unicode[STIX]{x2202}}_{2})[f(\text{}\underline{1},t)f(\text{}\underline{2},t)+g(\text{}\underline{1},\text{}\underline{2},t)]\nonumber\\ \displaystyle & & \displaystyle +\,[\boldsymbol{\unicode[STIX]{x2202}}_{1}\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{1},\overline{\text{}\underline{3}})h(\text{}\underline{1},\text{}\underline{2},\overline{\text{}\underline{3}},t)+(1\leftrightarrow 2)],\end{eqnarray}$$
(5.17c ) $$\begin{eqnarray}\displaystyle 0 & = & \displaystyle (\unicode[STIX]{x2202}_{t}+\text{i}\text{L}_{1}+\text{i}\text{L}_{2}+\text{i}\text{L}_{3})h(\text{}\underline{1},\text{}\underline{2},\text{}\underline{3},t)\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D750}_{12}\boldsymbol{\cdot }(q_{2}\boldsymbol{\unicode[STIX]{x2202}}_{1})[g(\text{}\underline{1},\text{}\underline{3})f(\text{}\underline{2})+f(\text{}\underline{1})g(\text{}\underline{2},\text{}\underline{3})+h(\text{}\underline{1},\text{}\underline{2},\text{}\underline{3})]\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D750}_{13}\boldsymbol{\cdot }(q_{3}\boldsymbol{\unicode[STIX]{x2202}}_{1})[g(\text{}\underline{1},\text{}\underline{2})f(\text{}\underline{3})+f(\text{}\underline{1})g(\text{}\underline{3},\text{}\underline{2})+h(\text{}\underline{1},\text{}\underline{2},\text{}\underline{3})]\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D750}_{21}\boldsymbol{\cdot }(q_{1}\boldsymbol{\unicode[STIX]{x2202}}_{2})[g(\text{}\underline{2},\text{}\underline{3})f(\text{}\underline{1})+f(\text{}\underline{2})g(\text{}\underline{1},\text{}\underline{3})+h(\text{}\underline{1},\text{}\underline{2},\text{}\underline{3})]\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D750}_{23}\boldsymbol{\cdot }(q_{3}\boldsymbol{\unicode[STIX]{x2202}}_{2})[g(\text{}\underline{2},\text{}\underline{1})f(\text{}\underline{3})+f(\text{}\underline{2})g(\text{}\underline{3},\text{}\underline{1})+h(\text{}\underline{1},\text{}\underline{2},\text{}\underline{3})]\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D750}_{31}\boldsymbol{\cdot }(q_{1}\boldsymbol{\unicode[STIX]{x2202}}_{3})[g(\text{}\underline{3},\text{}\underline{2})f(\text{}\underline{1})+f(\text{}\underline{3})g(\text{}\underline{1},\text{}\underline{2})+h(\text{}\underline{1},\text{}\underline{2},\text{}\underline{3})]\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D750}_{32}\boldsymbol{\cdot }(q_{2}\boldsymbol{\unicode[STIX]{x2202}}_{3})[g(\text{}\underline{3},\text{}\underline{1})f(\text{}\underline{2})+f(\text{}\underline{3})g(\text{}\underline{1},\text{}\underline{2})+h(\text{}\underline{1},\text{}\underline{2},\text{}\underline{3})]\nonumber\\ \displaystyle & & \displaystyle +\,\boldsymbol{\unicode[STIX]{x2202}}_{1}\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{1},\text{}\underline{4})[g(1,2)g(3,4)+g(1,3)g(2,4)+k(\text{}\underline{1},\ldots ,\,\text{}\underline{4})]\nonumber\\ \displaystyle & & \displaystyle +\,\boldsymbol{\unicode[STIX]{x2202}}_{2}\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{2},\text{}\underline{4})[g(2,1)g(3,4)+g(2,3)g(1,4)+k(\text{}\underline{1},\ldots ,\,\text{}\underline{4})]\nonumber\\ \displaystyle & & \displaystyle +\,\boldsymbol{\unicode[STIX]{x2202}}_{3}\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{3},\text{}\underline{4})[g(3,1)g(2,4)+g(3,2)g(1,4)+k(\text{}\underline{1},\ldots ,\,\text{}\underline{4})].\end{eqnarray}$$
In the above, the terms in  $\unicode[STIX]{x1D750}_{ij}$ are related to particle noise; see Rose (Reference Rose1979 and references therein) for discussion of this concept. If one retains terms only through  $O(\unicode[STIX]{x1D716}_{\text{p}})$ , the terms in  $h$ and  $k$ can be ignored.

5.3 Two-time, two phase-space point correlations

Now consider the extension of these results to two times. Introduce the extended generating functionalFootnote 55

(5.18) $$\begin{eqnarray}S_{2}[\unicode[STIX]{x1D702},\unicode[STIX]{x1D702}s_{+}\prime ]\doteq \left\langle \exp \left(\displaystyle \int \text{d}\overline{\boldsymbol{q}}\,\widetilde{f}(\overline{\boldsymbol{q}},t)\unicode[STIX]{x1D702}(\overline{\boldsymbol{q}})+\displaystyle \int \text{d}\overline{\boldsymbol{q}}s_{+}\prime \,\widetilde{f}(\overline{\boldsymbol{q}}s_{+}\prime ,ts_{+}\prime )\unicode[STIX]{x1D702}s_{+}\prime (\overline{\boldsymbol{q}}s_{+}\prime )\right)\right\rangle ,\end{eqnarray}$$

which involves two independent functions  $\unicode[STIX]{x1D702}(\boldsymbol{q})$ and  $\unicode[STIX]{x1D702}s_{+}\prime (\boldsymbol{q})$ . Generalized one-time cumulants are naturally defined from (5.10) by replacing  $S$ by  $S_{2}$ , and two-time cumulants follow as

(5.19) $$\begin{eqnarray}C^{(s,1)}(\text{}\underline{1},\ldots ,\,\text{}\underline{s},t,\text{}\underline{1}s_{+}\prime ,ts_{+}\prime )\doteq \frac{\unicode[STIX]{x1D6FF}\overline{f}^{(s)}(\text{}\underline{1},\ldots ,\,\text{}\underline{s},t)}{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D702}s_{+}\prime (\text{}\underline{1}s_{+}\prime )}.\end{eqnarray}$$

The superscript pair $(s,ss_{+}\prime )$ indicates the number of arguments associated with times  $t$ (namely  $s$ ) and  $ts_{+}\prime$ (namely  $ss_{+}\prime$ ). This notation is redundant when the argument list is displayed, but it fosters readability. Importantly, $C^{(1,1)}(1,1s_{+}\prime )=\langle \unicode[STIX]{x1D6FF}\widetilde{f}(1)\unicode[STIX]{x1D6FF}\widetilde{f}(1s_{+}\prime )\rangle \equiv C(1,1s_{+}\prime )$ – this is the fundamental two-time Klimontovich correlation function. The key result is that the $C^{(s,1)}$ correlation functions obey the linearization of the one-time BBGKY cumulant hierarchy – for example, in the $\unicode[STIX]{x2202}_{t}g$ equation replace $g(\text{}\underline{1},\text{}\underline{2},t)\rightarrow g(\text{}\underline{1},\text{}\underline{2},t)+\unicode[STIX]{x1D716}\,C^{(2,1)}(\text{}\underline{1},\text{}\underline{2},t,\text{}\underline{1}s_{+}\prime ,ts_{+}\prime )$ and collect the terms of first order in  $\unicode[STIX]{x1D716}$ to find $\unicode[STIX]{x2202}_{t}C^{(2,1)}=\cdots \,$ . This follows immediately from the definition of  $C^{(s,1)}$ as a functional derivative. Thus, the functional derivative of (5.17a ) is

(5.20) $$\begin{eqnarray}0=\unicode[STIX]{x2202}_{t}C^{(1,1)}(1,1s_{+}\prime )+\text{i}\text{L}_{1}[\,f]C^{(1,1)}+\boldsymbol{\unicode[STIX]{x2202}}_{1}\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{1},\overline{\text{}\underline{2}})C^{(2,1)}(\text{}\underline{1},\overline{\text{}\underline{2}},t,1s_{+}\prime ),\end{eqnarray}$$

with $C^{(1,1)}\equiv C$ . In deriving this result, it was crucial to not assume prematurely that the mean field  $\boldsymbol{E}$ vanishes; indeed, $\boldsymbol{E}\neq \mathbf{0}$ when $\unicode[STIX]{x1D702}\neq 0$ and the functional derivative of $\boldsymbol{E}[\,f]=\pmb{\mathbb{E}}(\text{}\underline{1},\overline{\text{}\underline{1}})f(\overline{\text{}\underline{1}},t)$ with respect to  $\unicode[STIX]{x1D702}s_{+}\prime$ is $\pmb{\mathbb{E}}(\text{}\underline{1},\overline{\text{}\underline{1}})C^{(1,1)}(\overline{\text{}\underline{1}},t,1s_{+}\prime )$ , which produces the last, self-consistent response or polarization term in the Landau operator (5.16). Only after performing all functional derivatives and setting $\unicode[STIX]{x1D702}=0$ may one assert that $\boldsymbol{E}=\mathbf{0}$ .

Equation (5.20) is not closed, as it involves the unknown function $C^{(2,1)}$ . It is clear that the closure problem cannot be solved merely by performing further functional differentiations. Instead, at some point one needs to express $C^{(n+1,1)}$ in terms of $\{C^{(m,1)}\mid m\leqslant n\}$ . When this is done for $n=1$ , the formal equation that results is called the Dyson equation. Martin et al. (Reference Martin, Siggia and Rose1973) showed how to do this for continuous classical fields,Footnote 56 and Rose (Reference Rose1979) provided an elegant generalization that handles particle discreteness as well. Rose’s work is very important, and his equations could be used as the basis for the subsequent discussion. Instead, I shall continue with an analysis of the two-time hierarchy, which for weak coupling is somewhat more transparent and leads rather directly to an approximate closure. Thus, the present discussion adds additional perspective to Rose’s general results.

Note that near thermal equilibrium one has $C^{(s,ss_{+}\prime )}=O(\unicode[STIX]{x1D716}_{\text{p}}^{s+ss_{+}\prime -1})$ . In particular, $C^{(1,1)}=O(\unicode[STIX]{x1D716}_{\text{p}})$ ; to find collisional corrections, one needs to work only to $O(\unicode[STIX]{x1D716}_{\text{p}}^{2})$ . To find a representation for  $C^{(2,1)}$ , consider its equation, which follows from the linearization of (5.17b ):

(5.21) $$\begin{eqnarray}\displaystyle & & \displaystyle (\unicode[STIX]{x2202}_{t}+\text{i}\text{L}_{1}[\,f]+\text{i}\text{L}_{2}[\,f])C^{(2,1)}(\text{}\underline{1},\text{}\underline{2},t,1s_{+}\prime )\nonumber\\ \displaystyle & & \displaystyle \quad =-\unicode[STIX]{x1D750}_{12}\boldsymbol{\cdot }(q_{2}\boldsymbol{\unicode[STIX]{x2202}}_{1}-q_{1}\boldsymbol{\unicode[STIX]{x2202}}_{2})[C^{(1,1)}(1,1s_{+}\prime )f(2)+f(1)C^{(1,1)}(2,1s_{+}\prime )]\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\boldsymbol{\unicode[STIX]{x2202}}_{1}g(\text{}\underline{1},\text{}\underline{2},t)\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{1},\overline{\text{}\underline{1}})C^{(1,1)}(\overline{\text{}\underline{1}},t,1s_{+}\prime )-\boldsymbol{\unicode[STIX]{x2202}}_{2}g(\text{}\underline{1},\text{}\underline{2},t)\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{2},\overline{\text{}\underline{2}})C^{(1,1)}(\overline{\text{}\underline{2}},t,1s_{+}\prime )\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\boldsymbol{\unicode[STIX]{x2202}}_{1}C^{(1;1)}(1;1s_{+}\prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{1},\overline{\text{}\underline{1}})g(\overline{\text{}\underline{1}},\text{}\underline{2},t)-\boldsymbol{\unicode[STIX]{x2202}}_{2}C^{(1;1)}(2;1s_{+}\prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{2},\overline{\text{}\underline{2}})g(\text{}\underline{1},\overline{\text{}\underline{2}},t)+O(\unicode[STIX]{x1D716}_{\text{p}}^{3}).\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Notation such as  $f(2)$ means $f(\text{}\underline{2},t)$ (i.e. $t_{2}=t$ ). The second and third lines after the equals sign in this equation arise from the linearization of the second and third terms in (5.16).

Because the integrals that determine the transport coefficients are integrated in  $\overline{\unicode[STIX]{x1D70F}}$ from 0 to  $\infty$ , we are interested only in the correlations for $t\geqslant ts_{+}\prime$ . Thus, consider the one-sided version of (5.21):

(5.22) $$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}+\text{i}\text{L}_{1}[\,f]+\text{i}\text{L}_{2}[\,f])C_{+}^{(2;1)}(\text{}\underline{1},\text{}\underline{2},t;1s_{+}\prime )=\unicode[STIX]{x1D6FF}(t-ts_{+}\prime )C^{(0,3)}(\text{}\underline{1},\text{}\underline{2},\text{}\underline{1}s_{+}\prime ,ts_{+}\prime )+s_{+}^{(2;1)}(\text{}\underline{1},\text{}\underline{2},t;1s_{+}\prime ),\end{eqnarray}$$

where

(5.23) $$\begin{eqnarray}\displaystyle & & \displaystyle s_{+}^{(2;1)}(\text{}\underline{1},\text{}\underline{2},t;1s_{+}\prime )\doteq -\unicode[STIX]{x1D750}_{12}\boldsymbol{\cdot }(q_{2}\boldsymbol{\unicode[STIX]{x2202}}_{1}-q_{1}\boldsymbol{\unicode[STIX]{x2202}}_{2})[C_{+}^{(1;1)}(1;1s_{+}\prime )f(2)+f(1)C_{+}^{(1;1)}(2;1s_{+}\prime )]\nonumber\\ \displaystyle & & \displaystyle \quad -\,\boldsymbol{\unicode[STIX]{x2202}}_{1}g(\text{}\underline{1},\text{}\underline{2},t)\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{1},\overline{\text{}\underline{1}})C_{+}^{(1;1)}(\overline{\text{}\underline{1}},t,1s_{+}\prime )-\boldsymbol{\unicode[STIX]{x2202}}_{2}g(\text{}\underline{1},\text{}\underline{2},t)\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{2},\overline{\text{}\underline{2}})C_{+}^{(1;1)}(\overline{\text{}\underline{2}},t,1s_{+}\prime )\nonumber\\ \displaystyle & & \displaystyle \quad -\,\boldsymbol{\unicode[STIX]{x2202}}_{1}C_{+}^{(1;1)}(\text{}\underline{1},t;\text{}\underline{1}s_{+}\prime ,ts_{+}\prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{1},\overline{\text{}\underline{1}})g(\overline{\text{}\underline{1}},\text{}\underline{2},t)-\boldsymbol{\unicode[STIX]{x2202}}_{2}C_{+}^{(1;1)}(\text{}\underline{2},t;\text{}\underline{1}s_{+}\prime ,ts_{+}\prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{2},\overline{\text{}\underline{2}})g(\text{}\underline{1},\overline{\text{}\underline{2}},t).\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(The semicolon is used to indicate one-sided functions.) Note that the equal-time function $C^{(0,3)}(ts_{+}\prime )$ has entered as an initial condition. Let the causal Green’s function for the linearized Vlasov equation obey

(5.24) $$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}+\text{i}\text{L}_{1})R(1;1s_{+}\prime )=\unicode[STIX]{x1D6FF}(1-1s_{+}\prime ),\end{eqnarray}$$

and represent the solution as

(5.25) $$\begin{eqnarray}R(1;1s_{+}\prime )=H(t-ts_{+}\prime )\unicode[STIX]{x1D6EF}(1,1s_{+}\prime ).\end{eqnarray}$$

Then the solution of (5.23) is

(5.26) $$\begin{eqnarray}\displaystyle C_{+}^{(2;1)}(\text{}\underline{1},\text{}\underline{2},t,1s_{+}\prime ) & = & \displaystyle H(t-ts_{+}\prime )\unicode[STIX]{x1D6EF}(\text{}\underline{1},t,\text{}\underline{\overline{1}},ts_{+}\prime )\unicode[STIX]{x1D6EF}(\text{}\underline{2},t,\text{}\underline{\overline{2}},ts_{+}\prime )C^{(0,3)}(\text{}\underline{\overline{1}},\text{}\underline{\overline{2}};1s_{+}\prime )\nonumber\\ \displaystyle & & \displaystyle \quad -\,\displaystyle \int _{ts_{+}\prime }^{t}\!\text{d}\overline{t}\,\unicode[STIX]{x1D6EF}(\text{}\underline{1},t,\text{}\underline{\overline{1}},\overline{t})\unicode[STIX]{x1D6EF}(\text{}\underline{2},t,\text{}\underline{\overline{2}},\overline{t})s_{+}^{(2;1)}(\overline{\text{}\underline{1}},\overline{\text{}\underline{2}},\overline{t};1s_{+}\prime ).\end{eqnarray}$$

If the first line of (5.23) were inserted into (5.26), the last line of (5.26) would have the same form as the solution for the pair correlation function  $g$ in standard Balescu–Lenard theory (reviewed in § G.1) except that the product $f(\text{}\underline{1},\overline{t})f(\text{}\underline{2},\overline{t})$ that appears in the classical derivation is replaced here by the $Cf$ terms in the square brackets in the first line of (5.23). Those terms emerge as the $O(\unicode[STIX]{x1D716})$ terms when $f$  is replaced by $f+\unicode[STIX]{x1D716}\,C_{+}^{(1;1)}$ . This suggests that the $s_{+}^{(2;1)}$ -driven contribution of  $C_{+}^{(2;1)}$ to (5.20) may be related to the linearized collision operator. However, this conclusion is not immediate because in the standard derivation of the Balescu–Lenard operator cancellations occur between the form of  $\unicode[STIX]{x1D6EF}$ , which contains an  $f$ , and the terms on which the $q_{2}\boldsymbol{\unicode[STIX]{x2202}}_{1}-q_{1}\boldsymbol{\unicode[STIX]{x2202}}_{2}$ operates; it is not immediately clear whether those cancellations still occur when it is $C_{+}^{(1;1)}$ that is differentiated. Furthermore, a formal linearization of the Balescu–Lenard operator ought to contain terms involving fluctuations in the strength of the dielectric shielding. Thus, one must do a serious calculation.

In the general case, the solution for the response function  $\unicode[STIX]{x1D6EF}$ of a system possessing weak background gradients will lead to the convective-amplification effect discussed by Kent & Taylor (Reference Kent and Taylor1969) and to the enhanced fluctuation spectrum that follows from the generalized test-particle superposition principle that those authors employ. From formulas such as (5.26), one could derive generalizations of the Balescu–Lenard operator. However, for later comparison with the work of Catto & Simakov, it is useful to ignore this effect, assume homogeneous statistics, and work with spatial Fourier transforms. With the conventions used in (4.16), the Fourier transform of (5.20) isFootnote 57

(5.27) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}C_{+;\boldsymbol{k}s_{+}\prime }^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )+\text{i}\text{L}_{\boldsymbol{k}s_{+}\prime }(\text{}\underline{1};\overline{\text{}\underline{1}})C_{+;\boldsymbol{k}s_{+}\prime }^{(1;1)}(\overline{\text{}\underline{1}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})C_{\boldsymbol{k}s_{+}\prime }^{(0,2)}(\text{}\underline{1},\text{}\underline{1}s_{+}\prime )-\boldsymbol{\unicode[STIX]{x2202}}\boldsymbol{\cdot }\displaystyle \int \frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\,\pmb{\mathbb{E}}_{\boldsymbol{k}}(\overline{\text{}\underline{2}})C_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime }^{(2;1)}(\text{}\underline{1},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ),\end{eqnarray}$$

and the Fourier transform of (5.23) is

(5.28) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}C_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime }^{(2;1)}(\text{}\underline{1},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )+\text{i}\text{L}_{\boldsymbol{k}+\boldsymbol{k}s_{+}\prime }(\text{}\underline{1},\overline{\text{}\underline{1}})C_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime }^{(2;1)}(\overline{\text{}\underline{1}},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )+\text{i}\text{L}_{-\boldsymbol{k}}(\text{}\underline{1},\overline{\text{}\underline{1}})C_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime }^{(2;1)}(\text{}\underline{1},\overline{\text{}\underline{2}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})C_{\boldsymbol{k},\boldsymbol{k}s_{+}\prime }^{(0,3)}(\text{}\underline{1},\text{}\underline{2},\text{}\underline{1}s_{+}\prime )+s_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime }^{(2;1)}(\text{}\underline{1},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ),\end{eqnarray}$$

where

(5.29) $$\begin{eqnarray}\displaystyle & & \displaystyle s_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime }^{(2;1)}(\text{}\underline{1},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\nonumber\\ \displaystyle & & \displaystyle \quad \doteq -(q_{2}\boldsymbol{\unicode[STIX]{x2202}}_{1}-q_{1}\boldsymbol{\unicode[STIX]{x2202}}_{2})\boldsymbol{\cdot }[\unicode[STIX]{x1D750}_{\boldsymbol{k}}C_{+;\boldsymbol{k}s_{+}\prime }^{(1;1)}(\text{}\underline{1};\text{}\underline{1}s_{+}\prime )f(2)+\unicode[STIX]{x1D716}_{\boldsymbol{k}+\boldsymbol{k}s_{+}\prime }f(1)C_{+;\boldsymbol{k}s_{+}\prime }^{(2;1)}(\text{}\underline{2};\text{}\underline{1}s_{+}\prime )]\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\boldsymbol{\unicode[STIX]{x2202}}_{1}g_{\boldsymbol{k}}(\text{}\underline{1},\text{}\underline{2},t)\boldsymbol{\cdot }\pmb{\mathbb{E}}_{\boldsymbol{k}s_{+}\prime }(\overline{\text{}\underline{1}})C_{+;\boldsymbol{k}s_{+}\prime }^{(1;1)}(\overline{\text{}\underline{1}},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )-\boldsymbol{\unicode[STIX]{x2202}}_{2}g_{\boldsymbol{k}+\boldsymbol{k}s_{+}\prime }(\text{}\underline{1},\text{}\underline{2},t)\boldsymbol{\cdot }\pmb{\mathbb{E}}_{\boldsymbol{k}s_{+}\prime }(\overline{\text{}\underline{2}})C_{+;\boldsymbol{k}s_{+}\prime }^{(1;1)}(\text{}\underline{1},\overline{\text{}\underline{2}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\boldsymbol{\unicode[STIX]{x2202}}_{1}C_{+;\boldsymbol{k}s_{+}\prime }^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}_{\boldsymbol{k}}(\overline{\text{}\underline{1}})g_{\boldsymbol{k}}(\overline{\text{}\underline{1}},\text{}\underline{2},t)-\boldsymbol{\unicode[STIX]{x2202}}_{2}C_{+;\boldsymbol{k}s_{+}\prime }^{(1;1)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\boldsymbol{\cdot }[\pmb{\mathbb{E}}_{\boldsymbol{k}+\boldsymbol{k}s_{+}\prime }(\overline{\text{}\underline{1}})g_{\boldsymbol{k}+\boldsymbol{k}s_{+}\prime }(\overline{\text{}\underline{1}},\text{}\underline{1},t)]^{\ast }.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

It was shown in § 3.2.1 that the two-time, two phase-space point correlation functions required for many of the transport coefficients can be written in terms of weighted velocity integrals of the basic Klimontovich correlation function  $C_{+;\mathbf{0}}^{(1;1)}(\unicode[STIX]{x1D70F})\equiv C_{+;\boldsymbol{k}s_{+}\prime =\mathbf{0}}^{(1;1)}(\unicode[STIX]{x1D70F})$ . Upon setting $\boldsymbol{k}s_{+}\prime =\mathbf{0}$ , one can solve (5.28) as

(5.30) $$\begin{eqnarray}\displaystyle C_{+\boldsymbol{k};\mathbf{0}}^{(2;1)}(\text{}\underline{1},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ) & = & \displaystyle H(\unicode[STIX]{x1D70F})\unicode[STIX]{x1D6EF}_{\boldsymbol{k}}(\text{}\underline{1},\overline{\text{}\underline{1}},\unicode[STIX]{x1D70F})\unicode[STIX]{x1D6EF}_{\boldsymbol{k}}^{\ast }(\text{}\underline{2},\overline{\text{}\underline{2}},\unicode[STIX]{x1D70F})C_{\boldsymbol{k},\mathbf{0}}^{(0,3)}(\overline{\text{}\underline{1}},\overline{\text{}\underline{2}},\text{}\underline{1}s_{+}\prime )\nonumber\\ \displaystyle & & \displaystyle +\,\displaystyle \int _{0}^{\unicode[STIX]{x1D70F}}\!\,\text{d}\overline{\unicode[STIX]{x1D70F}}\,\unicode[STIX]{x1D6EF}_{\boldsymbol{k}}(\text{}\underline{1},\overline{\text{}\underline{1}},\overline{\unicode[STIX]{x1D70F}})\unicode[STIX]{x1D6EF}_{\boldsymbol{k}}^{\ast }(\text{}\underline{2},\overline{\text{}\underline{2}},\overline{\unicode[STIX]{x1D70F}})s_{+,\boldsymbol{k};\mathbf{0}}^{(2;1)}(\overline{\text{}\underline{1}},\overline{\text{}\underline{2}},\unicode[STIX]{x1D70F}-\overline{\unicode[STIX]{x1D70F}};\text{}\underline{1}s_{+}\prime ),\end{eqnarray}$$

where

(5.31) $$\begin{eqnarray}\displaystyle & s_{+,\boldsymbol{k};\mathbf{0}}^{(2;1)}(\text{}\underline{1},\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\doteq -\unicode[STIX]{x1D750}_{\boldsymbol{k}}\boldsymbol{\cdot }(q_{2}\boldsymbol{\unicode[STIX]{x2202}}_{1}-q_{1}\boldsymbol{\unicode[STIX]{x2202}}_{2})[C_{+;\mathbf{0}}^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )f(2)+C_{+;\mathbf{0}}^{(1;1)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )f(1)] & \displaystyle \nonumber\\ \displaystyle & \quad -\,\boldsymbol{\unicode[STIX]{x2202}}_{1}C_{+;\mathbf{0}}^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}_{\boldsymbol{k}}(\overline{\text{}\underline{1}})g_{\boldsymbol{k}}(\overline{\text{}\underline{1}},\text{}\underline{2},t)-\boldsymbol{\unicode[STIX]{x2202}}_{2}C_{+;\mathbf{0}}^{(1;1)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\boldsymbol{\cdot }[\pmb{\mathbb{E}}_{\boldsymbol{k}}(\overline{\text{}\underline{1}})g_{\boldsymbol{k}}(\text{}\underline{1},\overline{\text{}\underline{1}},t)]^{\ast }. & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Here the terms in (5.29) involving $\pmb{\mathbb{E}}_{\boldsymbol{k}s_{+}\prime }C_{+;\boldsymbol{k}s_{+}\prime }^{(1;1)}|_{\boldsymbol{k}s_{+}\prime =\mathbf{0}}$ were assumed to vanish.

The $\unicode[STIX]{x1D6EF}$ functions describe Debye shielding clouds, so their characteristic time scale is the microscopic autocorrelation time  $\unicode[STIX]{x1D714}_{\text{p}}^{-1}$ . In contrast, all of the $\unicode[STIX]{x1D70F}$ -dependent terms in (5.31) involve $C_{+}(\unicode[STIX]{x1D70F})$ , which (as will be shown) varies on the collisional time scale. Thus, a Markovian approximation is appropriate and the $s_{+}^{(2;1)}$  contribution to  $C_{+}^{(2;1)}$ is approximately

(5.32) $$\begin{eqnarray}\left(\int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\,\unicode[STIX]{x1D6EF}_{\boldsymbol{ k}}(\text{}\underline{1},\overline{\text{}\underline{1}},\overline{\unicode[STIX]{x1D70F}})\unicode[STIX]{x1D6EF}_{\boldsymbol{k}}^{\ast }(\text{}\underline{2},\overline{\text{}\underline{2}},\overline{\unicode[STIX]{x1D70F}})\right)s_{+,\boldsymbol{k};\mathbf{0}}^{(2;1)}(\overline{\text{}\underline{1}},\overline{\text{}\underline{2}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ).\end{eqnarray}$$

The contribution of this term to (5.27) is analysed in § G.2; it is found that it leads to the linearized Balescu–Lenard operator  $\widehat{\text{C}}^{\text{BL}}$ acting on  $C_{+}^{(1;1)}$ . I shall subsequently drop the BL superscript. Thus, one has

(5.33) $$\begin{eqnarray}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}C_{+;\mathbf{0}}^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )+\widehat{\text{C}}C_{+;\mathbf{0}}^{(1;1)}=\unicode[STIX]{x1D6FF}(t-ts_{+}\prime )C_{+;\mathbf{0}}^{(0;2)}(\text{}\underline{1},\text{}\underline{1}s_{+}\prime ,ts_{+}\prime )+s_{+;\mathbf{0}}^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ),\end{eqnarray}$$

where

(5.34) $$\begin{eqnarray}s_{+}^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\doteq -\boldsymbol{\unicode[STIX]{x2202}}_{1}\boldsymbol{\cdot }\pmb{\mathbb{E}}(1,2)\unicode[STIX]{x1D6EF}(\text{}\underline{1},\unicode[STIX]{x1D70F};\overline{\text{}\underline{1}})\unicode[STIX]{x1D6EF}(\text{}\underline{2},\unicode[STIX]{x1D70F};\overline{\text{}\underline{2}})C^{(0,3)}(\overline{\text{}\underline{1}},\overline{\text{}\underline{2}},\text{}\underline{1}s_{+}\prime ,ts_{+}\prime ).\end{eqnarray}$$

For the evaluation of transport coefficients, one is required to integrate the solution from 0 to  $\infty$ in  $\unicode[STIX]{x1D70F}$ . An equation for $\int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\,C_{+;\mathbf{0}}^{(1;1)}(\overline{\unicode[STIX]{x1D70F}})$ can be obtained by integrating (5.33) from $\overline{\unicode[STIX]{x1D70F}}=0_{-}$ to  $\infty$ and assuming that correlations decay to 0 for large time lags:

(5.35) $$\begin{eqnarray}\widehat{\text{C}}\int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\,C_{+;\mathbf{0}}^{(1;1)}(\overline{\unicode[STIX]{x1D70F}})=C_{\boldsymbol{ 0}}^{(0,2)}(\boldsymbol{v}_{1},\boldsymbol{v}_{1s_{+}\prime },ts_{+}\prime )+\int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\,s_{+;\mathbf{0}}^{(1;1)}(\boldsymbol{v}_{1},\overline{\unicode[STIX]{x1D70F}};\boldsymbol{v}_{1}^{\prime }),\end{eqnarray}$$

where

(5.36) $$\begin{eqnarray}C_{\mathbf{0}}^{(0,2)}(\boldsymbol{v}_{1},\boldsymbol{v}_{1s_{+}\prime },ts_{+}\prime )=\overline{n}_{s_{1}}^{-1}\unicode[STIX]{x1D6FF}_{s_{1}s_{1s_{+}\prime }}\unicode[STIX]{x1D6FF}(\boldsymbol{v}_{1}-\boldsymbol{v}_{1s_{+}\prime })f_{s_{1s_{+}\prime }}(\boldsymbol{v}_{1s_{+}\prime }).\end{eqnarray}$$

If the $s_{+}$  term were negligible, one would find

(5.37) $$\begin{eqnarray}\int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\,C_{+;\mathbf{0}}^{(1;1)}(\overline{\unicode[STIX]{x1D70F}})=\widehat{\text{C}}^{-1}\overline{n}_{s}^{-1}\unicode[STIX]{x1D6FF}_{sss_{+}\prime }\unicode[STIX]{x1D6FF}(\boldsymbol{v}-\boldsymbol{v}s_{+}\prime )f_{ss_{+}\prime }(\boldsymbol{v}s_{+}\prime )\end{eqnarray}$$

and, with the aid of (3.8d ), formulas such as (4.15a ) would reduce to the standard velocity-space matrix elements that emerged in Part 1. A discussion that justifies the neglect of  $s_{+}$ is given in § H.3.

5.4 Two-time, three phase-space point correlations

For the integrals (4.15f )–(4.15k ), one requires two-time correlations with three phase-space points, i.e. $C_{+}^{(1;2)}(\text{}\underline{1},t;\text{}\underline{1}s_{+}\prime ,ts_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime ,ts_{+}\prime )$ . A natural way of proceeding is to introduce a third source  $\unicode[STIX]{x1D702}s_{+}\prime \prime$ , generate the equation for $C(t,ts_{+}\prime ,ts_{+}\prime \prime )$ , set $ts_{+}\prime \prime =ts_{+}\prime$ , and evolve forward from  $ts_{+}\prime$ to  $t$ . The functional derivative of (5.20) with respect to  $\unicode[STIX]{x1D702}s_{+}\prime \prime$ is

(5.38) $$\begin{eqnarray}0=\unicode[STIX]{x2202}_{t}C^{(1,1,1)}(1,1s_{+}\prime ,1s_{+}\prime \prime )+\text{i}\text{L}_{1}C^{(1,1,1)}+\boldsymbol{\unicode[STIX]{x2202}}_{1}\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{1},\overline{\text{}\underline{2}})C^{(2,1,1)}(\text{}\underline{1},\overline{\text{}\underline{2}},t,1s_{+}\prime ,1s_{+}\prime \prime ),\end{eqnarray}$$

and the functional derivative of (5.23) with respect to  $\unicode[STIX]{x1D702}s_{+}\prime \prime$ is

(5.39) $$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}+\text{i}\text{L}_{1}+\text{i}\text{L}_{2})C_{+}^{(2;1,1)}(\text{}\underline{1},\text{}\underline{2},t;1s_{+}\prime ,1s_{+}\prime \prime )=\unicode[STIX]{x1D6FF}(t-ts_{+}\prime )C^{(0,3,1)}(\text{}\underline{1},\text{}\underline{2},\text{}\underline{1}s_{+}\prime ,ts_{+}\prime ,ts_{+}\prime \prime )+s_{+}^{(2;1,1)}(\text{}\underline{1},\text{}\underline{2},t;1s_{+}\prime ,1s_{+}\prime \prime ),\end{eqnarray}$$

where

(5.40) $$\begin{eqnarray}\displaystyle & & \displaystyle s_{+}^{(2;1,1)}(\text{}\underline{1},\text{}\underline{2},t;1s_{+}\prime ,1s_{+}\prime \prime )\nonumber\\ \displaystyle & & \displaystyle \doteq -\unicode[STIX]{x1D750}_{12}\boldsymbol{\cdot }(q_{2}\boldsymbol{\unicode[STIX]{x2202}}_{1}-q_{1}\boldsymbol{\unicode[STIX]{x2202}}_{2})[C_{+}^{(1;1,1)}(1;1s_{+}\prime ,1s_{+}\prime \prime )f(2)+f(1)C_{+}^{(1;1,1)}(2;1s_{+}\prime ,1s_{+}\prime \prime )]\nonumber\\ \displaystyle & & \displaystyle ~~-\,\unicode[STIX]{x1D750}_{12}\boldsymbol{\cdot }(q_{2}\boldsymbol{\unicode[STIX]{x2202}}_{1}-q_{1}\boldsymbol{\unicode[STIX]{x2202}}_{2})[C_{+}^{(1;1)}(1;1s_{+}\prime )C_{+}^{(1;1)}(2;1s_{+}\prime \prime )+C_{+}^{(1;1)}(1;1s_{+}\prime \prime )C_{+}^{(1;1)}(2;1s_{+}\prime )]\nonumber\\ \displaystyle & & \displaystyle ~~-\,\boldsymbol{\unicode[STIX]{x2202}}_{1}C_{+}^{(1;1)}(1;1s_{+}\prime \prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{1},\overline{\text{}\underline{1}})C_{+}^{(2;1)}(\overline{\text{}\underline{1}},\text{}\underline{2},t;1s_{+}\prime )-\boldsymbol{\unicode[STIX]{x2202}}_{2}C_{+}^{(1;1)}(2;1s_{+}\prime \prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{2},\overline{\text{}\underline{2}})C_{+}^{(2;1)}(\text{}\underline{1},\overline{\text{}\underline{2}},t;1s_{+}\prime );\hspace{-3.99994pt}\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

the last two terms arise by differentiating the  $f$ in the last term of the Landau operator (5.16).

One may now set $ts_{+}\prime \prime =ts_{+}\prime$ . For stationary fluctuations, the one-sided version of (5.38) becomes

(5.41) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x2202}_{t}C_{+}^{(1;2)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )+\text{i}\text{L}_{1}(\text{}\underline{1},\overline{\text{}\underline{1}})C_{+}^{(1;2)}(\overline{\text{}\underline{1}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )\nonumber\\ \displaystyle & & \displaystyle =\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})C^{(0,3)}(\text{}\underline{1},\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime ,ts_{+}\prime )-\boldsymbol{\unicode[STIX]{x2202}}_{1}\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{1},\overline{\text{}\underline{2}})C_{+}^{(2;2)}(\text{}\underline{1},\overline{\text{}\underline{2}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime ),\end{eqnarray}$$

and (5.40) becomes

(5.42) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x2202}_{t}C_{+}^{(2;2)}(\text{}\underline{1},\text{}\underline{2},t;\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime ,ts_{+}\prime )+\text{i}\text{L}_{1}(\text{}\underline{1},\overline{\text{}\underline{1}})C_{+}^{(2;2)}(\overline{\text{}\underline{1}},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )+\text{i}\text{L}_{2}(\text{}\underline{2},\overline{\text{}\underline{2}})C_{+}^{(2;2)}(\text{}\underline{1},\overline{\text{}\underline{2}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )\nonumber\\ \displaystyle & & \displaystyle =\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})C_{+}^{(0,4)}(\text{}\underline{1},\text{}\underline{2},\text{}\underline{1}s_{+}\prime ,\text{}\underline{2}s_{+}\prime \prime ,ts_{+}\prime )+s_{+}^{(2;2)}(\text{}\underline{1},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime ),\end{eqnarray}$$

where

(5.43) $$\begin{eqnarray}\displaystyle & & \displaystyle \!\hspace{-6.0pt}s_{+}^{(2;2)}(\text{}\underline{1},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )\nonumber\\ \displaystyle & & \displaystyle \!\doteq -\unicode[STIX]{x1D750}_{12}\boldsymbol{\cdot }(q_{2}\boldsymbol{\unicode[STIX]{x2202}}_{1}-q_{1}\boldsymbol{\unicode[STIX]{x2202}}_{2})[C_{+}^{(1;2)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )f(2)+f(1)C_{+}^{(1;2)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )]\nonumber\\ \displaystyle & & \displaystyle \!\quad -\,[\boldsymbol{\unicode[STIX]{x2202}}_{1}C_{+}^{(1;2)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{1},\overline{\text{}\underline{1}})g(\overline{\text{}\underline{1}},\text{}\underline{2})+\boldsymbol{\unicode[STIX]{x2202}}_{2}C_{+}^{(1;2)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{2},\overline{\text{}\underline{2}})g(\text{}\underline{1},\overline{\text{}\underline{2}})]\nonumber\\ \displaystyle & & \displaystyle \!\quad \quad ~-\,\unicode[STIX]{x1D750}_{12}\boldsymbol{\cdot }(q_{2}\boldsymbol{\unicode[STIX]{x2202}}_{1}-q_{1}\boldsymbol{\unicode[STIX]{x2202}}_{2})[C_{+}^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )C_{+}^{(1;1)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )+C_{+}^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )C_{+}^{(1;1)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )]\nonumber\\ \displaystyle & & \displaystyle \!\quad -\, [\boldsymbol{\unicode[STIX]{x2202}}_{1}C_{+}^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{1},\overline{\text{}\underline{1}})C_{+}^{(2;1)}(\overline{\text{}\underline{1}},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )+(1s_{+}\prime \Leftrightarrow 1s_{+}\prime \prime )\nonumber\\ \displaystyle & & \displaystyle \!\quad \quad ~+\,\boldsymbol{\unicode[STIX]{x2202}}_{2}C_{+}^{(1;1)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{2},\overline{\text{}\underline{2}})C_{+}^{(2;1)}(\text{}\underline{1},\overline{\text{}\underline{2}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )+(1s_{+}\prime \Leftrightarrow 1s_{+}\prime \prime )]\nonumber\\ \displaystyle & & \displaystyle \!\quad -\, [\boldsymbol{\unicode[STIX]{x2202}}_{1}C_{+}^{(2;1)}(\text{}\underline{1},\text{}\underline{2},\unicode[STIX]{x1D70F};1s_{+}\prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{1},\overline{\text{}\underline{1}})C_{+}^{(1;1)}(\overline{\text{}\underline{1}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )+(1s_{+}\prime \Leftrightarrow 1s_{+}\prime \prime )\nonumber\\ \displaystyle & & \displaystyle \!\quad \quad ~+\,\boldsymbol{\unicode[STIX]{x2202}}_{2}C_{+}^{(2;1)}(\text{}\underline{1},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{2},\overline{\text{}\underline{2}})C_{+}^{(1;1)}(\overline{\text{}\underline{2}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )+(1s_{+}\prime \Leftrightarrow 1s_{+}\prime \prime )]\nonumber\\ \displaystyle & & \displaystyle \!\quad -\,[\boldsymbol{\unicode[STIX]{x2202}}_{1}g(\text{}\underline{1},\text{}\underline{2})\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{1},\overline{\text{}\underline{1}})C_{+}^{(1;2)}(\overline{\text{}\underline{1}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )+\boldsymbol{\unicode[STIX]{x2202}}_{2}g(\text{}\underline{1},\text{}\underline{2})\boldsymbol{\cdot }\pmb{\mathbb{E}}(\text{}\underline{2},\overline{\text{}\underline{2}})C_{+}^{(1;2)}(\overline{\text{}\underline{2}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )].\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

With the additional definition $\unicode[STIX]{x1D746}s_{+}\prime \prime \doteq \boldsymbol{x}_{1}-\boldsymbol{x}_{1}^{\prime \prime }$ , one has, for example,

(5.44a ) $$\begin{eqnarray}\displaystyle & & \displaystyle C_{+}^{(2;2)}(\text{}\underline{1},\text{}\underline{2},t;\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime ,ts_{+}\prime )=C_{+}^{(2;2)}(\unicode[STIX]{x1D746},\text{}\underline{1},\text{}\underline{2},\unicode[STIX]{x1D70F};\unicode[STIX]{x1D746}s_{+}\prime ,\text{}\underline{1}s_{+}\prime ,\unicode[STIX]{x1D746}s_{+}\prime \prime ,\text{}\underline{1}s_{+}\prime \prime )\end{eqnarray}$$
(5.44b ) $$\begin{eqnarray}\displaystyle & & \displaystyle \quad =\displaystyle \int \frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\displaystyle \int \frac{\text{d}\boldsymbol{k}s_{+}\prime }{(2\unicode[STIX]{x03C0})^{3}}\displaystyle \int \frac{\text{d}\boldsymbol{k}s_{+}\prime \prime }{(2\unicode[STIX]{x03C0})^{3}}\,\text{e}^{\text{i}\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x1D746}+\text{i}\boldsymbol{k}s_{+}\prime \boldsymbol{\cdot }\unicode[STIX]{x1D746}s_{+}\prime +\text{i}\boldsymbol{k}s_{+}\prime \prime \boldsymbol{\cdot }\unicode[STIX]{x1D746}s_{+}\prime \prime }C_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(2;2)}(\text{}\underline{1},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime ).\qquad\end{eqnarray}$$
As discussed in § 4.2, one requires $\text{i}\boldsymbol{\unicode[STIX]{x2202}}_{\boldsymbol{k}s_{+}\prime \prime }C_{+;\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )|_{\boldsymbol{k}s_{+}\prime =\mathbf{0},\,\boldsymbol{k}s_{+}\prime \prime =\mathbf{0}}$ (see (4.17b )). The function $C_{+;\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\unicode[STIX]{x1D70F})$ evolves according to the Fourier transform of (5.41):
(5.45) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}C_{+;\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )+\text{i}\text{L}_{\boldsymbol{k}s_{+}\prime +\boldsymbol{k}s_{+}\prime \prime }(\text{}\underline{1},\overline{\text{}\underline{1}})C_{+;\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\overline{\text{}\underline{1}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})C_{\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(0,3)}(\text{}\underline{1},\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )-\boldsymbol{\unicode[STIX]{x2202}}_{1}\boldsymbol{\cdot }\displaystyle \int \frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\,\pmb{\mathbb{E}}_{\boldsymbol{k}}^{\ast }(\overline{\text{}\underline{2}})C_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(2;2)}(\text{}\underline{1},\overline{\text{}\underline{2}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime ).\end{eqnarray}$$

Clearly, $C_{+}^{(2;2)}$ is crucial for determining the collisional dynamics of  $C_{+}^{(1;2)}$ . The Fourier transform of (5.43) is

(5.46) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}C_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(2;2)}(\text{}\underline{1},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\text{i}\text{L}_{\boldsymbol{k}+\boldsymbol{k}s_{+}\prime +\boldsymbol{k}s_{+}\prime \prime }(\text{}\underline{1},\overline{\text{}\underline{1}})C_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(2;2)}(\overline{\text{}\underline{1}},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )+\text{i}\text{L}_{\boldsymbol{k}}^{\ast }(\text{}\underline{2},\overline{\text{}\underline{2}})C_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(2;2)}(\boldsymbol{k},\text{}\underline{1},\overline{\text{}\underline{2}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})C_{\boldsymbol{k},\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(0,4)}(\text{}\underline{1},\text{}\underline{2},\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )+s_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(2;2)}(\text{}\underline{1},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime ),\end{eqnarray}$$

where

(5.47) $$\begin{eqnarray}\displaystyle & & \displaystyle s_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(2;2)}(\text{}\underline{1},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )\doteq \nonumber\\ \displaystyle & & \displaystyle \quad -\,(q_{2}\boldsymbol{\unicode[STIX]{x2202}}_{1}-q_{1}\boldsymbol{\unicode[STIX]{x2202}}_{2})\boldsymbol{\cdot }[\unicode[STIX]{x1D750}_{\boldsymbol{k}}C_{+;\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )f(\text{}\underline{2})+\unicode[STIX]{x1D750}_{\boldsymbol{k}+\boldsymbol{k}s_{+}\prime +\boldsymbol{k}s_{+}\prime \prime }C_{+;\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )f(\text{}\underline{1})]\nonumber\\ \displaystyle & & \displaystyle \quad -\,\{\boldsymbol{\unicode[STIX]{x2202}}_{1}C_{+;\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}_{\boldsymbol{k}}(\overline{\text{}\underline{1}})g_{\boldsymbol{k}}(\overline{\text{}\underline{1}},2)\nonumber\\ \displaystyle & & \displaystyle \quad \quad ~+\,\boldsymbol{\unicode[STIX]{x2202}}_{2}C_{+;\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )\boldsymbol{\cdot }[\pmb{\mathbb{E}}_{\boldsymbol{k}+\boldsymbol{k}s_{+}\prime +\boldsymbol{k}s_{+}\prime \prime }^{\ast }(\overline{\text{}\underline{2}})g_{\boldsymbol{k}+\boldsymbol{k}s_{+}\prime +\boldsymbol{k}s_{+}\prime \prime }(\text{}\underline{1},\overline{\text{}\underline{2}})]\}\nonumber\\ \displaystyle & & \displaystyle \quad -\,(q_{2}\boldsymbol{\unicode[STIX]{x2202}}_{1}-q_{1}\boldsymbol{\unicode[STIX]{x2202}}_{2})\boldsymbol{\cdot }[\unicode[STIX]{x1D750}_{\boldsymbol{k}+\boldsymbol{k}s_{+}\prime \prime }C_{+;\boldsymbol{k}s_{+}\prime }^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )C_{+;\boldsymbol{k}s_{+}\prime \prime }^{(1;1)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad \qquad \quad \quad ~+\,\unicode[STIX]{x1D750}_{\boldsymbol{k}+\boldsymbol{k}s_{+}\prime }C_{+;\boldsymbol{k}s_{+}\prime \prime }^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )C_{+;\boldsymbol{k}s_{+}\prime }^{(1;1)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )]\nonumber\\ \displaystyle & & \displaystyle \quad -\, [\boldsymbol{\unicode[STIX]{x2202}}_{1}C_{+;\boldsymbol{k}s_{+}\prime }^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}_{\boldsymbol{k}}(\overline{\text{}\underline{1}})C_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime \prime }^{(2;1)}(\overline{\text{}\underline{1}},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )\nonumber\\ \displaystyle & & \displaystyle \quad \quad ~+\,\boldsymbol{\unicode[STIX]{x2202}}_{1}C_{+;\boldsymbol{k}s_{+}\prime \prime }^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}_{\boldsymbol{k}}(\overline{\text{}\underline{1}})C_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime }^{(2;1)}(\overline{\text{}\underline{1}},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\nonumber\\ \displaystyle & & \displaystyle \quad \quad ~+\,\boldsymbol{\unicode[STIX]{x2202}}_{2}C_{+;\boldsymbol{k}s_{+}\prime }^{(1;1)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}_{\boldsymbol{k}+\boldsymbol{k}s_{+}\prime }^{\ast }(\overline{\text{}\underline{2}})C_{+,\boldsymbol{k}+\boldsymbol{k}s_{+}\prime ;\boldsymbol{k}s_{+}\prime \prime }^{(2;1)}(\text{}\underline{1},\overline{\text{}\underline{2}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )\nonumber\\ \displaystyle & & \displaystyle \quad \quad ~+\,\boldsymbol{\unicode[STIX]{x2202}}_{2}C_{+;\boldsymbol{k}s_{+}\prime \prime }^{(1;1)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}_{\boldsymbol{k}+\boldsymbol{k}s_{+}\prime }^{\ast }(\overline{\text{}\underline{2}})C_{+,\boldsymbol{k}+\boldsymbol{k}s_{+}\prime ;\boldsymbol{k}s_{+}\prime }^{(2;1)}(\text{}\underline{1},\overline{\text{}\underline{2}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )]\nonumber\\ \displaystyle & & \displaystyle \quad -\, [\boldsymbol{\unicode[STIX]{x2202}}_{1}C_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime }^{(2;1)}(\text{}\underline{1},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\boldsymbol{\cdot }\text{}\underline{\pmb{\mathbb{E}}_{\boldsymbol{k}s_{+}\prime \prime }(\overline{\text{}\underline{1}})C_{+;\boldsymbol{k}s_{+}\prime \prime }^{(1;1)}(\overline{\text{}\underline{1}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )}\nonumber\\ \displaystyle & & \displaystyle \quad \quad ~+\,\boldsymbol{\unicode[STIX]{x2202}}_{1}C_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime \prime }^{(2;1)}(\text{}\underline{1},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )\boldsymbol{\cdot }\text{}\underline{\pmb{\mathbb{E}}_{\boldsymbol{k}s_{+}\prime }(\overline{\text{}\underline{1}})C_{+;\boldsymbol{k}s_{+}\prime }^{(1;1)}(\overline{\text{}\underline{1}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )}\nonumber\\ \displaystyle & & \displaystyle \quad \quad ~+\,\boldsymbol{\unicode[STIX]{x2202}}_{2}C_{+,\boldsymbol{k}+\boldsymbol{k}s_{+}\prime \prime ;\boldsymbol{k}s_{+}\prime }^{(2;1)}(\text{}\underline{1},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\boldsymbol{\cdot }\text{}\underline{\pmb{\mathbb{E}}_{\boldsymbol{k}s_{+}\prime \prime }(\overline{\text{}\underline{2}})C_{+;\boldsymbol{k}s_{+}\prime \prime }^{(1;1)}(\overline{\text{}\underline{2}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )}\nonumber\\ \displaystyle & & \displaystyle \quad \quad ~+\,\boldsymbol{\unicode[STIX]{x2202}}_{2}C_{+,\boldsymbol{k}+\boldsymbol{k}s_{+}\prime ;\boldsymbol{k}s_{+}\prime \prime }^{(2;1)}(\text{}\underline{1},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )\boldsymbol{\cdot }\text{}\underline{\pmb{\mathbb{E}}_{\boldsymbol{k}s_{+}\prime }(\overline{\text{}\underline{2}})C_{+;\boldsymbol{k}s_{+}\prime }^{(1;1)}(\overline{\text{}\underline{2}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )}]\nonumber\\ \displaystyle & & \displaystyle \quad -\, [\boldsymbol{\unicode[STIX]{x2202}}_{1}g_{\boldsymbol{k}}(\text{}\underline{1},\text{}\underline{2})\boldsymbol{\cdot }\text{}\underline{\pmb{\mathbb{E}}_{\boldsymbol{k}s_{+}\prime +\boldsymbol{k}s_{+}\prime \prime }(\overline{\text{}\underline{1}})C_{+;\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\overline{\text{}\underline{1}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )}\nonumber\\ \displaystyle & & \displaystyle \quad \quad ~+\,\boldsymbol{\unicode[STIX]{x2202}}_{2}g_{\boldsymbol{k}+\boldsymbol{k}s_{+}\prime +\boldsymbol{k}s_{+}\prime \prime }(\text{}\underline{1},\text{}\underline{2})\boldsymbol{\cdot }\text{}\underline{\pmb{\mathbb{E}}_{\boldsymbol{k}s_{+}\prime +\boldsymbol{k}s_{+}\prime \prime }(\overline{\text{}\underline{2}})C_{+;\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\overline{\text{}\underline{2}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )}].\end{eqnarray}$$

For small  $\boldsymbol{k}s_{+}\prime$ and  $\boldsymbol{k}s_{+}\prime \prime$ , the underlined terms are negligible. Upon setting $\boldsymbol{k}s_{+}\prime =\mathbf{0}$ , one finds

(5.48) $$\begin{eqnarray}\displaystyle & & \displaystyle s_{+,\boldsymbol{k};\mathbf{0},\boldsymbol{k}s_{+}\prime \prime }^{(2;2)}(\text{}\underline{1},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )\doteq s_{+,\boldsymbol{k};\mathbf{0},\boldsymbol{k}s_{+}\prime \prime }^{(2;2)}(\text{}\underline{1},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )\nonumber\\ \displaystyle & & \displaystyle =-(q_{2}\boldsymbol{\unicode[STIX]{x2202}}_{1}-q_{2}\boldsymbol{\unicode[STIX]{x2202}}_{2})\boldsymbol{\cdot }\unicode[STIX]{x1D750}_{\boldsymbol{k}}[C_{+;\mathbf{0},\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )f(\text{}\underline{2})+C_{+;\mathbf{0},\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )f(\text{}\underline{1})]\nonumber\\ \displaystyle & & \displaystyle \quad -\,\{\boldsymbol{\unicode[STIX]{x2202}}_{1}C_{+;\mathbf{0},\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}_{\boldsymbol{k}}(\overline{\text{}\underline{1}})g_{\boldsymbol{k}}(\overline{\text{}\underline{1}},\text{}\underline{2})+\boldsymbol{\unicode[STIX]{x2202}}_{2}C_{+;\mathbf{0},\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )\boldsymbol{\cdot }[\pmb{\mathbb{E}}_{\boldsymbol{k}}^{\ast }(\overline{\text{}\underline{2}})g_{\boldsymbol{k}}(\text{}\underline{1},\overline{\text{}\underline{2}})]^{\ast }\}\nonumber\\ \displaystyle & & \displaystyle \quad -\,(q_{2}\boldsymbol{\unicode[STIX]{x2202}}_{1}-q_{1}\boldsymbol{\unicode[STIX]{x2202}}_{2})\boldsymbol{\cdot }\unicode[STIX]{x1D750}_{\boldsymbol{k}}[C_{+;\mathbf{0}}^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )C_{+;\boldsymbol{k}s_{+}\prime \prime }^{(1;1)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )+C_{+;\boldsymbol{k}s_{+}\prime \prime }^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )C_{+;\mathbf{0}}^{(1;1)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )]\nonumber\\ \displaystyle & & \displaystyle \quad - [\boldsymbol{\unicode[STIX]{x2202}}_{1}C_{+;\mathbf{0}}^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}_{\boldsymbol{k}}(\overline{\text{}\underline{1}})C_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime \prime }^{(2;1)}(\overline{\text{}\underline{1}},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )\nonumber\\ \displaystyle & & \displaystyle \quad \quad ~+\,\boldsymbol{\unicode[STIX]{x2202}}_{1}C_{+;\boldsymbol{k}s_{+}\prime \prime }^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}_{\boldsymbol{k}}(\overline{\text{}\underline{1}})C_{+,\boldsymbol{k};\mathbf{0}}^{(2;1)}(\overline{\text{}\underline{1}},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\nonumber\\ \displaystyle & & \displaystyle \quad \quad ~+\,\boldsymbol{\unicode[STIX]{x2202}}_{2}C_{+;\mathbf{0}}^{(1;1)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}_{\boldsymbol{k}}^{\ast }(\overline{\text{}\underline{2}})C_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime \prime }^{(2;1)}(\text{}\underline{1},\overline{\text{}\underline{2}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )\nonumber\\ \displaystyle & & \displaystyle \quad \quad ~+\,\boldsymbol{\unicode[STIX]{x2202}}_{2}C_{+;\boldsymbol{k}s_{+}\prime \prime }^{(1;1)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}_{\boldsymbol{k}}^{\ast }(\overline{\text{}\underline{2}})C_{+,\boldsymbol{k};\mathbf{0}}^{(2;1)}(\text{}\underline{1},\overline{\text{}\underline{2}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )].\end{eqnarray}$$

Upon following the same arguments as in § 5.3, one finds that the second and third lines of (5.40) lead to a term $-\widehat{\text{C}}C_{+;\mathbf{0},\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(1;\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime ,ts_{+}\prime )$ in (5.45). The remaining terms are analysed in § G.3, where it is shown that they are related to the nonlinear Balescu–Lenard collision operator. One finds

(5.49) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}C_{+;\mathbf{0},\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )+\text{i}\text{L}_{\boldsymbol{k}s_{+}\prime \prime }(\text{}\underline{1},\overline{\text{}\underline{1}})C_{+;\mathbf{0},\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\overline{\text{}\underline{1}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )+\widehat{\text{C}}C_{+;\mathbf{0},\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})C_{\mathbf{0},\boldsymbol{k}s_{+}\prime \prime }^{(0,3)}(\text{}\underline{1},\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )-\{\text{C}^{\text{BL}}[f;C_{+;\mathbf{0}}^{(1;1)}(\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ),\overline{C}_{+;\boldsymbol{k}s_{+}\prime \prime }^{(1;1)}(\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )]+(1s_{+}\prime \Leftrightarrow 1s_{+}\prime \prime )\}.\qquad\end{eqnarray}$$

Notice that while the last term of this equation does contain the (generalized, three-argument) nonlinear collision operator (for an explanation of the notation $\text{C}[a;b,\overline{c}]$ , see the discussion of (G 24)), that construction (involving $\unicode[STIX]{x1D70F}$ -dependent arguments) is not the $\text{C}[\,f_{1},\overline{f}_{1}]$ that appears in second-order Chapman–Enskog theory ( $f_{1}$  does not depend on  $\unicode[STIX]{x1D70F}$ ; it is an integral over all  $\unicode[STIX]{x1D70F}$ ). We shall see in the next section how $\text{C}[\,f_{1},\overline{f}_{1}]$ emerges from the solution of (5.49).

6 Comparison to the results of Catto & Simakov

For magnetized plasmas, the most complete calculation that involves some Burnett terms is the one by Catto & Simakov (Reference Catto and Simakov2004). I shall now discuss the relationship of their calculation to the present results.

6.1 Brief review of the calculation of Catto & Simakov (Reference Catto and Simakov2004)

For definiteness, I shall consider the ion version of the calculation of Catto & Simakov. They begin with the Landau kinetic equation

(6.1) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\,f(\boldsymbol{x},\boldsymbol{v},t)+\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}f+(\boldsymbol{E}+c^{-1}\boldsymbol{v}\times \boldsymbol{B}^{\text{ext}})\boldsymbol{\cdot }\boldsymbol{\unicode[STIX]{x2202}}f=-\text{C}[\,f,\overline{f}],\end{eqnarray}$$

where $\text{C}[\,f,\overline{f}]$ is the bilinear ion Landau collision operator $\text{C}_{ii}+\text{C}_{ie}$ . Use of the Landau operator restricts the calculation to weakly coupled plasma,Footnote 58 although, as is typical for work that extends the class of calculations reviewed by Braginskii (Reference Braginskii and Leontovich1965), this important point is not stressed. (Weak coupling is a good approximation for magnetically confined fusion plasmas.) Use of that operator also implies that the convective-amplification effects discussed by Kent & Taylor (Reference Kent and Taylor1969) have been ignored. With $\boldsymbol{w}\doteq \boldsymbol{v}-\boldsymbol{u}(\boldsymbol{x},t)$ , the variable transformation $(\boldsymbol{x},\boldsymbol{v},t)\rightarrow (\boldsymbol{x},\boldsymbol{w},t)$ is made; that transforms (6.1) to

(6.2) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x2202}_{t}\,f(\boldsymbol{x},\boldsymbol{w},t)+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}f+\boldsymbol{w}\boldsymbol{\cdot }\unicode[STIX]{x1D735}f+[(mn)^{-1}(\unicode[STIX]{x1D735}p+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D745}-\boldsymbol{R})-\boldsymbol{w}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{u})]\boldsymbol{\cdot }\unicode[STIX]{x2202}_{\boldsymbol{w}}f\nonumber\\ \displaystyle & & \displaystyle \quad +\,\text{i}\widehat{\text{M}}+\text{C}[\,f,\overline{f}]=0,\end{eqnarray}$$

where

(6.3) $$\begin{eqnarray}\text{i}\widehat{\text{M}}\doteq \unicode[STIX]{x1D714}_{\text{c}}\boldsymbol{w}\times \widehat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x2202}_{\boldsymbol{w}}\end{eqnarray}$$

and the exact form of the momentum equation was used to replace the  $\unicode[STIX]{x2202}_{t}\boldsymbol{u}$ that arises after the variable transformation. The distribution function is expanded as $f=\sum _{n=0}^{\infty }\unicode[STIX]{x1D716}^{n}f_{n}$ , where $\unicode[STIX]{x1D716}$  is an ordering parameter. Catto & Simakov follow the standard procedure of taking $\unicode[STIX]{x1D735}=O(\unicode[STIX]{x1D716})$ ,Footnote 59 but they do not explicitly describe a multiple-scale procedure as is explicated in appendix  1:A; I shall return to this point. Whereas Braginskii implicitly assumes that $\boldsymbol{u}=O(1)$ , Catto & Simakov follow Mikhaǐlovskiǐ & Tsypin (Reference Mikhaǐlovskiǐ and Tsypin1971) in taking $\boldsymbol{u}=O(\unicode[STIX]{x1D716})$ .Footnote 60 This implies that $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D745}=O(\unicode[STIX]{x1D716}^{3})$ . The term involving the ion friction force cancels with  $\text{C}_{ie}$ to lowest order in the mass ratio. Catto & Simakov are led to the sequence of equations

(6.4a ) $$\begin{eqnarray}\displaystyle \text{i}\widehat{\text{M}}f_{0}+\text{C}[f_{0},\overline{f}_{0}] & = & \displaystyle 0,\end{eqnarray}$$
(6.4b ) $$\begin{eqnarray}\displaystyle \text{i}\widehat{\text{M}}f_{1}+\widehat{\text{C}}f_{1} & = & \displaystyle \boldsymbol{w}\boldsymbol{\cdot }\unicode[STIX]{x1D735}f_{0}+(mn)^{-1}\unicode[STIX]{x1D735}p\boldsymbol{\cdot }\unicode[STIX]{x2202}_{\boldsymbol{w}}f_{0},\end{eqnarray}$$
(6.4c ) $$\begin{eqnarray}\displaystyle \text{i}\widehat{\text{M}}f_{2}+\widehat{\text{C}}f_{2} & = & \displaystyle \boldsymbol{w}\boldsymbol{\cdot }\unicode[STIX]{x1D735}f_{1}+(mn)^{-1}\unicode[STIX]{x1D735}p\boldsymbol{\cdot }\unicode[STIX]{x2202}_{\boldsymbol{w}}f_{1}\nonumber\\ \displaystyle & & \displaystyle +\,[\unicode[STIX]{x2202}_{t}f_{0}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}f_{0}-\boldsymbol{w}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{u})\boldsymbol{\cdot }\unicode[STIX]{x2202}_{\boldsymbol{w}}f_{0}]-\text{C}[\,f_{1},\overline{f}_{1}].\end{eqnarray}$$
Solution of (6.4a ) leads to the shifted Maxwellian
(6.5) $$\begin{eqnarray}f_{0}=n(2\unicode[STIX]{x03C0}\,v_{\text{t}}^{2})^{-3/2}\text{e}^{-w^{2}/2v_{\text{t}}^{2}},\end{eqnarray}$$

where $n=n(\boldsymbol{x}_{1},t_{1},\ldots )$ and similarly for the  $T$ in $v_{\text{t}}^{2}\doteq T/m$ . Use of the fluid equations gives

(6.6) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}f_{0}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}f_{0}\approx -\frac{2}{3}x^{2}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}-\left(\frac{2}{3}x^{2}-1\right)\frac{1}{p}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q},\end{eqnarray}$$

where $x^{2}\doteq w^{2}/2v_{\text{t}}^{2}$ .

Because Catto & Simakov do not explicitly describe a multiple-scale procedure, the origin of the terms on the right-hand sides of (6.4b ) and (6.4c ) may not be totally clear, so I shall provide a bit more detail. As discussed in appendix  1:A, a systematic discussion of Chapman–Enskog theory includes, in addition to expansion of the distribution function, a multiple-scale expansion in both spaceFootnote 61 and time:

(6.7) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t_{0}}+\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t_{1}}+\unicode[STIX]{x1D716}^{2}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t_{2}}+\cdots \,,\end{eqnarray}$$

and similarly for $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\boldsymbol{x}$ . Here $t_{0}$  is the kinetic time scale, $t_{1}$  is the transit time scale, and $t_{2}$  is the transport time scale; for the physics of those scales, see the discussion after (1:A 2). One then finds (see (1:A 4a)–(1:A 4c))

(6.8a ) $$\begin{eqnarray}\displaystyle \text{i}\widehat{\text{M}}f_{0} & = & \displaystyle -\text{C}[f_{0},\overline{f}_{0}],\end{eqnarray}$$
(6.8b ) $$\begin{eqnarray}\displaystyle \frac{\text{D}f_{0}}{\text{D}t_{1}} & = & \displaystyle -\widehat{\text{C}}f_{1},\end{eqnarray}$$
(6.8c ) $$\begin{eqnarray}\displaystyle \frac{\text{D}f_{0}}{\text{D}t_{2}}+\frac{\text{D}f_{1}}{\text{D}t_{1}} & = & \displaystyle -(\widehat{\text{C}}f_{2}+\text{C}[\,f_{1},\overline{f}_{1}]),\end{eqnarray}$$
which should be compared to the Catto & Simakov equations (6.4) above. Clearly, in those equations one should take $\unicode[STIX]{x1D735}\equiv \unicode[STIX]{x1D735}_{1}$ . In (6.2) written at first order, one identifies $t\equiv t_{1}$ ; the $\unicode[STIX]{x2202}_{t_{1}}f_{0}$ term that is part of the $\text{D}f_{0}/\text{D}t$ in (6.8b ) is eliminated by using (6.2), which ultimately leads to the right-hand side of (6.4b ). The same procedure applied to  $\unicode[STIX]{x2202}_{t_{1}}f_{1}$ leads to the right-hand side of (6.4c ), with the  $t$ in  $\unicode[STIX]{x2202}_{t}\,f_{0}$ being identified as $t\equiv t_{2}$ .

6.2 Comparison of results

Catto & Simakov focus more on obtaining approximate quantitative results (e.g. by means of variational methods) than on the general structure of the transport theory. The presence of a magnetic field complicates the formulas. Magnetic-field-related effects are obviously important for practical applications, as discussed by Catto & Simakov (Reference Catto and Simakov2004, Reference Catto and Simakov2005); some of the effects they calculate are essential for a proper determination of the radial electric field in toroidal devices. In the present paper, my interest is on the general structure of the theory, so I shall not discuss explicit results for magnetic-field corrections at Burnett order. However, a connection to the general unmagnetized formulas may be obtained by examining the result of Catto & Simakov (Reference Catto and Simakov2004) for the parallel viscosity tensor $\unicode[STIX]{x1D745}_{\Vert }$ . It is not hard to see that they obtain only a subset of the terms displayed in the general unmagnetized result (4.6); this is a natural consequence of the subsidiary ordering $\boldsymbol{u}=O(\unicode[STIX]{x1D716})$ . Terms quadratic in  $\boldsymbol{u}$ (the last four lines of (4.11)) are then negligible, being of third order. Furthermore, in this unmagnetized theory one must take the pressure gradient to be $O(\unicode[STIX]{x1D716}^{2})$ , as can be seen from the balance $\unicode[STIX]{x2202}_{t}\boldsymbol{u}\approx -(mn)^{-1}\unicode[STIX]{x1D735}p$ with the ordering $\boldsymbol{u}=O(\unicode[STIX]{x1D716})$ . (Time derivatives are at least of first order.) All Burnett terms involving  $\unicode[STIX]{x1D735}p$ are thus negligible, and (4.11) reduces to

(6.9) $$\begin{eqnarray}\displaystyle (nm)^{-1}\unicode[STIX]{x1D745}^{\text{wc}} & = & \displaystyle -2\unicode[STIX]{x1D707}\left(\frac{1}{2}[\unicode[STIX]{x1D735}\boldsymbol{u}+(\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}]-\frac{1}{d}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}\,\unicode[STIX]{x1D644}\right),\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D707}_{3}\left(\unicode[STIX]{x1D735}\unicode[STIX]{x1D735}T-\frac{1}{d}\unicode[STIX]{x1D6FB}^{2}T\,\unicode[STIX]{x1D644}\right)+\unicode[STIX]{x1D707}_{5}\left(\unicode[STIX]{x1D735}T\,\unicode[STIX]{x1D735}T-\frac{1}{d}|\unicode[STIX]{x1D735}T|^{2}\unicode[STIX]{x1D644}\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D707}$ , $\unicode[STIX]{x1D707}_{3}$ , and  $\unicode[STIX]{x1D707}_{5}$ are given by (4.7a ), (4.8c ) and (4.8e ), respectively. Those coefficients are defined (see §§ 4.1.2 and 4.2) by the integrals  $K^{\text{I}}$ , $K_{1}$ , and  $K_{20}$ , which are expressed in terms of two-time correlation functions involving two phase-space points, as well as by  and  which are expressed in terms of two-time correlation functions involving three phase-space points. This unmagnetized result should be compared to the result of Catto & Simakov,

(6.10) $$\begin{eqnarray}(nm)^{-1}\unicode[STIX]{x1D745}_{\Vert }=(nm)^{-1}(p_{\Vert }-p_{\bot })(\widehat{\boldsymbol{b}}\,\widehat{\boldsymbol{b}}-{\textstyle \frac{1}{3}}\unicode[STIX]{x1D644}),\end{eqnarray}$$

where

(6.11) $$\begin{eqnarray}\displaystyle & & \displaystyle (nm)^{-1}(p_{\Vert }-p_{\bot })\nonumber\\ \displaystyle & & \displaystyle \quad \approx -c_{1}\left(\frac{v_{\text{t}}^{2}}{\unicode[STIX]{x1D708}}\right)\left(\widehat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{u})\boldsymbol{\cdot }\widehat{\boldsymbol{b}}-\frac{1}{3}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}\right)\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\left(\frac{v_{\text{t}}^{2}}{\unicode[STIX]{x1D708}}\right)\frac{1}{p}\left[c_{2}\left(\widehat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{q})\boldsymbol{\cdot }\widehat{\boldsymbol{b}}-\frac{1}{3}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q}\right)+c_{3}\left(\widehat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{q}_{\Vert })\boldsymbol{\cdot }\widehat{\boldsymbol{b}}-\frac{1}{3}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q}_{\Vert }\right)\right]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,c_{4}\left(\frac{v_{\text{t}}^{2}}{\unicode[STIX]{x1D708}}\right)\frac{1}{p}\left(q_{\Vert }\unicode[STIX]{x1D6FB}_{\Vert }\ln p-\frac{1}{3}\boldsymbol{q}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\ln p\right)\nonumber\\ \displaystyle & & \displaystyle \qquad -\,c_{5}\left(\frac{v_{\text{t}}^{2}}{\unicode[STIX]{x1D708}}\right)\left(\frac{q_{\Vert }}{p}\right)\unicode[STIX]{x1D6FB}_{\Vert }\ln T\nonumber\\ \displaystyle & & \displaystyle \qquad -\,c_{6}\left(\frac{q}{p}\right)^{2}+c_{7}\left(\frac{q_{\Vert }}{p}\right)^{2}\end{eqnarray}$$

and the  $c_{i}$ are positive numerical coefficients – for example,

(6.12a,b ) $$\begin{eqnarray}c_{1}\doteq 3\cdot \frac{1025}{1068},\quad c_{7}\doteq \frac{1137}{17\,800}.\end{eqnarray}$$

(Such rational fractions arise from the evaluation of variational forms.) The last two terms of (6.11) arise from the term $\text{C}[\,f_{1},\overline{f}_{1}]$ in (6.4c ). The parallel component of the unmagnetized Navier–Stokes stress (the first line of (6.9)) agrees in form with the first line of (6.11). For the remainder of this discussion, let us examine the parallel physics by replacing in (6.9) $\unicode[STIX]{x1D735}\rightarrow \widehat{\boldsymbol{b}}\unicode[STIX]{x1D6FB}_{\Vert }$ ; then the tensor $\widehat{\boldsymbol{b}}\,\widehat{\boldsymbol{b}}-(1/3)\unicode[STIX]{x1D644}$ emerges, in agreement with (6.10). One anticipates that the  $c_{6}$ and  $c_{7}$ terms in (6.11), which arise from the nonlinear collision operator, are related to the contributions to (6.9) stemming from three-point correlations; I shall discuss this further below.

It is clear that both formalisms will generate the same Navier–Stokes coefficients when the background is assumed to be statistically homogeneous, so I shall focus on establishing consistency for the Burnett coefficients with that same assumption. One must compare

(6.13) $$\begin{eqnarray}{\textstyle \frac{2}{3}}\unicode[STIX]{x1D707}_{3}\unicode[STIX]{x1D6FB}_{\Vert }^{2}T+{\textstyle \frac{2}{3}}\unicode[STIX]{x1D707}_{5}(\unicode[STIX]{x1D6FB}_{\Vert }T)^{2}\quad \text{(Brey)}\end{eqnarray}$$

with

(6.14)

Unfortunately, term by term comparison is not possible because Brey expresses his results solely in terms of gradients of the thermodynamic forces whereas Catto & Simakov use a mixed representation in which gradients of the heat flux appear in addition to gradients of pressure and temperature. Since one has $q_{\Vert }=-\unicode[STIX]{x1D705}_{\Vert }(T)\unicode[STIX]{x1D6FB}_{\Vert }T$ , a contribution to $(\unicode[STIX]{x1D6FB}_{\Vert }T)^{2}$ arises from $\unicode[STIX]{x1D6FB}_{\Vert }\unicode[STIX]{x1D705}$ in addition to the explicit $q_{\Vert }\unicode[STIX]{x1D6FB}_{\Vert }T$ term in (6.14). The overall forms of the results, involving terms in $\unicode[STIX]{x1D6FB}_{\Vert }^{2}T$ and $(\unicode[STIX]{x1D6FB}_{\Vert }T)^{2}$ (having the proper scaling with the dimensional variables), clearly agree, but this is not surprising since it could have been predicted on the basis of symmetry considerations. Quantitative comparison is also not possible because the two-time formulas are formally exact, whereas the results of Catto & Simakov are approximate. However, one can return to the multiple-scale expansion employed by Catto & Simakov, inquire about the formal expressions (involving, for example, $\widehat{\text{C}}^{-1}$ ) that it predicts, and compare those to the predictions of the two-time formalism. That is done for a particular example in the next section.

6.3 Example: the Burnett viscosity coefficient of $\unicode[STIX]{x1D735}T\,\unicode[STIX]{x1D735}T$

In (4.11) for the dissipative momentum flux, it is seen that in Brey’s notation the coefficient of $\unicode[STIX]{x1D735}T\,\unicode[STIX]{x1D735}T$ is called  $\unicode[STIX]{x1D707}_{5}$ . Formula (4.8e ) for  $\unicode[STIX]{x1D707}_{5}$ reduces in the limit of weak coupling to

(6.15) $$\begin{eqnarray}\unicode[STIX]{x1D707}_{5}=4(nmT^{3})^{-1}K_{1}-2(nmT^{2})^{-1}(T^{-2}K_{5}-{\textstyle \frac{5}{2}}T^{-1}K_{3}-{\textstyle \frac{5}{2}}m^{-1}K_{20}).\end{eqnarray}$$

By solving the equations for the relevant two- and three-point correlation functions that determine the integrals  $K_{1}$ and  I shall make this formula more explicit and ultimately compare it successfully with the prediction of the Chapman–Enskog formalism.

6.3.1 The $K_{1}$ integral

The $K_{1}$ integral is defined in terms of the tensor integral (4.15e ), repeated here for convenience:

(6.16) $$\begin{eqnarray}I_{1,ijkl}\doteq \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,\langle \widehat{\unicode[STIX]{x1D70F}}_{ij}(\mathbf{0})\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}})\widehat{J}_{k}^{E}(\overline{\boldsymbol{x}})\rangle _{0}\overline{x}_{l}=K_{1}(\unicode[STIX]{x1D6FF}_{ik}\unicode[STIX]{x1D6FF}_{jl}+\unicode[STIX]{x1D6FF}_{il}\unicode[STIX]{x1D6FF}_{jk})+K_{2}\unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x1D6FF}_{kl}.\end{eqnarray}$$

It was shown in § 3.2.1 how to express the required expectations in terms of two- or three-point correlation functions. With $\overline{\boldsymbol{x}}=-\overline{\unicode[STIX]{x1D746}}$ , one has

(6.17a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D644}_{1} & \doteq & \displaystyle -\int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D746}}\,\langle \widehat{\unicode[STIX]{x1D749}}(\mathbf{0})\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}})\widehat{\boldsymbol{J}}\text{}^{E}(-\overline{\unicode[STIX]{x1D746}})\rangle _{0}\overline{\unicode[STIX]{x1D746}}\end{eqnarray}$$
(6.17b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle -T\int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D746}}\displaystyle \int \text{d}\boldsymbol{v}\displaystyle \int \text{d}\boldsymbol{v}s_{+}\prime \,\widehat{\unicode[STIX]{x1D749}}(\boldsymbol{v})C_{+}^{(1;1)}(\overline{\unicode[STIX]{x1D746}},\boldsymbol{v},\unicode[STIX]{x1D70F},\boldsymbol{v}s_{+}\prime )\unicode[STIX]{x1D737}(\boldsymbol{v}s_{+}\prime )\,\overline{\unicode[STIX]{x1D746}}\end{eqnarray}$$
(6.17c ) $$\begin{eqnarray}\displaystyle & = & \displaystyle T\int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\boldsymbol{v}\displaystyle \int \text{d}\boldsymbol{v}s_{+}\prime \,\widehat{\unicode[STIX]{x1D749}}(\boldsymbol{v})\unicode[STIX]{x1D737}(\boldsymbol{v}s_{+}\prime )\frac{\unicode[STIX]{x2202}C_{+;\boldsymbol{k}s_{+}\prime }^{(1;1)}(\boldsymbol{v},\overline{\unicode[STIX]{x1D70F}};\boldsymbol{v}s_{+}\prime )}{\unicode[STIX]{x2202}(-\text{i}\boldsymbol{k}s_{+}\prime )}\bigg|_{\boldsymbol{k}s_{+}\prime =\mathbf{0}},\end{eqnarray}$$
where
(6.18) $$\begin{eqnarray}\unicode[STIX]{x1D737}(\boldsymbol{v})\equiv \widehat{\boldsymbol{J}}\text{}^{E}(\boldsymbol{v})\doteq \left(\frac{mv^{2}/2}{T}-\frac{5}{2}\right)\boldsymbol{v}.\end{eqnarray}$$

The two-time correlation function is governed by (5.27), where the last term of that equation is replaced by $-\widehat{\text{C}}C_{+;\boldsymbol{k}s_{+}\prime }^{(1;1)}$ . The solution is

(6.19) $$\begin{eqnarray}C_{+;\boldsymbol{k}s_{+}\prime }^{(1;1)}(\overline{\unicode[STIX]{x1D70F}})=\text{e}^{-(\text{i}\boldsymbol{k}s_{+}\prime \boldsymbol{\cdot }\boldsymbol{v}+\widehat{\text{C}})\overline{\unicode[STIX]{x1D70F}}}\,C_{\boldsymbol{ k}s_{+}\prime }^{(0,2)}(\boldsymbol{v},\overline{\boldsymbol{v}}).\end{eqnarray}$$

The initial condition contains a singular term, displayed in (5.36), plus a term in $g_{\boldsymbol{k}s_{+}\prime }(\boldsymbol{v},\overline{\boldsymbol{v}})$ that can be shown to phase-mix away. The $\overline{\boldsymbol{v}}$ integration can be performed over the singular term. The $\overline{\unicode[STIX]{x1D70F}}$  integration can be done, leading to the operator $(\text{i}\boldsymbol{k}s_{+}\prime \boldsymbol{\cdot }\boldsymbol{v}+\widehat{\text{C}})^{-1}$ . The operator relation $\text{d}(\text{A}^{-1})=-\text{A}^{-1}(\text{d}\text{A})\text{A}^{-1}$ allows the $\boldsymbol{k}s_{+}\prime$  derivative to be evaluated. The final result is the velocity-space matrix element

(6.20) $$\begin{eqnarray}\unicode[STIX]{x1D644}_{1}=-T\langle \widehat{\unicode[STIX]{x1D749}}(\boldsymbol{v})\mid \widehat{\text{C}}^{-1}\boldsymbol{v}\widehat{\text{C}}^{-1}\mid \unicode[STIX]{x1D737}(\boldsymbol{v})\rangle _{0}.\end{eqnarray}$$

6.3.2 Three-point correlations and the nonlinear collision operator

I now turn to the evaluation of the remaining terms in (6.15), namely

(6.21)

From (4.15f ), (4.15g ), and (4.15l ), one has

(6.22) $$\begin{eqnarray}\displaystyle & & \displaystyle \left(T^{-2}K_{5}-\frac{5}{2}T^{-1}K_{3}-\frac{5}{2}m^{-1}K_{20}\right)(\unicode[STIX]{x1D6FF}_{ik}\unicode[STIX]{x1D6FF}_{jl}+\unicode[STIX]{x1D6FF}_{il}\unicode[STIX]{x1D6FF}_{jk})\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\left(T^{-2}K_{6}-\frac{5}{2}T^{-1}K_{4}-\frac{5}{2}m^{-1}K_{21}\right)\unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x1D6FF}_{kl}\nonumber\\ \displaystyle & & \displaystyle \quad =\int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,\left\langle \widehat{\unicode[STIX]{x1D70F}}_{ij}(\mathbf{0})\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}})(T^{-1}\widehat{\mathscr{J}}_{k}^{E})\left(\frac{1}{T}Es_{+}\prime (\overline{\boldsymbol{x}})-\frac{5}{2}Ns_{+}\prime (\overline{\boldsymbol{x}})\right)\right\rangle _{0}\overline{x}_{l}\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\frac{5}{2}\int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,\langle \widehat{\unicode[STIX]{x1D70F}}_{ij}(\mathbf{0})\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}})m^{-1}P_{k}(\overline{\boldsymbol{x}})\rangle _{0}\overline{x}_{l}.\end{eqnarray}$$

First consider the last integral, which stems from  $K_{20}$ . The same sequence of operations that led to (6.20) produces

(6.23) $$\begin{eqnarray}\unicode[STIX]{x1D644}_{20}=-{\textstyle \frac{5}{2}}\langle \widehat{\unicode[STIX]{x1D749}}(\boldsymbol{v})\mid \text{C}^{-1}\boldsymbol{v}\text{C}^{-1}\mid \boldsymbol{v}\rangle _{0}.\end{eqnarray}$$

Unfortunately, this integral is infinite for a one-component plasma since $|\boldsymbol{v}\!\rangle$ is in the null space of  $\widehat{\text{C}}$ . Upon inquiring into the origin of this term, one learns that it stems from the expectation $\unicode[STIX]{x1D65D}_{2}^{ee}[\widehat{\unicode[STIX]{x1D749}}](\unicode[STIX]{x1D707},t)$ (see (3.39d )), which contains a $\text{Q}$  operator. Brey’s integrals  and $K_{20}$ arise by writing $\text{Q}=1-\text{P}$ arises from the 1, and  $K_{20}$  arises from the $\text{P}$  operation. The role of  $\text{Q}$ is precisely to prevent such infinities from occurring; thus, one expects that  will contain a singular part that will be exactly cancelled by  $K_{20}$ , and this will be seen to be true.

Now consider the first line of (6.22),

(6.24) $$\begin{eqnarray}\unicode[STIX]{x1D644}_{5-3}\doteq -\int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D746}}\,\left\langle \widehat{\unicode[STIX]{x1D749}}(\mathbf{0})\text{R}_{0}(\overline{\unicode[STIX]{x1D70F}})\left(\frac{1}{T}\widehat{\pmb{\pmb{\mathscr{J}}}}^{E}\right)\left(\frac{1}{T}Es_{+}\prime (-\overline{\unicode[STIX]{x1D746}})-\frac{5}{2}Ns_{+}\prime (-\overline{\unicode[STIX]{x1D746}})\right)\right\rangle _{0}\overline{\unicode[STIX]{x1D746}},\end{eqnarray}$$

which can be expressed in terms of the triplet correlation function $C_{+;\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}$ . The presence of  $\widehat{\pmb{\pmb{\mathscr{J}}}}^{E}$ (the total amount of  $\widehat{\boldsymbol{J}}\text{}^{E}$ ) in this expression in (6.24) means that only the $\boldsymbol{k}s_{+}\prime =\mathbf{0}$ limit is required, and the $\overline{\unicode[STIX]{x1D746}}$  integral requires only $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}(-\text{i}\boldsymbol{k}s_{+}\prime \prime )|_{\boldsymbol{k}s_{+}\prime \prime =\mathbf{0}}$ . It was shown in § G.3 that (5.45) for the triplet correlation function  $C_{+}^{(1;2)}$ becomes approximately (5.49). The solution of (5.49) (which contains an inhomogeneous, $\unicode[STIX]{x1D70F}$ -dependent source term) is

(6.25) $$\begin{eqnarray}\displaystyle & & \displaystyle C_{+;\mathbf{0},\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )=\text{e}^{-(\text{i}\boldsymbol{k}s_{+}\prime \prime \boldsymbol{\cdot }\boldsymbol{v}+\widehat{\text{C}})\unicode[STIX]{x1D70F}}C_{\mathbf{0},\boldsymbol{k}s_{+}\prime \prime }^{(0,3)}(\text{}\underline{1},\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )\nonumber\\ \displaystyle & & \displaystyle \quad -\,\displaystyle \int _{0}^{\unicode[STIX]{x1D70F}}\!\text{d}\widehat{\unicode[STIX]{x1D70F}}\,\text{e}^{-(\text{i}\boldsymbol{k}s_{+}\prime \prime \boldsymbol{\cdot }\boldsymbol{v}+\widehat{\text{C}})(\unicode[STIX]{x1D70F}-\widehat{\unicode[STIX]{x1D70F}})}\{\text{C}[f;C_{+;\mathbf{0}}^{(1;1)}(\widehat{\unicode[STIX]{x1D70F}};\text{}\underline{1}s_{+}\prime ),\overline{C}_{+;\boldsymbol{k}s_{+}\prime \prime }^{(1;1)}(\widehat{\unicode[STIX]{x1D70F}};\text{}\underline{1}s_{+}\prime \prime )]+(1s_{+}\prime \Leftrightarrow 1s_{+}\prime \prime )\}.\end{eqnarray}$$

(For brevity, I shall drop the BL label on $\text{C}[\,f;a,\overline{b}]$ .) In addition to the  $\unicode[STIX]{x1D737}(\boldsymbol{v})$ defined by (6.18), let us define $\unicode[STIX]{x1D6FE}(\boldsymbol{v})\doteq {\textstyle \frac{1}{2}}mv^{2}/T-{\textstyle \frac{5}{2}}$ ; one has $\unicode[STIX]{x1D737}=\unicode[STIX]{x1D6FE}\boldsymbol{v}$ . To evaluate  $\unicode[STIX]{x1D644}_{5-3}$ , the solution (6.25) must be multiplied by  $\unicode[STIX]{x1D737}(\boldsymbol{v}s_{+}\prime )$ and  $\unicode[STIX]{x1D6FE}(\boldsymbol{v}s_{+}\prime \prime )$ ; integrated over  $\unicode[STIX]{x1D70F}$ , $\boldsymbol{v}$ , $\boldsymbol{v}s_{+}\prime$ , and  $\boldsymbol{v}s_{+}\prime \prime$ ; differentiated with respect to  $-\text{i}\boldsymbol{k}s_{+}\prime \prime$ ; and evaluated at $\boldsymbol{k}s_{+}\prime \prime =\mathbf{0}$ . Performing those operations first on the singular initial-condition term leadsFootnote 62 to the matrix element

(6.26) $$\begin{eqnarray}-\!\langle \widehat{\unicode[STIX]{x1D749}}\!|\widehat{\text{C}}^{-1}\boldsymbol{v}\widehat{\text{C}}^{-1} |\!\unicode[STIX]{x1D737}\unicode[STIX]{x1D6FE}\rangle _{0}.\end{eqnarray}$$

Write $\unicode[STIX]{x1D737}\unicode[STIX]{x1D6FE}=(\text{P}+\text{Q})\unicode[STIX]{x1D737}\unicode[STIX]{x1D6FE}$ . By symmetry, only the momentum projection contributes. One has

(6.27) $$\begin{eqnarray}\text{P}|\unicode[STIX]{x1D737}\unicode[STIX]{x1D6FE}\!\rangle \rightarrow |\boldsymbol{v}\!\rangle \frac{1}{v_{\text{t}}^{2}}\boldsymbol{\cdot }\langle \boldsymbol{v}\,\unicode[STIX]{x1D737}\unicode[STIX]{x1D6FE}\rangle =\frac{5}{2}|\boldsymbol{v}\!\rangle \!.\end{eqnarray}$$

This contribution is seen to exactly cancel  $\unicode[STIX]{x1D644}_{20}$ , as was predicted.Footnote 63 For future use, note that

(6.28) $$\begin{eqnarray}\text{Q}|\unicode[STIX]{x1D737}\unicode[STIX]{x1D6FE}\!\rangle =|\unicode[STIX]{x1D737}\unicode[STIX]{x1D6FE}\!\rangle -\text{P}|\unicode[STIX]{x1D737}\unicode[STIX]{x1D6FE}\!\rangle =(\unicode[STIX]{x1D6FE}^{2}-{\textstyle \frac{5}{2}})|\boldsymbol{v}\!\rangle \!.\end{eqnarray}$$

Now consider the processing of the source term, namely, the second line of (6.25). The time integral from 0 to  $\infty$ of a one-sided function is its $\unicode[STIX]{x1D714}=0$ Fourier component. Because the source term is in convolution form, its Fourier transform is the product of the individual Fourier transforms of $\exp [-\text{i}(\boldsymbol{k}s_{+}\prime \prime \boldsymbol{\cdot }\boldsymbol{v}+\widehat{\text{C}})\unicode[STIX]{x1D70F}]$ and $\text{C}[\,f;Cs_{+}\prime (\unicode[STIX]{x1D70F}),\overline{C}s_{+}\prime \prime (\unicode[STIX]{x1D70F})]$ . Because $\text{C}[\,f;a,\overline{b}]$ is multiplicative in its last two slots (involving $a(\boldsymbol{v})b(\overline{\boldsymbol{v}})$ , where $\overline{\boldsymbol{v}}$  is the integration variable in the collision operator), that $\unicode[STIX]{x1D70F}$ dependence is $\exp [-(\widehat{\text{C}}+\text{i}\boldsymbol{k}s_{+}\prime \prime \boldsymbol{\cdot }\overline{\boldsymbol{v}}+\overline{\widehat{\text{C}}})\unicode[STIX]{x1D70F}]+(\boldsymbol{v}\Leftrightarrow \overline{\boldsymbol{v}})$ . Therefore, the collisional contribution is

(6.29) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D644}_{5-3}^{\text{C}} & = & \displaystyle \displaystyle \int \text{d}\boldsymbol{v}\,\widehat{\unicode[STIX]{x1D749}}(\boldsymbol{v})(\,[\text{i}(\boldsymbol{k}s_{+}\prime \prime \boldsymbol{\cdot }\boldsymbol{v}+\widehat{\text{C}})]^{-1}\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\{\text{C}[f;(\widehat{\text{C}}+\text{i}\boldsymbol{k}s_{+}\prime \prime \boldsymbol{\cdot }\overline{\boldsymbol{v}}+\overline{\widehat{\text{C}}})^{-1}\,|\,\unicode[STIX]{x1D737}f,\overline{\unicode[STIX]{x1D6FE}}\overline{f}]+(\boldsymbol{v}\Leftrightarrow \overline{\boldsymbol{v}})\})\frac{\overleftarrow{\unicode[STIX]{x2202}}}{\unicode[STIX]{x2202}(-\text{i}\boldsymbol{ k}s_{+}\prime \prime )}\!\!\left.\vphantom{\left(\frac{1}{2}\right)}\right|_{\boldsymbol{k}s_{+}\prime \prime =\mathbf{0}}.\end{eqnarray}$$

Here I have introduced the notation $\text{C}[\,f;\unicode[STIX]{x1D706}\,|\,a,\overline{b}]$ to indicate that the scaling factor  $\unicode[STIX]{x1D706}$ operates on the product $a(\boldsymbol{v})b(\overline{\boldsymbol{v}})$ . The $\boldsymbol{k}s_{+}\prime \prime$  derivativeFootnote 64 generates two contributions, one from each of the operator inverses. I shall consider each in turn.

(i) Differentiating $\text{C}[\,f;\unicode[STIX]{x1D706}(\boldsymbol{k}s_{+}\prime \prime )\,|\,a,\overline{b}]$ : Upon performing the $\boldsymbol{k}s_{+}\prime \prime$  derivative on the second line of (6.29), one is led to the integrand

(6.30) $$\begin{eqnarray}I_{kl}\doteq (\widehat{\text{C}}+\overline{\widehat{\text{C}}})^{-1}[\overline{v}_{l}(\widehat{\text{C}}+\overline{\widehat{\text{C}}})^{-1}\unicode[STIX]{x1D6FD}_{k}\,f\overline{\unicode[STIX]{x1D6FE}}\overline{f}+(\boldsymbol{v}\Leftrightarrow \overline{\boldsymbol{v}})].\end{eqnarray}$$

Because $|\overline{\unicode[STIX]{x1D6FE}}\rangle$  is a null eigenfunction of  $\overline{\widehat{\text{C}}}$ , $I_{kl}$ reduces to

(6.31) $$\begin{eqnarray}I_{kl}=(\widehat{\text{C}}+\overline{\widehat{\text{C}}})^{-1}[(\widehat{\text{C}}^{-1}\unicode[STIX]{x1D6FD}_{k}\,f)(\overline{\unicode[STIX]{x1D6FD}}_{l}\,\overline{f})+(\boldsymbol{v}\Leftrightarrow \overline{\boldsymbol{v}})].\end{eqnarray}$$

Now use the identity

(6.32) $$\begin{eqnarray}(\widehat{\text{C}}+\overline{\widehat{\text{C}}})^{-1}=\overline{\widehat{\text{C}}}^{\,-1}-(\widehat{\text{C}}+\overline{\widehat{\text{C}}})^{-1}\widehat{\text{C}}\overline{\widehat{\text{C}}}^{\,-1}\end{eqnarray}$$

to find that

(6.33) $$\begin{eqnarray}I_{kl}=(\widehat{\text{C}}^{-1}\unicode[STIX]{x1D6FD}_{k}\,f)(\overline{\widehat{\text{C}}}^{-1}\overline{\unicode[STIX]{x1D6FD}}_{l}\,\overline{f})-(\widehat{\text{C}}+\overline{\widehat{\text{C}}})^{-1}[\unicode[STIX]{x1D6FD}_{k}\,f(\overline{\widehat{\text{C}}}^{-1}\overline{\unicode[STIX]{x1D6FD}}_{l}\,\overline{f})]+(\boldsymbol{v}\Leftrightarrow \overline{\boldsymbol{v}}).\end{eqnarray}$$

The construction

(6.34) $$\begin{eqnarray}|\unicode[STIX]{x1D73B}\!\rangle \doteq -\widehat{\text{C}}^{-1}|\unicode[STIX]{x1D737}\!\rangle\end{eqnarray}$$

can be recognized as the velocity-dependent part of the first-order distribution function driven by temperature gradients; see (1:A 21). Equation (6.33) thus reduces to

(6.35) $$\begin{eqnarray}I_{kl}=2\unicode[STIX]{x1D701}_{k}\unicode[STIX]{x1D701}_{l}-I_{lk},\end{eqnarray}$$

which determines the symmetric part of  $I_{kl}$ to be

(6.36) $$\begin{eqnarray}S_{kl}\doteq {\textstyle \frac{1}{2}}(I_{kl}+I_{lk})=\unicode[STIX]{x1D701}_{k}\unicode[STIX]{x1D701}_{l}.\end{eqnarray}$$

Only the symmetric part contributes to  $\unicode[STIX]{x1D644}_{5-3}^{\text{C}}$ . Thus, when (6.36) is inserted into (6.29), one obtains

(6.37) $$\begin{eqnarray}_{a}\unicode[STIX]{x1D644}_{5-3}^{\text{C}}=\displaystyle \int \text{d}\boldsymbol{v}\,\widehat{\unicode[STIX]{x1D749}}(\boldsymbol{v})\widehat{\text{C}}^{-1}\text{C}[\,f;\unicode[STIX]{x1D73B},\overline{\unicode[STIX]{x1D73B}}].\end{eqnarray}$$

(ii) Differentiating $[\text{i}(\boldsymbol{k}s_{+}\prime \prime \boldsymbol{\cdot }\boldsymbol{v}+\widehat{\text{C}})]^{-1}\text{C}$ : Upon differentiating the first term of (6.29), one is led to the contribution

(6.38a ) $$\begin{eqnarray}\displaystyle ~_{b}\unicode[STIX]{x1D644}_{5-3}^{\text{C}} & \doteq & \displaystyle \displaystyle \int \text{d}\boldsymbol{v}\,\widehat{\unicode[STIX]{x1D749}}(\boldsymbol{v})\widehat{\text{C}}^{-1}\boldsymbol{v}\widehat{\text{C}}^{-1}\{\text{C}[\,f;(\widehat{\text{C}}+\overline{\widehat{\text{C}}})^{-1}\,|\,\unicode[STIX]{x1D737}f,\overline{\unicode[STIX]{x1D6FE}}\overline{f}]+(\boldsymbol{v}\Leftrightarrow \overline{\boldsymbol{v}})\}\end{eqnarray}$$
(6.38b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle -\displaystyle \int \text{d}\boldsymbol{v}\,\widehat{\unicode[STIX]{x1D749}}(\boldsymbol{v})\widehat{\text{C}}^{-1}\boldsymbol{v}\widehat{\text{C}}^{-1}\{\text{C}[\,f;\unicode[STIX]{x1D73B},\overline{\unicode[STIX]{x1D6FE}}\overline{f}]+\text{C}[\,f;\unicode[STIX]{x1D6FE}f,\overline{\unicode[STIX]{x1D73B}}]\}.\end{eqnarray}$$
(Recall that $\unicode[STIX]{x1D73B}$ was defined by (6.34).) It is useful to manipulate this result as follows. Define $\unicode[STIX]{x1D6FC}(\boldsymbol{v})\doteq {\textstyle \frac{1}{2}}mv^{2}/T$ , then add and subtract a factor of  ${\textstyle \frac{1}{2}}$ to each of the $\unicode[STIX]{x1D6FE}$  factors (note that $\unicode[STIX]{x1D6FE}+{\textstyle \frac{1}{2}}=\unicode[STIX]{x1D6FC}-{\textstyle \frac{3}{2}}$ ):
(6.39) $$\begin{eqnarray}\displaystyle \text{C}[f;\unicode[STIX]{x1D73B},\overline{\unicode[STIX]{x1D6FE}}\overline{f}]+\text{C}[\,f;\unicode[STIX]{x1D6FE}f,\overline{\unicode[STIX]{x1D73B}}] & = & \displaystyle \text{C}[\,f;\unicode[STIX]{x1D73B},(\overline{\unicode[STIX]{x1D6FC}}-{\textstyle \frac{3}{2}})\overline{f}]+\text{C}[\,f;(\unicode[STIX]{x1D6FC}-{\textstyle \frac{3}{2}})f,\overline{\unicode[STIX]{x1D73B}}]\nonumber\\ \displaystyle & & \displaystyle -\,{\textstyle \frac{1}{2}}(\text{C}[\,f;\widehat{\text{C}}^{-1}\unicode[STIX]{x1D737}f,\overline{f}]+\text{C}[\,f;f,\widehat{\text{C}}^{-1}\overline{\unicode[STIX]{x1D737}}\,\overline{f}]).\end{eqnarray}$$

The expression in the second line can be recognized as $-{\textstyle \frac{1}{2}}\widehat{\text{C}}(\widehat{\text{C}}^{-1}|\unicode[STIX]{x1D737}\!\rangle )=-{\textstyle \frac{1}{2}}|\unicode[STIX]{x1D737}\!\rangle$ . When this term is inserted into (6.38b ), it produces a term that cancels half of the matrix element  $\unicode[STIX]{x1D644}_{1}$ , formula (6.20).

I shall now summarize these results in a form that can be directly compared with the predictions of Chapman–Enskog theory, which are obtained in the next section. Upon taking into account the coefficients in (6.15) and the 50 % cancellation noted above (reflected by the second factor of  ${\textstyle \frac{1}{2}}$ in the next equation), one needs

(6.40) $$\begin{eqnarray}\displaystyle & & \displaystyle {\textstyle \frac{1}{2}}\left[4\left({\textstyle \frac{1}{2}}\right)T^{-1}\unicode[STIX]{x1D644}_{1}-(\unicode[STIX]{x1D644}_{5-3}+\unicode[STIX]{x1D644}_{20})\right]=\underbrace{-\langle \widehat{\unicode[STIX]{x1D749}}|\widehat{\text{C}}^{-1}\boldsymbol{v}\widehat{\text{C}}^{-1}|\unicode[STIX]{x1D737}\rangle }_{\text{(a)}}+\underbrace{\langle \widehat{\unicode[STIX]{x1D749}}|\widehat{\text{C}}^{-1}\boldsymbol{v}\widehat{\text{C}}^{-1}|(\unicode[STIX]{x1D6FE}^{2}-{\textstyle \frac{5}{2}})\boldsymbol{v}\rangle }_{\text{(b)}}\nonumber\\ \displaystyle & & \displaystyle \quad +\underbrace{\langle \widehat{\unicode[STIX]{x1D749}}|\widehat{\text{C}}^{-1}\boldsymbol{v}\widehat{\text{C}}^{-1}|f^{-1}\{\text{C}[\,f;\unicode[STIX]{x1D73B},(\overline{\unicode[STIX]{x1D6FC}}-{\textstyle \frac{3}{2}})\overline{f}]+\text{C}[\,f;(\unicode[STIX]{x1D6FC}-{\textstyle \frac{3}{2}})f,\overline{\unicode[STIX]{x1D73B}}]\}\rangle }_{\text{(c)}}\nonumber\\ \displaystyle & & \displaystyle \quad \underbrace{-\;\langle \widehat{\unicode[STIX]{x1D749}}|\widehat{\text{C}}^{-1}|f^{-1}\text{C}[\,f;\unicode[STIX]{x1D73B},\overline{\unicode[STIX]{x1D73B}}]\rangle }_{\text{(d)}}.\end{eqnarray}$$

These tensors are to be contracted with $T^{-2}\unicode[STIX]{x1D735}T\,\unicode[STIX]{x1D735}T$ . By symmetry, that introduces a factor of 2, which is cancelled by the first factor of  ${\textstyle \frac{1}{2}}$ on the left-hand side of (6.40).

6.4 Explicit formulas from Chapman–Enskog theory

One may now compare these predictions from the two-time theory to those that follow from standard Chapman–Enskog theory applied to the Landau kinetic equation. One must solve (6.8c ),

(6.41) $$\begin{eqnarray}\widehat{\text{C}}f_{2}=-\frac{\text{D}f_{0}}{\text{D}t_{2}}-\frac{\text{D}f_{1}}{\text{D}t_{1}}-\text{C}[\,f_{1},\overline{f}_{1}].\end{eqnarray}$$

The solvability conditions for this equation were discussed in appendix  1:A; they amount to requiring that the equations are satisfied at Navier–Stokes order. The term in $\text{D}f_{0}/\text{D}t_{2}$ merely leads to second-order multiple-scale corrections to the first-order equation, so I focus on the last two terms. One has

(6.42) $$\begin{eqnarray}f_{1}=-\widehat{\text{C}}^{-1}\left[\frac{1}{T}\left(\unicode[STIX]{x1D6FC}-\frac{5}{2}\right)\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}Tf_{0}\right]\!.\end{eqnarray}$$

I shall evaluate $-\text{D}f_{1}/\text{D}t_{1}$ using the small-flow ordering of Catto & Simakov. Then $\unicode[STIX]{x2202}_{t}\,f_{1}$ is negligible to second order, as is the term in  $\boldsymbol{E}\boldsymbol{\cdot }\boldsymbol{\unicode[STIX]{x2202}}$ . Upon noting that $\unicode[STIX]{x2202}_{T}\unicode[STIX]{x1D6FC}=-T^{-1}\unicode[STIX]{x1D6FC}$ and that $\unicode[STIX]{x1D735}\ln f_{0}=(\unicode[STIX]{x1D6FC}-{\textstyle \frac{3}{2}})T^{-1}\unicode[STIX]{x1D735}T$ when only temperature gradients are considered, one has

(6.43) $$\begin{eqnarray}\displaystyle -\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}f_{1} & = & \displaystyle \frac{1}{T^{2}}\boldsymbol{v}\boldsymbol{\cdot }\widehat{\text{C}}^{-1}\left\{\vphantom{\left(\frac{1}{2}\right)}\right.\!\!\left[\vphantom{\left(\frac{1}{2}\right)}\right.\!\underbrace{-\left(\unicode[STIX]{x1D6FC}-\frac{5}{2}\right)}_{\text{(a)}}f_{0}\underbrace{-\unicode[STIX]{x1D6FC}+\left(\unicode[STIX]{x1D6FC}-\frac{5}{2}\right)\left(\unicode[STIX]{x1D6FC}-\frac{3}{2}\right)}_{\text{(b)}}\!\left.\vphantom{\left(\frac{1}{2}\right)}\right]\boldsymbol{v}f_{0}\!\!\left.\vphantom{\left(\frac{1}{2}\right)}\right\}\boldsymbol{\cdot }\unicode[STIX]{x1D735}T\,\unicode[STIX]{x1D735}T\nonumber\\ \displaystyle & & \displaystyle +\,\underbrace{\boldsymbol{v}\widehat{\text{C}}^{-1}(\unicode[STIX]{x1D735}\widehat{\text{C}})f_{1}\boldsymbol{\cdot }\unicode[STIX]{x1D735}T}_{\text{(c)}}+\,O(\unicode[STIX]{x1D6FB}^{2}T).\end{eqnarray}$$

Note that it was necessary to differentiate the collision operator because it is a functional of  $f_{0}$ .Footnote 65 I shall not calculate the term in $\unicode[STIX]{x1D6FB}^{2}T$ because in this example I am restricting my attention to just the coefficient of $(\unicode[STIX]{x1D735}T)^{2}$ .

Upon noting that the previously defined  $\unicode[STIX]{x1D737}(\boldsymbol{v})$ is equal to $[\unicode[STIX]{x1D6FC}(\boldsymbol{v})-\frac{5}{2}]\boldsymbol{v}$ , one finds that the second-order perturbation driven by term (a) is

(6.44) $$\begin{eqnarray}|_{\text{a}}\unicode[STIX]{x1D712}_{2}\!\rangle =-T^{-2}\widehat{\text{C}}^{-1}\boldsymbol{v}\widehat{\text{C}}^{-1}|\unicode[STIX]{x1D737}\!\rangle \boldsymbol{ : }\unicode[STIX]{x1D735}T\,\unicode[STIX]{x1D735}T.\end{eqnarray}$$

Term (b) can be rewritten more conveniently as

(6.45) $$\begin{eqnarray}\text{(b)}=(\unicode[STIX]{x1D6FC}-{\textstyle \frac{5}{2}})^{2}-{\textstyle \frac{5}{2}}=(\unicode[STIX]{x1D6FE}^{2}-{\textstyle \frac{5}{2}}),\end{eqnarray}$$

which gives rise to the contribution

(6.46) $$\begin{eqnarray}|_{\text{b}}\unicode[STIX]{x1D712}_{2}\!\rangle =\frac{1}{T^{2}}\widehat{\text{C}}^{-1}\boldsymbol{v}\widehat{\text{C}}^{-1}\left|\left(\unicode[STIX]{x1D6FE}^{2}-\frac{5}{2}\right)\boldsymbol{v}\right\rangle \boldsymbol{ : }\unicode[STIX]{x1D735}T\,\unicode[STIX]{x1D735}T.\end{eqnarray}$$

Term (c) can be reduced by explicitly performing the gradient operation on  $f_{0}$ to find

(6.47) $$\begin{eqnarray}(\unicode[STIX]{x1D735}\widehat{\text{C}})f_{1}=T^{-1}\{\text{C}[(\unicode[STIX]{x1D6FC}-{\textstyle \frac{3}{2}})f_{0},\overline{\unicode[STIX]{x1D73B}}]+\text{C}[\unicode[STIX]{x1D73B},(\overline{\unicode[STIX]{x1D6FC}}-{\textstyle \frac{3}{2}})\overline{f}]\}\boldsymbol{\cdot }\unicode[STIX]{x1D735}T.\end{eqnarray}$$

Finally, the nonlinear collision operator drives the correction

(6.48) $$\begin{eqnarray}|_{\text{d}}\unicode[STIX]{x1D712}_{2}\!\rangle =-\widehat{\text{C}}^{-1}|f_{0}^{-1}\text{C}[\,f_{1},\overline{f}_{1}]\!\rangle \!.\end{eqnarray}$$

It can now be seen that when one forms the matrix element $\langle \widehat{\unicode[STIX]{x1D749}}|\unicode[STIX]{x1D712}_{2}\rangle$ , each of the contributions (a)–(d) matches the corresponding ones of (6.40), which summarizes the predictions of the two-time theory for the coefficient of $T^{-2}\unicode[STIX]{x1D735}T\,\unicode[STIX]{x1D735}T$ . Thus, we have obtained agreement between the two formalisms at least for the coefficients of $\unicode[STIX]{x1D735}T\,\unicode[STIX]{x1D735}T$ . It should be clear on conceptual grounds that if one is careful – very careful – agreement for the other myriad of Burnett coefficients will follow as well.

It is important to note that agreement has been found only when the Landau collision operator is used. I point out in § G.3.2 that in the more complete Balescu–Lenard theory arises a second-order cross-term that involves fluctuations in the dielectric shielding. Formally, that effect is of the same order as that driven by $\text{C}[\,f_{1},\overline{f}_{1}]$ . It could easily be incorporated into the Chapman–Enskog expansion by adding to the right-hand side of (6.41) the last line of (G 47). Calculation of the analogous effect in two-time theory involves a tedious evaluation of the contributions of the last four lines of (G 46), as discussed in § G.3; that is left as an exercise for the future. At the very least, such an effect will quantitatively influence the values of some of the Burnett coefficients; whether it has a more interesting qualitative significance remains to be understood.

7 Summary and discussion

The principal contributions and limitations of this paper are as follows:

  1. (1) An introduction to Burnett effects was given.

    1. (i) A physical picture of one particular mechanism, arising from unbalanced viscous forces in the presence of a temperature gradient, that contributes to the coefficient of $\unicode[STIX]{x1D735}T\,\unicode[STIX]{x1D735}T$ in the momentum equation was described.

    2. (ii) It was emphasized that the Burnett effects arise from gradient-induced symmetry breaking that lead to non-Gaussian statistics.

    3. (iii) Burnett effects arise in contexts more general than many-body theory. In appendix  A I describe a simple stochastic model (containing no discrete-particle effects) that exhibits Burnett effects due to a certain kind of non-Gaussian statistics; it demonstrates the role of symmetry breaking. Knowledge of that appendix is not necessary in order to appreciate the body of the paper, but perhaps it adds some additional useful perspectives.

  2. (2) The time-independent projection-operator formalism of Brey et al. (Reference Brey, Zwanzig and Dorfman1981) was extended to include multiple species and a magnetic field.

    1. (i) In its general form, this formalism produces formulas for the regular parts of the first- and second-order transport coefficients valid for arbitrarily strong coupling.

    2. (ii) First-order momentum exchange between species was expressed as a two-time correlation, and an algebraically non-trivial calculation was done to demonstrate the equivalence of the hydrodynamic part of that formula to the corresponding one derived by Braginskii.

    3. (iii) Although it was pointed out where Burnett-level exchange effects arise in the formalism, specific formulas for those effects were not displayed.

    4. (iv) Also not displayed in detail were the additional Burnett-level formulas for perpendicular transport coefficients in the magnetized limit. (It is clear how to proceed; one must merely relax the symmetry assumptions that hold for $\boldsymbol{B}^{\text{ext}}=\mathbf{0}$ . In the usual limit of small  $\unicode[STIX]{x1D708}/\unicode[STIX]{x1D714}_{\text{c}}$ , straightforward perturbation theory can be done. However, the effects are very small in that limit.)

  3. (3) A formalism appropriate for evaluating the relevant two-time correlation functions, involving either two or three phase-space points, in the weakly coupled limit was described.

  4. (4) Solutions of the two-time equations (based on homogeneous background statistics) were used to evaluate a representative Burnett parallel viscosity, and agreement with one-time Chapman–Enskog theory (applied to the Landau kinetic equation) was obtained.

    1. (i) It was shown how the effects calculated from the Chapman–Enskog expansion employed by Catto & Simakov arise in the two-time formalism.

    2. (ii) In particular, nonlinear noise terms are responsible for effects relating to the nonlinear collision operator.

    3. (iii) An additional second-order effect involving fluctuations in the dielectric shielding was not evaluated either by Catto & Simakov or in the present paper. This effect will at least influence the values of some of the Burnett coefficients; qualitative implications are presently not understood.

  5. (5) While convective amplification of fluctuations in a weakly inhomogeneous background is included in the general formalism, I did not pursue any calculations related to that effect.

Clearly, setting up and working out the two-time formalism of Brey et al. involves a fair amount of effort. In view of the agreement between that formalism and standard Chapman–Enskog theory for the particular case of homogeneous background fluctuations, which is relatively straightforward, a natural question is, why should one bother? The answer depends on one’s goals as well as the physical situation. First assume that convective amplification of fluctuations can be neglected. If one is interested solely in the regular parts of the transport coefficients in the limit of weak coupling, it seems clear that Chapman–Enskog calculations, for example as implemented approximately by Catto & Simakov (Reference Catto and Simakov2004), are more direct. Indeed, some of the manipulations that were done in the present paper in working out the weakly coupled two-time formulas appear to be almost redundant. For example, the algebra that was done in § G.2 to obtain the linearized Balescu–Lenard operator from the two-time hierarchy and in appendix E to obtain the interspecies momentum transfer involves repeated instances of familiar manipulations involving the Vlasov response function – the same class of manipulations used in deriving the original nonlinear Balescu–Lenard operator (§ G.1). This is not surprising because, as I discussed, the two-time equations follow from the one-time ones by functional differentiation. If one already knows the one-time kinetic equation, there is no need to rederive its implications at the two-time levelFootnote 66 – although the calculations done here provide important confidence-building consistency checks.

However, one knows the one-time kinetic equation only in the limit of weak coupling and with the neglect of long-ranged collisional correlations and convective wave effects. More generally, in order to proceed via Chapman–Enskog expansion, at the very least one would have to derive a collision operator that is more complete than the Balescu–Lenard or Landau operators. If that operator were local in physical space (involving correlation lengths shorter than the mean free path), then the usual Chapman–Enskog formulas would apply; complication would arise only in the quantitative evaluation of certain matrix elements. But that locality may not hold. Even if non-local wave effects are neglected, in either one- or two-time theory one must ultimately face up to the concerns expressed in § 1 about possible divergences of the transport coefficients due to collision-induced non-local collective effects. Some sort of renormalized formalism is required even at the level of discrete particles. The most complete such description is by Rose (Reference Rose1979), who shows how to handle the effects of both discrete particles and continuum turbulence within a unified framework and presents a so-called particle direct-interaction approximation (PDIA). However, mere possession of a renormalized theory of two-time correlation functions will not solve all issues relating to divergences of the Burnett coefficients. Although the formulas in § 4.2 involve correlation functions that could be calculated from a renormalized theory, those formulas follow from the assumption that it is permissible to proceed with a regular perturbation expansion in the gradients. However, various authorsFootnote 67 have demonstrated that it is necessary to expand in fractional powers of the gradients or to assume more general non-analytic dependence. Considerable further work is required in order to establish the detailed connections between such research and Rose’s formalism.

It is fortunate that the classical weak-coupling limit has outsized importance in many plasma physics applications, including magnetic fusion. Although in that limit it is unnecessary to calculate collisional transport coefficients from two-time formulas, proceeding in that way does have value beyond consistency checking. The calculations in § 6.3.2, where I showed how the weakly coupled, nonlinear collision operator emerges in second-order kinetic theory, can be viewed as providing additional perspective on the nonlinear noise terms that have been extensively discussed in standard turbulence theory – they are generally treated in terms of renormalized approximations such as the direct-interaction approximation (DIA) – and appear as well in Rose’s more-encompassing formalism. In turbulence theory, in particular the DIA and related approximations, the noise terms describe the internal stochastic forcing associated with the nonlinear interactions; that forcing is required in order to maintain the fluctuation level against the tendency for it to decay by nonlinear scrambling. More generally, that balance shows up in the consistency that must be enforced between the one- and two-time functions (Rose Reference Rose1979); see appendix H for further discussion. Contributions from singular initial-condition terms were crucial in establishing in § 6.3 the agreement between Chapman–Enskog calculations and two-time theory. Also, as I showed, nonlinear noise terms involving discrete particles lead directly to contributions to transport coefficients involving the nonlinear collision operator. The key, stochastic forcing role of nonlinear noise makes it seem inevitable that those nonlinear $\text{C}[\,f_{1},\overline{f}_{1}]$ effects must be present, although they were missed by plasma physicists prior to Catto & Simakov. The proper treatment of nonlinear noise also enables one to discuss stable and turbulent regimes in a unified way (Krommes Reference Krommes2007), although that was not discussed in this paper.

Neither the standard Landau collision operator nor statistically homogeneous two-time theory takes into account the convective amplification of fluctuations discussed by Kent & Taylor (Reference Kent and Taylor1969). However, the general spatially inhomogeneous two-time formalism accommodates that, and a superior collision operator could be developed by beginning with a generalized test-particle superposition principle. Wave-induced transport (Rosenbluth & Liu Reference Rosenbluth and Liu1976) should also be incorporated into the formalism. Further implications at Burnett order – for example, to the physics of a tokamak pedestal – remain to be explored. Given that the Burnett equations have issues with the proper representation of the Knudsen boundary layer, which could conceivably also be relevant to the pedestal, it is clear that a great amount of practical work remains to be done. It is likely that some sort of renormalized theory (see footnote 1 on page 4) will be required, but the challenge is daunting.

Fundamentally, the discussions and calculations in Part 1 and the present Part 2 of this series are intended to raise awareness in the plasma physics community of the significant utility of projection-operator methods. Applied to many-body theory, they provide a beautiful, compact, and unified representation of dissipative transport by mapping the orthogonal subspace, containing rapid fluctuations, into the slow, hydrodynamic subspace. The formalism evokes poetry:

They were so amply beautiful, the maps, With their blue rivers winding to the sea, So calmly beautiful, who could have blamed Us for believing, bowed to our drawing boards, In a large and ultimate equivalence, One map that challenged and replaced the world?[6pt] – from Projection, by H. Nemerov (Reference Nemerov1967).

Acknowledgements

This paper is written in memory of the late Professor C. Oberman, who introduced me to plasma kinetic theory. Carl taught me many things, one of which has fostered patience in numerous calculations: He observed that, as one struggles to analytically understand the generic theory research problem, one often first incorrectly finds infinity; then (again incorrectly) zero; then the correct, finite answer. Clearly, infinities and zeros abound in projection-operator manipulations. Carl would have enjoyed the present ones.

I am grateful to G. Hammett for useful discussions about the physical and mathematical difficulties with the Burnett equations, and to E. Valeo for stimulating remarks about wave-enhanced transport. P. Catto graciously provided some background to his 2004 paper with Simakov; he also asked insightful questions about an early draft of the manuscript that led me to include some clarifying remarks. This work was supported by the U.S. Department of Energy Contract DE-AC02-09CH11466.

Supplementary material

Supplementary material (Krommes Reference Krommes2018c ) is available at https://doi.org/10.1017/S0022377818000892.

Appendix A. A simple stochastic model that exhibits Burnett effects

The only way to obtain a detailed description of Burnett effects for a plasma is to attack the many-body problem head on with the methods of non-equilibrium statistical mechanics, as was done in the body of the paper. However, the general problem is difficult because of dynamical nonlinearity. Here I shall discuss aspects of a simpler model that, although dynamically linear, is stochastically nonlinear and thus displays under statistical averaging many of the features of the full problem. The model is called the stochastic oscillator; variations of it have been frequently used to illustrate aspects of statistical closure.Footnote 68 The primitive amplitude equation is taken to be

(A 1) $$\begin{eqnarray}\frac{\text{d}\widetilde{\unicode[STIX]{x1D713}}}{\text{d}t}+\unicode[STIX]{x1D708}\widetilde{\unicode[STIX]{x1D713}}+\text{i}\boldsymbol{k}\boldsymbol{\cdot }\widetilde{\boldsymbol{V}}\widetilde{\unicode[STIX]{x1D713}}=\widetilde{f}(t),\end{eqnarray}$$

where  $\widetilde{\boldsymbol{V}}$ and  $\widetilde{f}$ are random variables whose statistics are prescribed as follows.

The random velocity  $\widetilde{\boldsymbol{V}}$ is assumed to be independent of both space and time, with specified (passive) centred, stationary, non-Gaussian Footnote 69 statistics:

(A 2a-c ) $$\begin{eqnarray}\langle \widetilde{\boldsymbol{V}}\rangle =\mathbf{0},\quad \langle \unicode[STIX]{x1D6FF}\widetilde{\boldsymbol{V}}\unicode[STIX]{x1D6FF}\widetilde{\boldsymbol{V}}\rangle =U\unicode[STIX]{x1D644},\quad \langle \unicode[STIX]{x1D6FF}\widetilde{\boldsymbol{V}}\unicode[STIX]{x1D6FF}\widetilde{\boldsymbol{V}}\unicode[STIX]{x1D6FF}\widetilde{\boldsymbol{V}}\rangle =\unicode[STIX]{x1D64F},\end{eqnarray}$$

where $U$ and  $\unicode[STIX]{x1D64F}$ are specified. Because $U$  has the dimensions of velocity squared, it can be written as $U=\overline{u}^{2}$ , where $\overline{u}$  is a characteristic root-mean-square velocity fluctuation. In the absence of a preferred direction, the fully symmetric third-rank tensor  $\unicode[STIX]{x1D64F}$ would vanish.Footnote 70 However, the goal is to emulate features of the hydrodynamic transport problem, in which long-wavelength gradients break the symmetry of the equilibrium state. Therefore, I chooseFootnote 71

(A 3) $$\begin{eqnarray}T_{ijk}=W(\unicode[STIX]{x1D6FF}_{ij}w_{k}+\unicode[STIX]{x1D6FF}_{jk}w_{i}+\unicode[STIX]{x1D6FF}_{ki}w_{j}),\end{eqnarray}$$

where the coefficient  $W$ also has the dimensions of $\overline{u}^{2}$ . The significance of the constant vector  $\boldsymbol{w}$ will become clear momentarily.

The random forcing $\widetilde{f}(t)$  is taken to be a time series (independent of  $\widetilde{\boldsymbol{V}}$ ) with white noise statistics:

(A 4a,b ) $$\begin{eqnarray}\langle \,\widetilde{f}(t)\rangle =0,\quad \langle \unicode[STIX]{x1D6FF}f(t+\unicode[STIX]{x1D70F})\unicode[STIX]{x1D6FF}f(t)\rangle =2D_{v}\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F}).\end{eqnarray}$$

Finally, the non-random relaxation rate  $\unicode[STIX]{x1D708}$ (analogous to the collision frequency in kinetic theory) is assumed to vanish in the equation for $\langle \widetilde{\unicode[STIX]{x1D713}}\rangle$ . (This emulates the hydrodynamic conservation properties of the collision operator.)

With these assumptions, the mean and fluctuating equations become

(A 5a ) $$\begin{eqnarray}\displaystyle \frac{\text{d}\langle \widetilde{\unicode[STIX]{x1D713}}\rangle }{\text{d}t}+\text{i}\boldsymbol{k}\boldsymbol{\cdot }\langle \unicode[STIX]{x1D6FF}\widetilde{\boldsymbol{V}}(t)\unicode[STIX]{x1D6FF}\widetilde{\unicode[STIX]{x1D713}}(t)\rangle & = & \displaystyle 0,\end{eqnarray}$$
(A 5b ) $$\begin{eqnarray}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713}}{\text{d}t}+\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713}+\text{i}\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\widetilde{\boldsymbol{V}}(t)\langle \widetilde{\unicode[STIX]{x1D713}}\rangle +\text{i}\boldsymbol{k}\boldsymbol{\cdot }[\unicode[STIX]{x1D6FF}\widetilde{\boldsymbol{V}}(t)\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713}(t)-\langle \ldots \rangle ] & = & \displaystyle \unicode[STIX]{x1D6FF}f(t).\end{eqnarray}$$
At $k=0$ , the equation for fluctuations reduces to the well-known classical Langevin equation for the velocity of a Brownian test particle (Wang & Uhlenbeck Reference Wang and Uhlenbeck1945). Thus, it is clear that a statistically steady state can be reached in which the collisional dissipation balances in mean square against the random forcing, the strength of which is measured by the diffusion coefficient  $D_{v}$ .

The $\boldsymbol{k}$ -dependent terms drive corrections to that steady state. In terms of the infinitesimal response function (Green’s function)

(A 6) $$\begin{eqnarray}R(t;ts_{+}\prime )\doteq H(t-ts_{+}\prime )\text{e}^{-\unicode[STIX]{x1D708}(t-ts_{+}\prime )},\end{eqnarray}$$

where $H(t)$  is the Heaviside unit step function, an exact integral equation is

(A 7) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713}(t) & = & \displaystyle \displaystyle \int _{-\infty }^{t}\!\text{d}\overline{t}\,R(t;\overline{t})\unicode[STIX]{x1D6FF}f(\overline{t})-\displaystyle \int _{-\infty }^{t}\!\text{d}\overline{t}\,R(t;\overline{t})\text{i}\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\widetilde{\boldsymbol{V}}(\overline{t})\langle \unicode[STIX]{x1D713}\rangle (\overline{t})\nonumber\\ \displaystyle & & \displaystyle -\,\displaystyle \int _{-\infty }^{t}\!\text{d}\overline{t}\,R(t;\overline{t})\text{i}\boldsymbol{k}\boldsymbol{\cdot }[\unicode[STIX]{x1D6FF}\widetilde{\boldsymbol{V}}(\overline{t})\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713}(\overline{t})-\langle \ldots \rangle ].\end{eqnarray}$$

At first order in  $k$ (not first order in the size of the fluctuations), one has

(A 8) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713}^{(1)}(t)=-\displaystyle \int _{-\infty }^{t}\!\text{d}\overline{t}\,R(t;\overline{t})\text{i}\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\widetilde{\boldsymbol{V}}(\overline{t})\langle \unicode[STIX]{x1D713}\rangle (\overline{t}),\end{eqnarray}$$

which gives an $O(k^{2})$ contribution to (A 5a ):

(A 9a ) $$\begin{eqnarray}\displaystyle \text{i}\boldsymbol{k}\boldsymbol{\cdot }\langle \unicode[STIX]{x1D6FF}\widetilde{\boldsymbol{V}}(t)\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713}^{(1)}(t)\rangle & = & \displaystyle k^{2}\displaystyle \int _{-\infty }^{t}\!\text{d}\overline{t}\,R(t;\overline{t})U(t,\overline{t})\langle \unicode[STIX]{x1D713}\rangle (\overline{t})\end{eqnarray}$$
(A 9b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle k^{2}\int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\,R(\overline{\unicode[STIX]{x1D70F}})U(\overline{\unicode[STIX]{x1D70F}})\langle \unicode[STIX]{x1D713}\rangle (t-\overline{\unicode[STIX]{x1D70F}})\end{eqnarray}$$
(A 9c ) $$\begin{eqnarray}\displaystyle & {\approx} & \displaystyle k^{2}D^{(1)}\langle \unicode[STIX]{x1D713}\rangle (t),\end{eqnarray}$$
where
(A 10) $$\begin{eqnarray}D^{(1)}\doteq \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\,R(\overline{\unicode[STIX]{x1D70F}})U(\overline{\unicode[STIX]{x1D70F}}).\end{eqnarray}$$

Since the velocity-correlation function  $U(\unicode[STIX]{x1D70F})$ has been chosen to be time independent, one has $D^{(1)}=\overline{u}^{2}/\unicode[STIX]{x1D708}$ . This agrees with the scaling of the classical Navier–Stokes transport coefficients if $\overline{u}$  is identified with the thermal velocity  $v_{\text{t}}$ .

At second order in  $k$ , one has

(A 11a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713}^{(2)}(t) & = & \displaystyle -\displaystyle \int _{-\infty }^{t}\!\text{d}\overline{t}\,R(t;\overline{t})\text{i}\boldsymbol{k}\boldsymbol{\cdot }[\unicode[STIX]{x1D6FF}\widetilde{\boldsymbol{V}}(\overline{t})\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713}^{(1)}(\overline{t})-\langle \ldots \rangle ]\end{eqnarray}$$
(A 11b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle -\displaystyle \int _{-\infty }^{t}\!\text{d}\overline{t}\,R(t;\overline{t})\boldsymbol{k}\boldsymbol{\cdot }\left(\unicode[STIX]{x1D6FF}\widetilde{\boldsymbol{V}}(\overline{t})\displaystyle \int _{-\infty }^{\overline{t}}\!\text{d}\overline{t}s_{+}\prime \,R(\overline{t};\overline{t}s_{+}\prime )\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\widetilde{\boldsymbol{V}}(\overline{t}s_{+}\prime )\langle \unicode[STIX]{x1D713}\rangle (\overline{t}s_{+}\prime )-\langle \ldots \rangle \right);\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
this contributes to (A 5a )
(A 12a ) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{i}\boldsymbol{k}\boldsymbol{\cdot }\langle \unicode[STIX]{x1D6FF}\widetilde{\boldsymbol{V}}(t)\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713}^{(2)}(t)\rangle \nonumber\\ \displaystyle & & \displaystyle \quad =-\text{i}\displaystyle \int _{-\infty }^{t}\!\text{d}\overline{t}\,\left\langle \boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\widetilde{\boldsymbol{V}}(t)R(t;\overline{t})\left(\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\widetilde{\boldsymbol{V}}(\overline{t})\displaystyle \int _{-\infty }^{\overline{t}}\!\text{d}\overline{t}s_{+}\prime \,R(\overline{t};\overline{t}s_{+}\prime )\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\widetilde{\boldsymbol{V}}(\overline{t}s_{+}\prime )\langle \unicode[STIX]{x1D713}\rangle (\overline{t}s_{+}\prime )-\langle \ldots \rangle \right)\right\rangle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
(A 12b ) $$\begin{eqnarray}\displaystyle & & \displaystyle \quad =-3\text{i}k^{2}W\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{w}\displaystyle \int _{-\infty }^{t}\!\text{d}\overline{t}\,R(t;\overline{t})\displaystyle \int _{-\infty }^{\overline{t}}\!\text{d}\overline{t}s_{+}\prime \,R(\overline{t};\overline{t}s_{+}\prime )\langle \unicode[STIX]{x1D713}\rangle (\overline{t}s_{+}\prime ).\end{eqnarray}$$
With the changes of variables $\overline{\unicode[STIX]{x1D70F}}\doteq t-\overline{t}$ and $\overline{\unicode[STIX]{x1D70F}}s_{+}\prime \doteq \overline{t}-\overline{t}s_{+}\prime$ , one has
(A 13a ) $$\begin{eqnarray}\displaystyle \displaystyle \int _{-\infty }^{t}\!\text{d}\overline{t}\,R(t;\overline{t})\displaystyle \int _{-\infty }^{\overline{t}}\!\text{d}\overline{t}s_{+}\prime \,R(\overline{t};\overline{t}s_{+}\prime )\langle \unicode[STIX]{x1D713}\rangle (\overline{t}s_{+}\prime ) & = & \displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\,R(\overline{\unicode[STIX]{x1D70F}})\int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}s_{+}\prime \,R(\overline{\unicode[STIX]{x1D70F}}s_{+}\prime )\langle \unicode[STIX]{x1D713}\rangle (t-\overline{\unicode[STIX]{x1D70F}}-\overline{\unicode[STIX]{x1D70F}}s_{+}\prime )\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
(A 13b ) $$\begin{eqnarray}\displaystyle & {\approx} & \displaystyle \left(\int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\,R(\overline{\unicode[STIX]{x1D70F}})\right)\left(\int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}s_{+}\prime \,R(\overline{\unicode[STIX]{x1D70F}}s_{+}\prime )\right)\langle \unicode[STIX]{x1D713}\rangle (t).\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
Therefore, one has
(A 14) $$\begin{eqnarray}\text{i}\boldsymbol{k}\boldsymbol{\cdot }\langle \unicode[STIX]{x1D6FF}\widetilde{\boldsymbol{V}}(t)\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713}^{(2)}(t)\rangle \approx -\text{i}kD^{(2)}(\boldsymbol{k}\boldsymbol{\cdot }\widehat{\boldsymbol{w}})(k\langle \unicode[STIX]{x1D713}\rangle ),\end{eqnarray}$$

where

(A 15) $$\begin{eqnarray}D^{(2)}\propto (\overline{u}^{2}/\unicode[STIX]{x1D708})(\overline{u}/\unicode[STIX]{x1D708}).\end{eqnarray}$$

This effect is analogous to the unmagnetized Burnett transport coefficients that I discuss in the body of the paper. They scale with  $\unicode[STIX]{x1D708}^{-2}$ . (The factor $\overline{u}/\unicode[STIX]{x1D708}$ is analogous to the collisional mean free path  $\unicode[STIX]{x1D706}_{\text{mfp}}$ .) The Burnett fluxes, represented here by $-D^{(2)}(\boldsymbol{k}\boldsymbol{\cdot }\widehat{\boldsymbol{w}})(k\langle \unicode[STIX]{x1D713}\rangle )$ , are of second order in the gradients and are smaller than the Navier–Stokes ones by a factor of $k\unicode[STIX]{x1D706}_{\text{mfp}}$ . (In many-body theory, the limit $k\rightarrow 0$ is taken and the reduction factor becomes  $\unicode[STIX]{x1D706}_{\text{mfp}}/L$ , where $L$  is the system size.) In reality, those fluxes involve products of hydrodynamic variables such as $T^{-1}|\unicode[STIX]{x1D735}T|^{2}$ . Here $\unicode[STIX]{x1D735}\ln T$ is replaced by the factor $\boldsymbol{k}\boldsymbol{\cdot }\widehat{\boldsymbol{w}}$ because the stochastic model is passive, so the advecting velocity is not linearly proportional to  $\unicode[STIX]{x1D713}$ .

This calculation demonstrates several important points that generalize to the full many-body problem:

  1. (i) Transport effects in the equations for mean fields arise from the symmetry breaking of the equilibrium state by the hydrodynamic gradients.

  2. (ii) The Burnett coefficients arise from non-Gaussian statistics. (The many-body theory described in the body of the paper expresses the non-Gaussian effects in terms of two-time Klimontovich correlations involving either two or three phase-space points. It is shown in § 6.4 that the contributions calculated by Catto & Simakov (Reference Catto and Simakov2004) from the nonlinear collision operator are related to three-point Klimontovich correlations.)

Appendix B. Microscopic and macroscopic forces

To find the time evolution of the microscopic momentum density, differentiate the definition (2.11) to find

(B 1a ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\widetilde{\boldsymbol{P}}_{s}(\boldsymbol{r},t)}{\unicode[STIX]{x2202}t} & = & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\mathop{\sum }_{i\in s}m_{s}\boldsymbol{v}_{i}(t)\unicode[STIX]{x1D6FF}\boldsymbol{(}\boldsymbol{r}-\boldsymbol{x}_{i}(t)\boldsymbol{)}\end{eqnarray}$$
(B 1b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle \mathop{\sum }_{i\in s}m_{s}\boldsymbol{v}_{i}[-\boldsymbol{v}_{i}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FF}(\boldsymbol{r}-\boldsymbol{x}_{i})]\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{i\in s}\left(q_{s}(\boldsymbol{E}^{\text{ext}}+c^{-1}\boldsymbol{v}_{i}\times \boldsymbol{B}^{\text{ext}})-\frac{\unicode[STIX]{x2202}\widetilde{U}_{i}}{\unicode[STIX]{x2202}\boldsymbol{x}_{i}}\right)\unicode[STIX]{x1D6FF}(\boldsymbol{r}-\boldsymbol{x}_{i}),\end{eqnarray}$$
where $\boldsymbol{E}^{\text{ext}}$ and  $\boldsymbol{B}^{\text{ext}}$ are externally imposed electric and magnetic fields (I shall assume that $\boldsymbol{E}^{\text{ext}}=\mathbf{0}$ ) and $\widetilde{U}_{i}\doteq \sum _{j\neq i}^{\mathscr{N}}q_{i}|\boldsymbol{x}_{i}-\boldsymbol{x}_{j}|^{-1}q_{j}$ is the interparticle potential energy. The first term of (B 1b ) is $-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\widetilde{\unicode[STIX]{x1D749}}_{K,s}$ , where the kinetic momentum flux is $\widetilde{\unicode[STIX]{x1D749}}_{K,s}\doteq \sum _{i\in s}m_{s}\boldsymbol{v}_{i}\boldsymbol{v}_{i}\unicode[STIX]{x1D6FF}(\boldsymbol{r}-\boldsymbol{x}_{i})$ . The Lorentz force term is easily written as $\unicode[STIX]{x1D714}_{\text{c}s}\widetilde{\boldsymbol{P}}_{s}\times \widehat{\boldsymbol{b}}$ , where $\unicode[STIX]{x1D714}_{\text{c}s}\doteq (qB/mc)_{s}$ , $B\doteq |\boldsymbol{B}^{\text{ext}}|$ , and $\widehat{\boldsymbol{b}}\doteq \boldsymbol{B}^{\text{ext}}/B$ .

While the kinetic term is in the form of a divergence, the potential-energy term is not (yet). A well-known trick is to symmetrize the potential term to the extent possible. It is convenient to work in $\boldsymbol{k}$  space. Upon Fourier transforming, that term becomes $-\sum _{i\in s}\sum _{ss_{+}\prime }\sum _{j\neq i}^{\mathscr{N}_{ss_{+}\prime }}(\unicode[STIX]{x2202}\widetilde{U}_{ij}/\unicode[STIX]{x2202}\boldsymbol{x}_{i})\text{e}^{-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{i}}$ . The contributions with $ss_{+}\prime =s$ can be symmetrized by interchanging  $i$ and  $j$ ; one finds

(B 2a ) $$\begin{eqnarray}\displaystyle -\frac{1}{2}\mathop{\sum }_{ij}^{\mathscr{N}_{s}}\left(\frac{\unicode[STIX]{x2202}\widetilde{U}_{ij}}{\unicode[STIX]{x2202}\boldsymbol{x}_{i}}\,\text{e}^{-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{i}}+\frac{\unicode[STIX]{x2202}\widetilde{U}_{ji}}{\unicode[STIX]{x2202}\boldsymbol{x}_{j}}\,\text{e}^{-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{j}}\right) & = & \displaystyle -\frac{1}{2}\mathop{\sum }_{ij}^{\mathscr{N}_{s}}\left(\frac{\unicode[STIX]{x2202}\widetilde{U}_{ij}}{\unicode[STIX]{x2202}\boldsymbol{x}_{i}}\text{e}^{-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{i}}-\frac{\unicode[STIX]{x2202}\widetilde{U}_{ij}}{\unicode[STIX]{x2202}\boldsymbol{x}_{i}}\text{e}^{-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{j}}\right)\end{eqnarray}$$
(B 2b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle -\frac{1}{2}\mathop{\sum }_{ij}^{\mathscr{N}_{s}}\frac{\unicode[STIX]{x2202}\widetilde{U}_{ij}}{\unicode[STIX]{x2202}\boldsymbol{x}_{i}}(1-\text{e}^{\text{i}\boldsymbol{k}\boldsymbol{\cdot }(\boldsymbol{x}_{i}-\boldsymbol{x}_{j})})\text{e}^{-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{i}}\end{eqnarray}$$
(B 2c ) $$\begin{eqnarray}\displaystyle & = & \displaystyle -\frac{1}{2}\mathop{\sum }_{ij}^{\mathscr{N}_{s}}\frac{\unicode[STIX]{x2202}\widetilde{U}_{ij}}{\unicode[STIX]{x2202}\boldsymbol{x}_{ij}}(-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{ij})\left(\frac{1-\text{e}^{\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{ij}}}{-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{ij}}\right)\text{e}^{-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{i}}\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
(B 2d ) $$\begin{eqnarray}\displaystyle & = & \displaystyle -\text{i}\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x0394}\widetilde{\unicode[STIX]{x1D749}}_{s}(\boldsymbol{k}),\end{eqnarray}$$
where
(B 3) $$\begin{eqnarray}\unicode[STIX]{x0394}\widetilde{\unicode[STIX]{x1D749}}_{s}(\boldsymbol{k})\doteq -\frac{1}{2}\mathop{\sum }_{ij}^{\mathscr{N}_{s}}\boldsymbol{x}_{ij}\frac{\unicode[STIX]{x2202}\widetilde{U}_{ij}}{\unicode[STIX]{x2202}\boldsymbol{x}_{ij}}\left(\frac{\text{e}^{\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{ij}}-1}{\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{ij}}\right)\text{e}^{-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{i}}.\end{eqnarray}$$

The limit of the parenthesized expressionFootnote 72 as $\boldsymbol{k}\rightarrow \mathbf{0}$ is 1. The tensor $\unicode[STIX]{x0394}\widetilde{\unicode[STIX]{x1D749}}_{s}(\boldsymbol{k})$ is symmetric for central forces since

(B 4) $$\begin{eqnarray}\boldsymbol{r}\,\frac{\unicode[STIX]{x2202}U_{ij}(\boldsymbol{r})}{\unicode[STIX]{x2202}\boldsymbol{r}}=\widehat{\boldsymbol{r}}\left(\frac{\unicode[STIX]{x2202}U_{ij}(r)}{\unicode[STIX]{x2202}r}\,r\right)\widehat{\boldsymbol{r}}.\end{eqnarray}$$

Note that for the Coulomb potential $U_{ij}(r)=q_{i}q_{j}/r$ , one has $-(\unicode[STIX]{x2202}U_{ij}/\unicode[STIX]{x2202}r)r=U_{ij}(r)$ . The terms $ss_{+}\prime \neq s$ cannot be symmetrized.

From these formulas, one wants to distil a macroscopic electrostatic potential  $\unicode[STIX]{x1D719}(\boldsymbol{r})$ and microscopic fluctuating stresses for use in (2.26b ), with $\boldsymbol{E}\doteq -\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}$ being the macroscopic (collective or internal) electric field. To do so, it is necessary to be more precise about the nature of the statistical ensemble. One possibility is to assume that one works in a large box of volume  $\mathscr{V}$ containing $\mathscr{N}=\sum _{s}\mathscr{N}_{s}$  particles with zero net charge. However, this obscures the distinction between short-ranged correlations and long-ranged forces. This is crucial for the plasma, which can support non-trivial, spatially varying electric fields  $\boldsymbol{E}(\boldsymbol{r},t)$ over distances much larger than the microscopic correlation length  $\unicode[STIX]{x1D706}_{\text{D}}$ . (The paradigmatic examples are the Langmuir oscillations $\unicode[STIX]{x1D714}\approx \pm \unicode[STIX]{x1D714}_{\text{p}}$ .) One would like to write the microscopic forces as an average plus a fluctuating piece, $-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x0394}\widetilde{\unicode[STIX]{x1D749}}_{s}(\boldsymbol{r},t)=-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\langle \unicode[STIX]{x0394}\unicode[STIX]{x1D749}_{s}\rangle (\boldsymbol{r},t)-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D749}_{s}(\boldsymbol{r},t)$ , where notably $\langle \unicode[STIX]{x0394}\unicode[STIX]{x1D749}_{s}\rangle$ depends on  $\boldsymbol{r}$ . An average with the exact non-equilibrium ensemble accomplishes this; however, one wants to express that average in terms of a local reference distribution. A uniform distribution of particles over the entire box is not satisfactory since symmetry considerations would lead to a spatially constant  $\langle \unicode[STIX]{x0394}\unicode[STIX]{x1D749}\rangle$ . One could imagine dividing the system into boxes whose sides are larger than  $\unicode[STIX]{x1D706}_{\text{D}}$ , and local charge imbalances in those boxes would lead to internal electric fields $\boldsymbol{E}^{\text{int}}=-\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}$ . However, such boxes are not necessarily in local thermal equilibrium due to high-frequency oscillations. Instead, the basic coarse-graining procedure discussed in § 2.1 uses boxes  $\unicode[STIX]{x0394}\mathscr{V}$ of side large compared to the collisional mean free path (but small compared to the macroscopic gradient scale length). Now an average over the charge distribution of particles in  $\unicode[STIX]{x0394}\mathscr{V}$ produces from the terms with $ss_{+}\prime =s$ a $\langle \unicode[STIX]{x0394}\unicode[STIX]{x1D749}_{s}\rangle (\boldsymbol{r},t)=(\overline{n}q)_{s}\unicode[STIX]{x1D719}_{s}(\boldsymbol{r},t)\unicode[STIX]{x1D644}$ , where $\unicode[STIX]{x1D719}_{s}$  is the contribution to the total low-frequency electrostatic potential due to species  $s$ . Note that $\sum _{j\in s}=\sum _{j\in s}^{\unicode[STIX]{x0394}\mathscr{V}(\boldsymbol{r})}+\sum _{j\in s}^{\unicode[STIX]{x0394}\mathscr{V}s_{+}\prime \neq \unicode[STIX]{x0394}\mathscr{V}}$ . The last term gives non-local contributions to the mean potential.Footnote 73

The terms $ss_{+}\prime \neq s$ can be broken into two kinds of effects: those arising from unlike-species particles all within  $\unicode[STIX]{x0394}\mathscr{V}$ , and contributions from other cells  $\unicode[STIX]{x0394}\mathscr{V}s_{+}\prime$ . The former generate local random momentum and energy exchange effects as well as a local contribution to the mean potential; the latter gives the non-local unlike-species contribution to the mean potential.

Apparently both coarse-graining procedures mentioned above support collective electric fields. The choice of box size is not irrelevant, however; it affects the approximate procedure (involving Novikov’s theorem) that is used in appendix D to establish formulas for the subtracted fluxes in terms of certain thermodynamic derivatives. For a discussion of low-frequency, long-wavelength classical transport, scaling the box side to  $\unicode[STIX]{x1D706}_{\text{mfp}}$ is the correct choice.

Thus, the momentum flux obeys (2.26b ). With a prime denoting the fluctuation from the mean of the potential-energy terms, the portion of the microscopic stress tensor that contributes to local transport coefficients is defined by

(B 5a,b ) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D749}}_{s}^{\prime }(\boldsymbol{r},t)=\displaystyle \int \frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\,\text{e}^{\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{r}}\widetilde{\unicode[STIX]{x1D749}}_{s}^{\prime }(\boldsymbol{k},t),\quad \widetilde{\unicode[STIX]{x1D749}}_{s}(\boldsymbol{k},t)=\mathop{\sum }_{i\in s}[m_{s}\boldsymbol{v}_{i}\boldsymbol{v}_{i}+\unicode[STIX]{x0394}\widetilde{\unicode[STIX]{x1D749}}_{s,i}(\boldsymbol{k})]\text{e}^{-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{i}},\end{eqnarray}$$

and

(B 6) $$\begin{eqnarray}\unicode[STIX]{x0394}\widetilde{\unicode[STIX]{x1D749}}_{s,i}(\boldsymbol{k})\doteq -\frac{1}{2}\mathop{\sum }_{j\in s}\frac{\unicode[STIX]{x2202}\widetilde{U}_{ij}}{\unicode[STIX]{x2202}\boldsymbol{x}_{ij}}\,\boldsymbol{x}_{ij}\left(\frac{\text{e}^{\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{ij}}-1}{\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{ij}}\right).\end{eqnarray}$$

The total amount of this flux is

(B 7) $$\begin{eqnarray}\widetilde{\pmb{\pmb{\mathscr{T}}}}_{s}^{\prime }(t)=\displaystyle \int \text{d}\boldsymbol{r}\,\widetilde{\unicode[STIX]{x1D749}}_{s}^{\prime }(\boldsymbol{r},t)=\widetilde{\unicode[STIX]{x1D749}}_{s}^{\prime }(\boldsymbol{k}=\mathbf{0},t)=\mathop{\sum }_{i\in s}[m_{s}\boldsymbol{v}_{i}\boldsymbol{v}_{i}+\unicode[STIX]{x0394}\widetilde{\unicode[STIX]{x1D749}}_{s,i}^{\prime }(\mathbf{0})],\end{eqnarray}$$

where for the Coulomb interaction $\unicode[STIX]{x0394}\widetilde{\unicode[STIX]{x1D749}}_{s,i}(\mathbf{0})={\textstyle \frac{1}{2}}\sum _{j\in s}\widehat{\boldsymbol{x}}_{ij}U_{ij}\widehat{\boldsymbol{x}}_{ij}$ .

When the stress tensor is used in formulas for transport coefficients, it invariably appears in its subtracted form (see appendix D). That subtraction naturally removes any mean potential.

For the exchange terms, one has

(B 8) $$\begin{eqnarray}\dot{\boldsymbol{P}}s_{+}\prime _{\unicode[STIX]{x0394},s}=-\mathop{\sum }_{i\in s}\mathop{\sum }_{ss_{+}\prime \neq s}\mathop{\sum }_{j\in ss_{+}\prime }\unicode[STIX]{x1D735}_{i}U_{ij}\unicode[STIX]{x1D6FF}(\boldsymbol{r}-\boldsymbol{x}_{i}).\end{eqnarray}$$

Since the mean field has already been extracted (denoted by the prime), the $j$  sum should in principle be restricted to particles lying in the same coarse-graining cell. The statistical effects of  $\dot{\widetilde{\boldsymbol{P}}}_{\unicode[STIX]{x0394}}$ will be discussed in appendix E.

Similar considerations lead to (2.26c ) for the evolution of the microscopic energy density.

Appendix C. The conjugate variables

Here I give some details about the choice (2.34) of the conjugate variables  $\boldsymbol{B}$ . The constraint (2.33) that the mean hydrodynamic variables  $\boldsymbol{a}_{s}$ are given by the local equilibrium average of the microscopic densities leads, according to (2.32a ), to a set of functional differential equations that determine  $\boldsymbol{B}$ in terms of  $\boldsymbol{a}$ :

(C 1) $$\begin{eqnarray}\boldsymbol{a}_{s}=\left(\begin{array}{@{}c@{}}n_{s}\\ \boldsymbol{ p}_{s}\\ e_{s}\end{array}\right)=\left(\begin{array}{@{}c@{}}\unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D6FF}\boldsymbol{B}_{n,s}\\ \unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D6FF}\boldsymbol{B}_{\boldsymbol{p},s}\\ \unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D6FF}\boldsymbol{B}_{e,s}\end{array}\right)\ln Z_{\boldsymbol{B}}[\boldsymbol{B}],\end{eqnarray}$$

where, upon using the definitions of the microscopic variables, performing the $\overline{\boldsymbol{r}}$  integration implied in the $\star$  operation, and defining $\unicode[STIX]{x1D6FD}_{s}\doteq -B_{e,s}$ for convenience, one finds

(C 2) $$\begin{eqnarray}Z_{\boldsymbol{B}}=\displaystyle \int \text{d}\unicode[STIX]{x1D6E4}\,\exp \left\{\vphantom{\left(\frac{1}{2}\right)}\right.\!\!\mathop{\sum }_{\overline{s}}\mathop{\sum }_{i\in \overline{s}}\left[B_{n,\overline{s}}(\boldsymbol{x}_{i})+m_{\overline{s}}\boldsymbol{v}_{i}\boldsymbol{\cdot }\boldsymbol{B}_{\boldsymbol{p},\overline{s}}(\boldsymbol{x}_{i})-\left(\frac{1}{2}m_{\overline{s}}v_{i}^{2}+U_{i}\right)\unicode[STIX]{x1D6FD}_{\overline{s}}(\boldsymbol{x}_{i})\right]\!\!\left.\vphantom{\left(\frac{1}{2}\right)}\right\}.\end{eqnarray}$$

Although $U_{i}$ implicitly depends on all $\boldsymbol{x}_{j\neq i}$ , which implies that $Z_{\boldsymbol{B}}$  contains two-point spatial correlations, all of the dependence on  $\boldsymbol{v}_{i}$ is explicit and uncoupled. The integrations over each of the  $\boldsymbol{v}_{i}$ can therefore be performed by completing the square:

(C 3) $$\begin{eqnarray}\displaystyle Z_{\boldsymbol{B}} & = & \displaystyle \displaystyle \int \text{d}X\,\exp \left[\vphantom{\left(\frac{1}{2}\right)}\right.\!\!\mathop{\sum }_{\overline{s}}\mathop{\sum }_{i\in \overline{s}}\left(\frac{3}{2}\ln [2\unicode[STIX]{x03C0}\,\unicode[STIX]{x1D70E}_{\overline{s}}^{2}(\boldsymbol{x}_{i})]\right.\nonumber\\ \displaystyle & & \displaystyle +\!\!\left.\vphantom{\left(\frac{1}{2}\right)}\left.B_{n\overline{s}}(\boldsymbol{x}_{i})+\frac{1}{2}m_{\overline{s}}|\boldsymbol{B}_{\boldsymbol{p}\overline{s}}|^{2}(\boldsymbol{x}_{i})\unicode[STIX]{x1D6FD}_{\overline{s}}^{-1}(\boldsymbol{x}_{i})-U_{i}\unicode[STIX]{x1D6FD}_{\overline{s}}(\boldsymbol{x}_{i})\right)\right],\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}^{2}\doteq (m\unicode[STIX]{x1D6FD})^{-1}$ . Upon using the basic result

(C 4) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FF}B_{n\overline{s}}(\boldsymbol{x}_{i})}{\unicode[STIX]{x1D6FF}B_{ns}(\boldsymbol{r})}=\unicode[STIX]{x1D6FF}_{\overline{s}s}\unicode[STIX]{x1D6FF}(\boldsymbol{x}_{i}-\boldsymbol{r}),\end{eqnarray}$$

one finds from the density component of (C 1) that

(C 5) $$\begin{eqnarray}n_{s}(\boldsymbol{r},t)=\langle \widetilde{N}_{s}(\boldsymbol{r},t)\rangle _{\boldsymbol{B}},\end{eqnarray}$$

where the expectation is taken with the normalized exponential in (C 3). That is,

(C 6) $$\begin{eqnarray}\displaystyle F_{\boldsymbol{B}}(\unicode[STIX]{x1D6E4}) & = & \displaystyle \exp \left[\vphantom{\left(\frac{1}{2}\right)}\right.\!\!-\!\ln Z_{\boldsymbol{B}}+\mathop{\sum }_{\overline{s}}\mathop{\sum }_{i\in \overline{s}}\left(\frac{3}{2}\ln [2\unicode[STIX]{x03C0}\,\unicode[STIX]{x1D70E}_{\overline{s}}^{2}(\boldsymbol{x}_{i})]\right.\nonumber\\ \displaystyle & & \displaystyle +\!\!\left.\vphantom{\left(\frac{1}{2}\right)}\left.B_{n\overline{s}}(\boldsymbol{x}_{i})+\frac{1}{2}m_{\overline{s}}|\boldsymbol{B}_{\boldsymbol{p}\overline{s}}|^{2}(\boldsymbol{x}_{i})\unicode[STIX]{x1D6FD}_{\overline{s}}^{-1}(\boldsymbol{x}_{i})-U_{i}\unicode[STIX]{x1D6FD}_{\overline{s}}(\boldsymbol{x}_{i})\right)\right].\end{eqnarray}$$

Similarly, upon evaluating the functional derivative of  $Z_{\boldsymbol{B}}$ with respect to  $\boldsymbol{B}_{\boldsymbol{p}s}(\boldsymbol{r})$ and making use of (C 5) and the definition (2.13) of the flow velocity, one finds

(C 7) $$\begin{eqnarray}\boldsymbol{B}_{\boldsymbol{p},s}(\boldsymbol{r},t)=(\unicode[STIX]{x1D6FD}\boldsymbol{u})_{s}(\boldsymbol{r},t).\end{eqnarray}$$

Next, the functional derivative with respect to $B_{es}=-\unicode[STIX]{x1D6FD}_{s}$ provides contributions to (C 1) from each of the $\unicode[STIX]{x1D70E}^{2}$ , $\unicode[STIX]{x1D6FD}^{-1}$ and $\unicode[STIX]{x1D6FD}$  terms. Upon using (2.20) and (2.21), one ultimately finds that the energy component of (C 1) expresses the mean internal energy as

(C 8) $$\begin{eqnarray}u_{s}(\boldsymbol{r},t)=\langle \widetilde{U}_{s}\rangle _{\boldsymbol{B}}\end{eqnarray}$$

provided that $\unicode[STIX]{x1D6FD}_{s}^{-1}(\boldsymbol{r},t)$ is identified with the local temperature $T_{s}(\boldsymbol{r},t)$ . With this crucial result, the analogy to the thermal-equilibrium Gibbs distribution is sufficiently close so that one can identify

(C 9) $$\begin{eqnarray}B_{ns}(\boldsymbol{r},t)=\unicode[STIX]{x1D6FD}_{s}(\boldsymbol{r},t)[\unicode[STIX]{x1D707}_{s}(\boldsymbol{r},t)-{\textstyle \frac{1}{2}}m_{s}u_{s}^{2}(\boldsymbol{r},t)],\end{eqnarray}$$

where $\unicode[STIX]{x1D707}$  is the chemical potentialFootnote 74 per particle.

Appendix D. The subtracted fluxes

In order to calculate the subtracted fluxes $\widehat{\boldsymbol{J}}\doteq \text{Q}\boldsymbol{J}=(1-\text{P})\boldsymbol{J}$ , it is necessary to understand the implications of the projection operator  $\text{P}$ defined by (2.45). First observe that in the reference distribution  $F_{0}$ there are no cross-correlations between the relative velocities  $\boldsymbol{w}_{i}$ and particle positions, and also that $F_{0}$  is an isotropic function of  $\{\boldsymbol{w}_{i}\}$ . Thus, the covariance matrix  $\unicode[STIX]{x1D648}_{}$ decomposes into a $2\times 2$ density–energy submatrix  $\unicode[STIX]{x1D648}_{2}$ and

(D 1a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D648}_{sss_{+}\prime }^{\boldsymbol{p}\boldsymbol{p}}(\boldsymbol{x},\boldsymbol{x}s_{+}\prime ) & \doteq & \displaystyle \langle \boldsymbol{P}_{s}^{\prime }(\boldsymbol{x})\boldsymbol{P}_{ss_{+}\prime }^{\prime }(\boldsymbol{x}s_{+}\prime )\rangle _{0}\end{eqnarray}$$
(D 1b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle \mathop{\sum }_{i\in s}m_{s}\mathop{\sum }_{j\in ss_{+}\prime }m_{ss_{+}\prime }\langle \boldsymbol{w}_{i}\boldsymbol{w}_{j}\rangle \langle \unicode[STIX]{x1D6FF}(\boldsymbol{x}-\boldsymbol{x}_{i})\unicode[STIX]{x1D6FF}(\boldsymbol{x}s_{+}\prime -\boldsymbol{x}_{j})\rangle _{0}\end{eqnarray}$$
(D 1c ) $$\begin{eqnarray}\displaystyle & = & \displaystyle m_{s}T_{s}(\boldsymbol{r},t)\unicode[STIX]{x1D644}\mathop{\sum }_{i\in s}\langle \unicode[STIX]{x1D6FF}(\boldsymbol{x}-\boldsymbol{x}_{i})\rangle _{0}\unicode[STIX]{x1D6FF}(\boldsymbol{x}-\boldsymbol{x}s_{+}\prime )\unicode[STIX]{x1D6FF}_{sss_{+}\prime }\end{eqnarray}$$
(D 1d ) $$\begin{eqnarray}\displaystyle & = & \displaystyle (m\overline{n}T)_{s}(\boldsymbol{r},t)\unicode[STIX]{x1D6FF}(\boldsymbol{x}-\boldsymbol{x}s_{+}\prime )\unicode[STIX]{x1D6FF}_{sss_{+}\prime }\unicode[STIX]{x1D644}.\end{eqnarray}$$
Here the mean density  $\overline{n}$ rather than the true density  $n(\boldsymbol{r},t)$ has entered because the expectation is taken in the reference ensemble, which is translationally invariant; see (2.8).

The specific form of  $\unicode[STIX]{x1D648}_{2}$ will not be needed. Note, however, that it is not delta correlated because of spatial correlations arising from internal-energy corrections.

Consider a quantity  $\widetilde{\unicode[STIX]{x1D712}}$ that is even in the $\boldsymbol{w}$  quantities (particularly, $\widetilde{\unicode[STIX]{x1D712}}=\widetilde{N}$ or  $\widetilde{E}$ ) and whose average in the reference ensemble is $\unicode[STIX]{x1D712}\doteq \langle \widetilde{\unicode[STIX]{x1D712}}\rangle _{0}$ . I shall prove that

(D 2) $$\begin{eqnarray}\text{P}\widetilde{\unicode[STIX]{x1D712}}(\boldsymbol{r},t)\approx \unicode[STIX]{x1D712}(\boldsymbol{r},t)+\mathop{\sum }_{\overline{s}}\widetilde{N}_{\overline{s}}(\boldsymbol{r},t)\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D712}}{\unicode[STIX]{x2202}n_{\overline{s}}}\right)_{e}+\widetilde{E}_{\overline{s}}(\boldsymbol{r},t)\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D712}}{\unicode[STIX]{x2202}e_{\overline{s}}}\right)_{n},\end{eqnarray}$$

where the thermodynamic derivatives are taken in the reference ensemble. This result was quoted by Brey (Reference Brey1983) for the one-component fluid. The argument relies on the assumption (used throughout this work) that in a local region the system is close to local thermal equilibrium. Thus, one can imagine dividing the system into cells whose sides are much larger than the spatial correlation length but much smaller than the macroscopic gradient scale length. One expects that all of the cumulants of  $\widetilde{N}s_{+}\prime$ and  $\widetilde{E}s_{+}\prime$ will be small when at least two of their spatial arguments lie in different cells. This implies that the coarse-grained  $\widetilde{N}s_{+}\prime$ and  $\widetilde{E}s_{+}\prime$ can be considered to be Gaussian random variables to lowest order. Then the expectations required to evaluate  $\text{P}\widetilde{\unicode[STIX]{x1D712}}$ can be calculated with the aid of Novikov’s theorem (Krommes Reference Krommes2015, appendix  B and references therein), whose basic form for a centred random scalar field  $\widetilde{\unicode[STIX]{x1D719}}s_{+}\prime$ is

(D 3) $$\begin{eqnarray}\langle \unicode[STIX]{x1D719}s_{+}\prime (\boldsymbol{x})\mathscr{F}[\unicode[STIX]{x1D719}s_{+}\prime ]\rangle =\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,C(\boldsymbol{x},\overline{\boldsymbol{x}})\left\langle \frac{\unicode[STIX]{x1D6FF}\mathscr{F}[\unicode[STIX]{x1D719}s_{+}\prime ]}{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}s_{+}\prime (\overline{\boldsymbol{x}})}\right\rangle ,\end{eqnarray}$$

where $\mathscr{F}$  denotes an arbitrary functional, $C(\boldsymbol{x},\boldsymbol{x}s_{+}\prime )\doteq \langle \unicode[STIX]{x1D719}s_{+}\prime (\boldsymbol{x})\unicode[STIX]{x1D719}s_{+}\prime (\boldsymbol{x}s_{+}\prime )\rangle$ , and $\unicode[STIX]{x1D6FF}$  denotes a functional derivative. In the present context with several species-dependent fields, this generalizes to

(D 4) $$\begin{eqnarray}\boldsymbol{y}(\unicode[STIX]{x1D707})\doteq \left\langle \left(\begin{array}{@{}c@{}}Ns_{+}\prime (\unicode[STIX]{x1D707})\\ Es_{+}\prime (\unicode[STIX]{x1D707})\end{array}\right)\widetilde{\unicode[STIX]{x1D712}}\right\rangle _{0}=\unicode[STIX]{x1D648}_{2}(\unicode[STIX]{x1D707},\overline{\unicode[STIX]{x1D707}})\star \left(\begin{array}{@{}c@{}}\langle \unicode[STIX]{x1D6FF}\widetilde{\unicode[STIX]{x1D712}}/\unicode[STIX]{x1D6FF}Ns_{+}\prime (\overline{\unicode[STIX]{x1D707}})\rangle _{0}\\ \langle \unicode[STIX]{x1D6FF}\widetilde{\unicode[STIX]{x1D712}}/\unicode[STIX]{x1D6FF}Es_{+}\prime (\overline{\unicode[STIX]{x1D707}})\rangle _{0}\end{array}\right).\end{eqnarray}$$

Then, upon noting that $\unicode[STIX]{x1D648}_{2}^{-1}\star \unicode[STIX]{x1D648}_{2}=\unicode[STIX]{x1D644}$ and omitting dependence on  $t$ , one finds

(D 5) $$\begin{eqnarray}\text{P}\widetilde{\unicode[STIX]{x1D712}}(\boldsymbol{r})=\unicode[STIX]{x1D712}(\boldsymbol{r})+(Ns_{+}\prime ,\;Es_{+}\prime )\star \unicode[STIX]{x1D648}_{2}^{-1}\star \,\boldsymbol{y}=\mathop{\sum }_{\overline{s}}\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,(Ns_{+}\prime ,\;Es_{+}\prime )_{\overline{s}}(\overline{\boldsymbol{x}})\boldsymbol{\cdot }\left(\begin{array}{@{}c@{}}\langle \unicode[STIX]{x1D6FF}\widetilde{\unicode[STIX]{x1D712}}(\boldsymbol{r})/\unicode[STIX]{x1D6FF}Ns_{+}\prime _{\overline{s}}(\overline{\boldsymbol{x}})\rangle _{0}\\ \langle \unicode[STIX]{x1D6FF}\widetilde{\unicode[STIX]{x1D712}}(\boldsymbol{r})/\unicode[STIX]{x1D6FF}Es_{+}\prime _{\overline{s}}(\overline{\boldsymbol{x}})\rangle _{0}\end{array}\right).\end{eqnarray}$$

By definition of the coarse graining, the support of the functional derivatives lies essentially within the cell centred on  $\boldsymbol{r}$ , so the first  $\overline{\boldsymbol{x}}$ under the integral can be replaced by  $\boldsymbol{x}$ . The remaining $\overline{\boldsymbol{x}}$  integration changes the functional derivatives to ordinary partial derivatives, and one is led to (D 2).

One can now construct the subtracted fluxes.

  1. (i) The subtracted density flux $\widehat{\boldsymbol{J}}_{s}\text{}^{n}\doteq m_{s}^{-1}\widehat{\boldsymbol{P}}_{s}$ vanishes because $\boldsymbol{P}_{s}$  lies in the hydrodynamic subspace.Footnote 75

  2. (ii) For the subtracted momentum flux, note that the velocities of particles of species  $s$ are in the frame moving with  $\boldsymbol{u}_{s}$ , so $\langle \widetilde{\unicode[STIX]{x1D749}}_{s}^{\prime }\rangle =p_{s}\unicode[STIX]{x1D644}$ . Therefore, in that frame one has, with the aid of (D 2), that

    (D 6) $$\begin{eqnarray}\widehat{\unicode[STIX]{x1D749}}_{s}\approx \unicode[STIX]{x1D749}_{s}-\unicode[STIX]{x1D644}\left[\vphantom{\left(\frac{1}{2}\right)}\right.\!\!p_{s}+\mathop{\sum }_{\overline{s}}N_{\overline{s}}^{\prime }\left(\frac{\unicode[STIX]{x2202}p_{s}}{\unicode[STIX]{x2202}n_{\overline{s}}}\right)_{e}+\mathop{\sum }_{\overline{s}}E_{\overline{s}}^{\prime }\left(\frac{\unicode[STIX]{x2202}p_{s}}{\unicode[STIX]{x2202}e_{\overline{s}}}\right)_{n}\!\!\left.\vphantom{\left(\frac{1}{2}\right)}\right].\end{eqnarray}$$
    Without the frame change, terms of $O(u_{s}^{2})$ would appear.
  3. (iii) Now consider the subtracted energy flux. In the local frame, symmetry considerations and the result (D 1d ) lead to

    (D 7) $$\begin{eqnarray}\text{P}\widetilde{\boldsymbol{J}}_{s}\text{}^{e}(\boldsymbol{r})=\langle \widetilde{\boldsymbol{J}}_{s}\text{}^{e}\rangle _{0}+\mathop{\sum }_{\overline{s}}\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,\boldsymbol{P}_{\overline{s}}^{\prime }(\overline{\boldsymbol{x}})(m\overline{n}T)_{\overline{s}}^{-1}\langle \boldsymbol{P}_{\overline{s}}^{\prime }(\overline{\boldsymbol{x}})\widetilde{\boldsymbol{J}}_{s}\text{}^{e}(\boldsymbol{r})\rangle _{0}.\end{eqnarray}$$
    It is not difficult to show thatFootnote 76
    (D 8) $$\begin{eqnarray}\langle \widetilde{\boldsymbol{J}}_{s}\text{}^{e}\rangle _{0}=h_{s}\boldsymbol{u}_{s}=\left(\frac{h}{mn}\right)_{s}\boldsymbol{p}_{s},\end{eqnarray}$$
    where $h\doteq e+p$ is the enthalpy, and thatFootnote 77
    (D 9) $$\begin{eqnarray}\unicode[STIX]{x1D653}_{\overline{s}s}(\boldsymbol{r},\overline{\boldsymbol{x}})\doteq \langle \boldsymbol{P}_{\overline{s}}^{\prime }(\overline{\boldsymbol{x}})\widetilde{\boldsymbol{J}}_{s}\text{}^{e}(\boldsymbol{r})\rangle _{0}=T\unicode[STIX]{x1D644}\displaystyle \int \frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\,\text{e}^{\text{i}\boldsymbol{k}\boldsymbol{\cdot }(\overline{\boldsymbol{x}}-\boldsymbol{r})}\left(e_{0}+\overline{n}T+\frac{1}{3}\overline{n}^{\text{T}}\unicode[STIX]{x0394}\unicode[STIX]{x1D749}_{0}(\boldsymbol{k})\right)_{s}.\end{eqnarray}$$
    The $\boldsymbol{k}$ -independent terms are proportional to $\unicode[STIX]{x1D6FF}(\overline{\boldsymbol{x}}-\boldsymbol{r})$ . The $\unicode[STIX]{x0394}\unicode[STIX]{x1D749}(\boldsymbol{k})$ term decays within one cell; thus,
    (D 10) $$\begin{eqnarray}\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,\boldsymbol{P}s_{+}\prime (\overline{\boldsymbol{x}})\boldsymbol{\cdot }\unicode[STIX]{x1D653}(\boldsymbol{r}-\overline{\boldsymbol{x}})\approx \boldsymbol{P}s_{+}\prime (\boldsymbol{r})\boldsymbol{\cdot }\displaystyle \int \text{d}\unicode[STIX]{x1D746}\,\unicode[STIX]{x1D653}(\unicode[STIX]{x1D746}).\end{eqnarray}$$
    The last integral extracts the $\boldsymbol{k}=\mathbf{0}$ component, whereupon $\int \!\text{d}\unicode[STIX]{x1D746}\,\unicode[STIX]{x1D653}(\unicode[STIX]{x1D746})=Th(\overline{n}/n)\unicode[STIX]{x1D644}$ . Finally,
    (D 11) $$\begin{eqnarray}\text{P}\widetilde{\boldsymbol{J}}_{s}\text{}^{e}(\boldsymbol{r})=\left(\frac{h}{mn}\right)_{s}\boldsymbol{p}_{s}(\boldsymbol{r})+\left(\frac{h}{mn}\right)_{s}\boldsymbol{P}s_{+}\prime (\boldsymbol{r})=\left(\frac{h}{mn}\right)_{s}\boldsymbol{P}_{s}(\boldsymbol{r}),\end{eqnarray}$$
    so
    (D 12) $$\begin{eqnarray}\widehat{\boldsymbol{J}}\text{}_{s}^{E}=\widetilde{\boldsymbol{J}}_{s}\text{}^{E}-\left(\frac{h}{mn}\right)_{s}\widetilde{\boldsymbol{P}}_{s}.\end{eqnarray}$$

Appendix E. Aspects of the first-order exchange terms

In this section I consider aspects of the first-order exchange term

(E 1a ) $$\begin{eqnarray}\displaystyle X_{s}^{\unicode[STIX]{x1D6FC}} & \doteq & \displaystyle -k_{1\unicode[STIX]{x0394}}^{\overline{\unicode[STIX]{x1D6FD}}}[{\dot{A}}_{\unicode[STIX]{x0394}s}^{\prime \unicode[STIX]{x1D6FC}}](\unicode[STIX]{x1D707},t)B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t)\end{eqnarray}$$
(E 1b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle -\int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}\,\langle \text{Q}{\dot{A}}_{\unicode[STIX]{x0394}s}^{\prime \unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D707})\text{e}^{-\text{i}\text{Q}\mathscr{L}\overline{\unicode[STIX]{x1D70F}}}{\dot{A}}_{\unicode[STIX]{x0394}}^{\prime \overline{\unicode[STIX]{x1D6FD}}}(\overline{\unicode[STIX]{x1D707}})\rangle _{0}B_{\overline{\unicode[STIX]{x1D6FD}},\overline{s}}(\boldsymbol{r},t).\end{eqnarray}$$
There is no density contribution to this term since ${\dot{A}}_{\unicode[STIX]{x0394}}^{\prime n}=0$ . For $\unicode[STIX]{x1D6FC}=\boldsymbol{p}$ or $\unicode[STIX]{x1D6FC}=e$ , one can divide  $X^{\unicode[STIX]{x1D6FC}}$ into a hydrodynamic and a non-hydrodynamic part, $X^{\unicode[STIX]{x1D6FC}}=X_{\text{h}}^{\unicode[STIX]{x1D6FC}}+X_{\text{nh}}^{\unicode[STIX]{x1D6FC}}$ , corresponding to the two terms associated with the $\text{Q}=1-\text{P}$ in the propagator. I shall not give a complete discussion of all of the exchange effects in this appendix, but as an example of the manipulations I shall show that for a statistically homogeneous, weakly coupled plasma $X_{\text{h}}^{\boldsymbol{p}}$ reduces to the hydrodynamic contribution to the exchange term calculated in Part 1, namely (for two-species plasma)
(E 2) $$\begin{eqnarray}X_{s}^{\unicode[STIX]{x1D6FC}}=-(nm)_{s}\unicode[STIX]{x1D708}_{sss_{+}\prime }(\boldsymbol{u}_{s}-\boldsymbol{u}_{ss_{+}\prime })\end{eqnarray}$$

(see (1:3.28) and (1:3.29b)).

Symmetry in velocity space constrains  $\overline{\unicode[STIX]{x1D6FD}}$ to equal  $\unicode[STIX]{x1D6FC}$ . I shall calculate the hydrodynamic momentum exchange ( $\unicode[STIX]{x1D6FC}=\boldsymbol{p}$ ). The random exchange force  $\dot{\boldsymbol{P}}\text{}s_{+}\prime _{\unicode[STIX]{x0394}}$ is given by (B 8). When only the kinetic parts of  $\boldsymbol{A}$ are retained, it is easy to see that $\text{P}\dot{\boldsymbol{P}}s_{+}\prime _{\unicode[STIX]{x0394},s}$ vanishes by symmetry. Therefore, one must calculate

(E 3) $$\begin{eqnarray}-X_{\text{h}}^{\boldsymbol{p}}\doteq \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}\,\langle \dot{\boldsymbol{P}}\text{}s_{+}\prime _{\unicode[STIX]{x0394}s}(\unicode[STIX]{x1D707})\text{e}^{-\text{i}\mathscr{L}\overline{\unicode[STIX]{x1D70F}}}\dot{\boldsymbol{P}}\text{}s_{+}\prime _{\unicode[STIX]{x0394}\overline{s}}(\overline{\unicode[STIX]{x1D707}})\rangle _{0}\boldsymbol{\cdot }\boldsymbol{B}_{\boldsymbol{p},\overline{s}},\end{eqnarray}$$

where $\boldsymbol{B}_{\boldsymbol{p},\overline{s}}\doteq T_{\overline{s}}^{-1}\boldsymbol{u}_{\overline{s}}$ . Upon writing this in terms of the Klimontovich microdensity, one finds

(E 4) $$\begin{eqnarray}\displaystyle -X_{\text{h}}^{\boldsymbol{p}} & = & \displaystyle (\overline{n}q)_{s}\mathop{\sum }_{ss_{+}\prime \neq s}(\overline{n}q)_{ss_{+}\prime }\mathop{\sum }_{\overline{s}}(\overline{n}q)_{\overline{s}}\mathop{\sum }_{\overline{s}s_{+}\prime \neq \overline{s}}(\overline{n}q)_{\overline{s}s_{+}\prime }\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\,\displaystyle \int \text{d}\boldsymbol{v}\,\displaystyle \int \text{d}\boldsymbol{r}s_{+}\prime \displaystyle \int \text{d}\boldsymbol{v}s_{+}\prime \displaystyle \int \text{d}\overline{\boldsymbol{r}}\displaystyle \int \text{d}\overline{\boldsymbol{v}}\displaystyle \int \text{d}\overline{\boldsymbol{r}}s_{+}\prime \displaystyle \int \text{d}\overline{\boldsymbol{v}}s_{+}\prime \unicode[STIX]{x1D750}(\boldsymbol{r}-\boldsymbol{r}s_{+}\prime )\,\unicode[STIX]{x1D750}(\overline{\boldsymbol{r}}-\overline{\boldsymbol{r}}s_{+}\prime )\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\langle \,\widetilde{f}_{s}(\boldsymbol{r},\boldsymbol{v},\overline{\unicode[STIX]{x1D70F}})\widetilde{f}_{ss_{+}\prime }(\boldsymbol{r}s_{+}\prime ,\boldsymbol{v}s_{+}\prime ,\overline{\unicode[STIX]{x1D70F}})\widetilde{f}_{\overline{s}}(\overline{\boldsymbol{r}},\overline{\boldsymbol{v}},0)\widetilde{f}_{\overline{s}s_{+}\prime }(\overline{\boldsymbol{r}}s_{+}\prime ,\overline{\boldsymbol{v}}s_{+}\prime ,0)\rangle _{0}\boldsymbol{\cdot }\boldsymbol{B}_{\boldsymbol{p},\overline{s}},\end{eqnarray}$$

where

(E 5) $$\begin{eqnarray}\unicode[STIX]{x1D750}(\boldsymbol{r})\doteq -\unicode[STIX]{x1D735}(r^{-1})=\displaystyle \int \frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\,\text{e}^{\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{r}}\unicode[STIX]{x1D750}_{\boldsymbol{k}}\quad (\text{with }\unicode[STIX]{x1D750}_{\boldsymbol{k}}\doteq -4\unicode[STIX]{x03C0}\text{i}\boldsymbol{k}/k^{2})\end{eqnarray}$$

gives the electric field due to a unit point charge at the origin. Cumulant expansion of the four-point correlation function gives schematically

(E 6) $$\begin{eqnarray}\langle \,\widetilde{f}\,\widetilde{f}\,\widetilde{f}\,\widetilde{f}\rangle =f\,f\,f\,f+6f\,fC_{2}+4fC_{3}+3C_{2}C_{2}+C_{4},\end{eqnarray}$$

where $C_{n}$ denotes the $n$ th-order cumulant. It is assumed that $f$  is a Maxwellian  $f_{\text{M}}$ to lowest order; this is equivalent to subtracting out the contributions due to a long-ranged mean potential. Then all of the terms involving  $f$ vanish under spatial integration due to the presence of an  $\unicode[STIX]{x1D750}$ and the isotropy of  $f_{\text{M}}$ . Of the three terms involving $CC$ , where $C\equiv C_{2}$ , one involves the equal-time correlations $C_{sss_{+}\prime }(\boldsymbol{r}-\boldsymbol{r}s_{+}\prime ,\boldsymbol{v},\boldsymbol{v}s_{+}\prime ,0)C_{\overline{s}\,\overline{s}s_{+}\prime }(\overline{\boldsymbol{r}}-\overline{\boldsymbol{r}}s_{+}\prime ,\overline{\boldsymbol{v}},\overline{\boldsymbol{v}}s_{+}\prime ,0)$ . That term also does not contribute because to lowest order $C(\boldsymbol{r}-\boldsymbol{r}s_{+}\prime )=C(|\boldsymbol{r}-\boldsymbol{r}s_{+}\prime |)$ and $\int \text{d}\unicode[STIX]{x1D746}\,\unicode[STIX]{x1D750}(\unicode[STIX]{x1D746})C(\unicode[STIX]{x1D70C})=\mathbf{0}$ due to isotropy. Because for thermal noise $C_{n}=O(\unicode[STIX]{x1D716}_{\text{p}}^{n-1})$ , $C_{4}$  is negligible for weak coupling. This leaves

(E 7) $$\begin{eqnarray}\displaystyle -X_{\text{h}}^{\boldsymbol{p}} & = & \displaystyle (\overline{n}q)_{s}\mathop{\sum }_{ss_{+}\prime \neq s}(\overline{n}q)_{ss_{+}\prime }\mathop{\sum }_{\overline{s}}(\overline{n}q)_{\overline{s}}\mathop{\sum }_{\overline{s}s_{+}\prime \neq \overline{s}}(\overline{n}q)_{\overline{s}s_{+}\prime }\nonumber\\ \displaystyle & & \displaystyle \times \,\int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\,\displaystyle \int \text{d}\boldsymbol{v}\,\displaystyle \int \text{d}\boldsymbol{r}s_{+}\prime \displaystyle \int \text{d}\boldsymbol{v}s_{+}\prime \displaystyle \int \text{d}\overline{\boldsymbol{r}}\displaystyle \int \text{d}\overline{\boldsymbol{v}}\displaystyle \int \text{d}\overline{\boldsymbol{r}}s_{+}\prime \displaystyle \int \text{d}\overline{\boldsymbol{v}}s_{+}\prime \unicode[STIX]{x1D750}(\boldsymbol{r}-\boldsymbol{r}s_{+}\prime )\,\unicode[STIX]{x1D750}(\overline{\boldsymbol{r}}-\overline{\boldsymbol{r}}s_{+}\prime )\nonumber\\ \displaystyle & & \displaystyle \times \,[C_{s\overline{s}}(\boldsymbol{r}-\overline{\boldsymbol{r}},\boldsymbol{v},\overline{\unicode[STIX]{x1D70F}};\overline{\boldsymbol{v}})C_{ss_{+}\prime \overline{s}s_{+}\prime }(\boldsymbol{r}s_{+}\prime -\overline{\boldsymbol{r}}s_{+}\prime ,\boldsymbol{v}s_{+}\prime ,\overline{\unicode[STIX]{x1D70F}};\overline{\boldsymbol{v}}s_{+}\prime )\nonumber\\ \displaystyle & & \displaystyle \quad ~+\,C_{s\overline{s}s_{+}\prime }(\boldsymbol{r}-\overline{\boldsymbol{r}}s_{+}\prime ,\boldsymbol{v},\overline{\unicode[STIX]{x1D70F}};\overline{\boldsymbol{v}}s_{+}\prime )C_{ss_{+}\prime \overline{s}}(\boldsymbol{r}s_{+}\prime -\overline{\boldsymbol{r}},\boldsymbol{v}s_{+}\prime ,\overline{\unicode[STIX]{x1D70F}};\overline{\boldsymbol{v}})]\boldsymbol{\cdot }\boldsymbol{B}_{\boldsymbol{p},\overline{s}}.\end{eqnarray}$$

In the plasma ordering (Rostoker & Rosenbluth Reference Rostoker and Rosenbluth1960), one has $\overline{n}=O(\unicode[STIX]{x1D716}_{\text{p}}^{-1})$ , $q=O(\unicode[STIX]{x1D716}_{\text{p}})$ , and $T=O(\unicode[STIX]{x1D716}_{\text{p}})$ ; I have already noted that $C=O(\unicode[STIX]{x1D716}_{\text{p}})$ . The coefficient of  $\boldsymbol{u}$ is therefore $O(\unicode[STIX]{x1D716}_{\text{p}})$ . Thus, it is adequate to calculate  $C$ only to lowest order in  $\unicode[STIX]{x1D716}_{\text{p}}$ (i.e. to use just the collisionless Vlasov response). Since the integration is extended only from 0 to  $\infty$ , only the one-sided correlation function $C_{+}(\overline{\unicode[STIX]{x1D70F}})\doteq H(\overline{\unicode[STIX]{x1D70F}})C(\overline{\unicode[STIX]{x1D70F}})$ enters; its evolution is expressed by the Vlasov response function  $R^{(0)}$ :

(E 8) $$\begin{eqnarray}C_{+}(\overline{\unicode[STIX]{x1D70F}})\approx R^{(0)}(\overline{\unicode[STIX]{x1D70F}})\ast C(0),\end{eqnarray}$$

where again $\ast$  denotes convolution. The initial condition is

(E 9) $$\begin{eqnarray}C_{sss_{+}\prime }(\boldsymbol{r},\boldsymbol{v},\boldsymbol{r}s_{+}\prime ,\boldsymbol{v}s_{+}\prime ,0)=\unicode[STIX]{x1D6FF}_{sss_{+}\prime }\unicode[STIX]{x1D6FF}(\boldsymbol{r}-\boldsymbol{r}s_{+}\prime )\unicode[STIX]{x1D6FF}(\boldsymbol{v}-\boldsymbol{v}s_{+}\prime )\overline{n}_{ss_{+}\prime }^{-1}f_{ss_{+}\prime }(\boldsymbol{r}s_{+}\prime ,\boldsymbol{v}s_{+}\prime ,0)+g_{sss_{+}\prime }(\boldsymbol{r},\boldsymbol{v},\boldsymbol{r}s_{+}\prime ,\boldsymbol{v}s_{+}\prime ,0),\end{eqnarray}$$

where $g$  is the pair correlation function.Footnote 78 Consistent with the assumptions used in deriving the Balescu–Lenard collision operator, I shall use the  $g$ appropriate for Debye length scales; one may use the thermal-equilibrium result since $X^{\unicode[STIX]{x1D6FC}}$  is calculated as a linearization from a Maxwellian. That formula is well known;Footnote 79 one finds (for Vlasov-scale wavenumbers)

(E 10) $$\begin{eqnarray}g_{sss_{+}\prime }(\boldsymbol{k},\boldsymbol{v},\boldsymbol{v}s_{+}\prime ,0)\approx -\left(\frac{q_{s}q_{ss_{+}\prime }}{T}\right)\frac{4\unicode[STIX]{x03C0}/k^{2}}{\mathscr{D}(\boldsymbol{k},0)},\end{eqnarray}$$

where the static dielectric function is

(E 11) $$\begin{eqnarray}\mathscr{D}(\boldsymbol{k},\unicode[STIX]{x1D714}=0)\doteq 1+\frac{k_{\text{D}}^{2}}{k^{2}}.\end{eqnarray}$$

Here the Debye wavenumber  $k_{\text{D}}$ obeys $k_{\text{D}}^{2}=\sum _{s}k_{\text{D}s}^{2}$ with $k_{\text{D}s}\doteq (4\unicode[STIX]{x03C0}\overline{n}q^{2}/T)_{s}^{1/2}$ . Upon Fourier transformation, one is led to

(E 12) $$\begin{eqnarray}\displaystyle -X_{\text{h}}^{\boldsymbol{p}} & = & \displaystyle (\overline{n}q)_{s}\mathop{\sum }_{ss_{+}\prime \neq s}(\overline{n}q)_{ss_{+}\prime }\mathop{\sum }_{\overline{s}}(\overline{n}q)_{\overline{s}}\mathop{\sum }_{\overline{s}s_{+}\prime \neq \overline{s}}(\overline{n}q)_{\overline{s}s_{+}\prime }\displaystyle \int \text{d}\boldsymbol{v}\displaystyle \int \text{d}\boldsymbol{v}s_{+}\prime \displaystyle \int \text{d}\overline{\boldsymbol{v}}\displaystyle \int \text{d}\overline{\boldsymbol{v}}s_{+}\prime \displaystyle \int \frac{\text{d}\unicode[STIX]{x1D714}}{2\unicode[STIX]{x03C0}}\displaystyle \int \frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\nonumber\\ \displaystyle & & \displaystyle \times \,\unicode[STIX]{x1D750}_{\boldsymbol{k}}\,\unicode[STIX]{x1D750}_{\boldsymbol{k}}^{\ast }[C_{+,s\overline{s}}^{\ast }(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{v};\overline{\boldsymbol{v}})C_{+,ss_{+}\prime \overline{s}s_{+}\prime }(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{v}s_{+}\prime ;\overline{\boldsymbol{v}}s_{+}\prime )-C_{+,s\overline{s}s_{+}\prime }^{\ast }(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{v};\overline{\boldsymbol{v}}s_{+}\prime )C_{+,ss_{+}\prime \overline{s}}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{v}s_{+}\prime ;\overline{\boldsymbol{v}})]\nonumber\\ \displaystyle & & \displaystyle \boldsymbol{\cdot }\,\boldsymbol{B}_{\boldsymbol{p},\overline{s}}.\end{eqnarray}$$

To be specific, I consider $s=e$ and a single species of ions. The expression (E 12) then reduces to

(E 13) $$\begin{eqnarray}\displaystyle -_{a}X_{\text{h}}^{\boldsymbol{p}} & = & \displaystyle (\overline{n}q)_{e}^{2}(\overline{n}q)_{i}^{2}\displaystyle \int \text{d}\boldsymbol{v}\displaystyle \int \text{d}\boldsymbol{v}s_{+}\prime \displaystyle \int \text{d}\overline{\boldsymbol{v}}\displaystyle \int \text{d}\overline{\boldsymbol{v}}s_{+}\prime \displaystyle \int \frac{\text{d}\unicode[STIX]{x1D714}}{2\unicode[STIX]{x03C0}}\displaystyle \int \frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\,\unicode[STIX]{x1D750}_{\boldsymbol{k}}\,\unicode[STIX]{x1D750}_{\boldsymbol{k}}^{\ast }\nonumber\\ \displaystyle & & \displaystyle \times \,[C_{+,ee}^{\ast }(\boldsymbol{k},\boldsymbol{v};\overline{\boldsymbol{v}})C_{+,ii}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{v}s_{+}\prime ;\overline{\boldsymbol{v}}s_{+}\prime )-C_{+,ei}^{\ast }(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{v};\overline{\boldsymbol{v}}s_{+}\prime )C_{+,ie}(\boldsymbol{v};\overline{\boldsymbol{v}})]\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{B}_{\boldsymbol{p}},\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where

(E 14) $$\begin{eqnarray}\unicode[STIX]{x0394}\boldsymbol{B}_{\boldsymbol{p}}\doteq \boldsymbol{B}_{\boldsymbol{p},e}-\boldsymbol{B}_{\boldsymbol{p},i}=(\unicode[STIX]{x1D6FD}\boldsymbol{u})_{e}-(\unicode[STIX]{x1D6FD}\boldsymbol{u})_{i}\approx T^{-1}(\boldsymbol{u}_{e}-\boldsymbol{u}_{i}).\end{eqnarray}$$

To obtain (E 13), I interchanged  $\overline{\boldsymbol{v}}$ and  $\overline{\boldsymbol{v}}s_{+}\prime$ after expanding out the species dependence of the  $\boldsymbol{B}_{i}$ term. The approximation of a common temperature in the last form of (E 14) is justified because I am ignoring second-order exchange effects.

To complete the calculation, one needs according to (E 8) an expression for the response function  $R^{(0)}$ . It is well known (Krommes Reference Krommes, Galeev and Sudan1984, Reference Krommes2002, Reference Krommes2015) that operational methods lead to the general expression for the fully renormalized electrostatic response function

(E 15) $$\begin{eqnarray}R=r-r\,\boldsymbol{\unicode[STIX]{x2202}}f\,\mathscr{D}^{-1}\boldsymbol{\cdot }\pmb{\mathbb{E}}r,\end{eqnarray}$$

where $r$  is the single-particle response function, $\boldsymbol{\unicode[STIX]{x2202}}\doteq (q/m)\unicode[STIX]{x2202}/\unicode[STIX]{x2202}_{\boldsymbol{v}}$ , $\mathscr{D}$  is the dielectric function

(E 16) $$\begin{eqnarray}\mathscr{D}=1+\pmb{\mathbb{E}}r\boldsymbol{\cdot }\boldsymbol{\unicode[STIX]{x2202}}f\end{eqnarray}$$

and $\pmb{\mathbb{E}}$ is the electric-field operator whose kernel is $\pmb{\mathbb{E}}_{s,\overline{s}}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{v};\overline{\boldsymbol{v}})=\unicode[STIX]{x1D750}_{\boldsymbol{k}}(\overline{n}q)_{\overline{s}}$ . In the presence of fluctuations, the actual evaluation of  $r$ and  $\mathscr{D}$ is entirely non-trivial and occupies a good portion of the formal discussion of plasma turbulence theory (Krommes Reference Krommes2002). But, as noted above, one only requires the collisionless approximation  $R^{(0)}$ . In that case

(E 17) $$\begin{eqnarray}r_{s\bar{s}}^{(0)}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{v};\overline{\boldsymbol{v}})=\frac{\unicode[STIX]{x1D6FF}_{s\overline{s}}\unicode[STIX]{x1D6FF}(\boldsymbol{v}-\overline{\boldsymbol{v}})}{-\text{i}(\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}+\text{i}\unicode[STIX]{x1D716})}\end{eqnarray}$$

(the positive infinitesimal  $\unicode[STIX]{x1D716}$ enforces causality) and

(E 18) $$\begin{eqnarray}\mathscr{D}(\boldsymbol{k},\unicode[STIX]{x1D714})=1+\mathop{\sum }_{\overline{s}}\unicode[STIX]{x1D712}_{\overline{s}}(\boldsymbol{k},\unicode[STIX]{x1D714}),\end{eqnarray}$$

where the zeroth-order susceptibility is

(E 19) $$\begin{eqnarray}\unicode[STIX]{x1D712}_{\overline{s}}^{(0)}(\boldsymbol{k},\unicode[STIX]{x1D714})=\frac{\unicode[STIX]{x1D714}_{\text{p}\overline{s}}^{2}}{k^{2}}\displaystyle \int \text{d}\overline{\boldsymbol{v}}\,\frac{\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x2202}_{\overline{\boldsymbol{v}}}f_{\overline{s}}(\overline{\boldsymbol{v}})}{\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{v}}+\text{i}\unicode[STIX]{x1D716}}.\end{eqnarray}$$

The delta functions in (E 17) then allow (E 15) to be simplified to

(E 20) $$\begin{eqnarray}\displaystyle R_{s\overline{s}}^{(0)}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{v};\overline{\boldsymbol{v}}) & = & \displaystyle \frac{\unicode[STIX]{x1D6FF}_{s\overline{s}}\unicode[STIX]{x1D6FF}(\boldsymbol{v}-\overline{\boldsymbol{v}})}{-\text{i}(\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}+\text{i}\unicode[STIX]{x1D716})}\nonumber\\ \displaystyle & & \displaystyle -\!\left(\frac{\unicode[STIX]{x1D750}_{\boldsymbol{k}}\boldsymbol{\cdot }\boldsymbol{\unicode[STIX]{x2202}}f_{s}}{-\text{i}(\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}+\text{i}\unicode[STIX]{x1D716})}\right)\!\frac{1}{\mathscr{D}^{(0)}(\boldsymbol{k},\unicode[STIX]{x1D714})}\!\left(\frac{(\overline{n}q)_{\overline{s}}}{-\text{i}(\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{v}}+\text{i}\unicode[STIX]{x1D716})}\right)\!.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

This expression can now be used in conjunction with (E 8) to evaluate (E 13). After somewhat lengthy but straightforward algebra that includes a number of cancellations, one finds

(E 21) $$\begin{eqnarray}\displaystyle -_{a}X_{\text{h}}^{\boldsymbol{p}} & = & \displaystyle (\overline{n}q^{2})_{e}(\overline{n}q^{2})_{i}\displaystyle \int \text{d}\boldsymbol{v}\displaystyle \int \text{d}\boldsymbol{v}s_{+}\prime \displaystyle \int \frac{\text{d}\unicode[STIX]{x1D714}}{2\unicode[STIX]{x03C0}}\displaystyle \int \frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\,\frac{1}{\mathscr{D}(\boldsymbol{k},0)}\,\unicode[STIX]{x1D750}_{\boldsymbol{k}}\,\unicode[STIX]{x1D750}_{\boldsymbol{k}}^{\ast }\nonumber\\ \displaystyle & & \displaystyle \times \,\left(\frac{1+\unicode[STIX]{x1D712}_{i}^{(0)\ast }(\boldsymbol{k},\unicode[STIX]{x1D714})+\unicode[STIX]{x1D712}_{e}^{(0)}(\boldsymbol{k},\unicode[STIX]{x1D714})}{|\mathscr{D}^{(0)}(\boldsymbol{k},\unicode[STIX]{x1D714})|^{2}}\right)\left(\frac{f_{e}(\boldsymbol{v})f_{i}(\boldsymbol{v}s_{+}\prime )}{(\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}-\text{i}\unicode[STIX]{x1D716})(\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}s_{+}\prime +\text{i}\unicode[STIX]{x1D716})}\right).\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The static shielding factor  $\mathscr{D}(\boldsymbol{k},0)^{-1}$ that appears in this expression is built from two contributions corresponding to the two terms of (E 9):

(E 22) $$\begin{eqnarray}\frac{1}{\mathscr{D}(\boldsymbol{k},0)}=1-\frac{k_{\text{D}}^{2}/k^{2}}{\mathscr{D}(\boldsymbol{k},0)}.\end{eqnarray}$$

It reflects the shielding of a test particle (the first term of (E 9)) by polarization (the second term of (E 9)).

The following remarks are relevant to the goal of obtaining expression (E 2): (i) (E 21) cannot be immediately reduced by residue methods applied to the $\unicode[STIX]{x1D714}$  integration because of the factor $|\mathscr{D}_{0}(\boldsymbol{k},\unicode[STIX]{x1D714})|^{-2}$ , which contains structure in both of the top and bottom halves of the complex $\unicode[STIX]{x1D714}$  plane; (ii) the expression (E 2) was derived from the Landau collision operator linearized around a Maxwellian; (iii) the Landau operator is an approximation to the more fundamental Balescu–Lenard operator. It is rather clear that from the present approach one should ultimately obtain the linearized-Balescu–Lenard generalization of (E 2). However, (E 21) holds for arbitrary  $f$ ; to obtain agreement, one needs to specialize that to $f=f_{\text{M}}$ and is free to use special properties of the Maxwellian.

One algebraic route is as follows. Because all susceptibilities from this point forward will be evaluated at zeroth order, I shall drop the $(0)$ superscripts in order to unclutter the notation. Note that

(E 23) $$\begin{eqnarray}1+\unicode[STIX]{x1D712}_{i}^{\ast }+\unicode[STIX]{x1D712}_{e}=1+\unicode[STIX]{x1D712}_{i}^{\ast }+\unicode[STIX]{x1D712}_{e}^{\ast }+(\unicode[STIX]{x1D712}_{e}-\unicode[STIX]{x1D712}_{e}^{\ast })=\mathscr{D}^{\ast }+2\text{i}\,\text{Im}\unicode[STIX]{x1D712}_{e}.\end{eqnarray}$$

The $\mathscr{D}^{\ast }$ term cancels one factor of the $|\mathscr{D}|^{-2}$ , leaving the integral

(E 24a ) $$\begin{eqnarray}\displaystyle I_{1} & \doteq & \displaystyle \displaystyle \int \frac{\text{d}\unicode[STIX]{x1D714}}{2\unicode[STIX]{x03C0}}\frac{1}{\mathscr{D}(\boldsymbol{k},\unicode[STIX]{x1D714})(\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}-\text{i}\unicode[STIX]{x1D716})(\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}s_{+}\prime +\text{i}\unicode[STIX]{x1D716})}\end{eqnarray}$$
(E 24b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle \frac{\text{i}}{\mathscr{D}(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{ v})[\boldsymbol{k}\boldsymbol{\cdot }(\boldsymbol{v}-\boldsymbol{v}s_{+}\prime )+\text{i}\unicode[STIX]{x1D716}]}\end{eqnarray}$$
(E 24c ) $$\begin{eqnarray}\displaystyle & = & \displaystyle \text{i}\left(\frac{\mathscr{D}s_{+}\prime (\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v})-\text{i}\mathscr{D}s_{+}\prime \prime (\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v})}{|\mathscr{D}(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v})|^{2}}\right)\left[\Pr \left(\frac{1}{\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{v}}\right)-\text{i}\unicode[STIX]{x03C0}\,\unicode[STIX]{x1D6FF}(\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{v})\right],\end{eqnarray}$$
where prime and double prime denote real and imaginary parts. Only the real part of this expression is required:
(E 25) $$\begin{eqnarray}\displaystyle \text{Re}\,I_{1} & = & \displaystyle \frac{1}{|\mathscr{D}(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v})|^{2}}\left[\vphantom{\left(\frac{1}{2}\right)}\right.\![\underbrace{1}_{(\text{a}_{1})}+\underbrace{\unicode[STIX]{x1D712}_{e}^{\prime }(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v})}_{(\text{b}_{1})}+\underbrace{\unicode[STIX]{x1D712}_{i}^{\prime }(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v})}_{(\text{c}_{1})}]\unicode[STIX]{x03C0}\,\unicode[STIX]{x1D6FF}(\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{v})\nonumber\\ \displaystyle & & \displaystyle +[\underbrace{\unicode[STIX]{x1D712}_{e}^{\prime \prime }(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v})}_{(\text{d}_{1})}+\underbrace{\unicode[STIX]{x1D712}_{i}^{\prime \prime }(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v})}_{(\text{e}_{1})}]\Pr \left(\frac{1}{\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{v}}\right)\!\!\left.\vphantom{\left(\frac{1}{2}\right)}\right].\end{eqnarray}$$

The last term of (E 23) contributes

(E 26) $$\begin{eqnarray}I_{2}\doteq 2\text{i}\displaystyle \int \frac{\text{d}\unicode[STIX]{x1D714}}{2\unicode[STIX]{x03C0}}\,\left(\frac{\unicode[STIX]{x1D712}_{e}^{\prime \prime }(\boldsymbol{k},\unicode[STIX]{x1D714})}{|\mathscr{D}(\boldsymbol{k},\unicode[STIX]{x1D714})|^{2}}\right)\left(\frac{1}{(\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}-\text{i}\unicode[STIX]{x1D716})(\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}s_{+}\prime +\text{i}\unicode[STIX]{x1D716})}\right),\end{eqnarray}$$

the real part of which is

(E 27) $$\begin{eqnarray}\text{Re}\,I_{2}=-\!\Pr \left(\frac{1}{\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{v}}\right)\left(\vphantom{\left(\frac{1}{2}\right)}\right.\!\underbrace{\frac{\unicode[STIX]{x1D712}_{e}^{\prime \prime }(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v})}{|\mathscr{D}(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v})|^{2}}}_{(\text{d}_{2})}+\underbrace{\frac{\unicode[STIX]{x1D712}_{e}^{\prime \prime }(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}s_{+}\prime )}{|\mathscr{D}(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}s_{+}\prime )|^{2}}}_{(\text{f}_{2})}\!\left.\vphantom{\left(\frac{1}{2}\right)}\right).\end{eqnarray}$$

Term (d $_{2}$ ) cancels with term (d $_{1}$ ).

The remaining terms must be averaged over  $f_{e}(\boldsymbol{v})$ and  $f_{i}(\boldsymbol{v}s_{+}\prime )$ according to (E 21). Progress can be made upon specializing to $f=f_{\text{M}}$ and using the property $\unicode[STIX]{x2202}_{\boldsymbol{v}}f_{\text{M}}=-(\boldsymbol{v}/v_{\text{t}}^{2})f_{\text{M}}$ . It is then easy to show that

(E 28) $$\begin{eqnarray}\unicode[STIX]{x1D712}_{s}(\boldsymbol{k},\unicode[STIX]{x1D714})=\frac{k_{\text{D}s}^{2}}{k^{2}}-\unicode[STIX]{x1D714}\left(\frac{k_{\text{D}s}^{2}}{k^{2}}\right)\displaystyle \int \text{d}\overline{\boldsymbol{v}}\,\frac{f_{s}(\overline{\boldsymbol{v}})}{\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{v}}+\text{i}\unicode[STIX]{x1D716}},\end{eqnarray}$$

or

(E 29a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}_{s}^{\prime }(\boldsymbol{k},\unicode[STIX]{x1D714}) & = & \displaystyle \frac{k_{\text{D}s}^{2}}{k^{2}}-\unicode[STIX]{x1D714}\left(\frac{k_{\text{D}s}^{2}}{k^{2}}\right)\displaystyle \int \text{d}\overline{\boldsymbol{v}}\,\Pr \left(\frac{1}{\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{v}}}\right)f_{s}(\overline{\boldsymbol{v}}),\end{eqnarray}$$
(E 29b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}_{s}^{\prime \prime }(\boldsymbol{k},\unicode[STIX]{x1D714}) & = & \displaystyle \unicode[STIX]{x03C0}\unicode[STIX]{x1D714}\left(\frac{k_{\text{D}s}^{2}}{k^{2}}\right)\displaystyle \int \text{d}\overline{\boldsymbol{v}}\,\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{v}})f_{s}(\overline{\boldsymbol{v}}).\end{eqnarray}$$
Upon using (E 29b ) to evaluate term (f $_{2}$ ), then interchanging  $\boldsymbol{v}$ and  $\overline{\boldsymbol{v}}$ , one can show that
(E 30) $$\begin{eqnarray}\displaystyle & & \displaystyle -\displaystyle \int \text{d}\boldsymbol{v}\,\text{d}\boldsymbol{v}s_{+}\prime \,\Pr \left(\frac{1}{\boldsymbol{k}\boldsymbol{\cdot }(\boldsymbol{v}-\boldsymbol{v}s_{+}\prime )}\right)\underbrace{\frac{\unicode[STIX]{x1D712}_{e}^{\prime \prime }(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}s_{+}\prime )}{|\mathscr{D}(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}s_{+}\prime )|^{2}}}_{(\text{f}_{2})}f_{e}(\boldsymbol{v})f_{i}(\boldsymbol{v}s_{+}\prime )\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x03C0}\displaystyle \int \text{d}\boldsymbol{v}\,\text{d}\boldsymbol{v}s_{+}\prime \,\left(\vphantom{\left(\frac{1}{2}\right)}\right.\!\underbrace{\frac{k_{\text{D}e}^{2}}{k^{2}}}_{(\text{a}_{e})}-\underbrace{\unicode[STIX]{x1D712}_{e}^{\prime }(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v})}_{(\text{b}_{2})}\!\left.\vphantom{\left(\frac{1}{2}\right)}\right)\frac{\unicode[STIX]{x1D6FF}\boldsymbol{(}\boldsymbol{k}\boldsymbol{\cdot }(\boldsymbol{v}-\boldsymbol{v}s_{+}\prime )\boldsymbol{)}}{|\mathscr{D}(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v})|^{2}}f_{e}(\boldsymbol{v})f_{i}(\boldsymbol{v}s_{+}\prime ).\end{eqnarray}$$

Term (a $_{e}$ ) adds to term (a $_{1}$ ), while term (b $_{2}$ ) cancels with term (b $_{1}$ ). Similarly, the contribution of term (e $_{1}$ ) can be evaluated by interchanging  $\boldsymbol{v}s_{+}\prime$ and  $\overline{\boldsymbol{v}}$ ; one finds

(E 31) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \int \text{d}\boldsymbol{v}\,\text{d}\boldsymbol{v}s_{+}\prime \,\Pr \left(\frac{1}{\boldsymbol{k}\boldsymbol{\cdot }(\boldsymbol{v}-\boldsymbol{v}s_{+}\prime )}\right)\underbrace{\frac{\unicode[STIX]{x1D712}_{e}^{\prime \prime }(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}s_{+}\prime )}{|\mathscr{D}(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}s_{+}\prime )|^{2}}}_{(\text{e}_{1})}f_{e}(\boldsymbol{v})f_{i}(\boldsymbol{v}s_{+}\prime )\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x03C0}\displaystyle \int \text{d}\boldsymbol{v}\,\text{d}\boldsymbol{v}s_{+}\prime \,\left(\vphantom{\left(\frac{1}{2}\right)}\right.\!\underbrace{\frac{k_{\text{D}i}^{2}}{k^{2}}}_{(\text{a}_{i})}-\underbrace{\unicode[STIX]{x1D712}_{i}^{\prime }(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v})}_{(\text{g}_{1})}\!\left.\vphantom{\left(\frac{1}{2}\right)}\right)\frac{\unicode[STIX]{x1D6FF}\boldsymbol{(}\boldsymbol{k}\boldsymbol{\cdot }(\boldsymbol{v}-\boldsymbol{v}s_{+}\prime )\boldsymbol{)}}{|\mathscr{D}(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v})|^{2}}f_{e}(\boldsymbol{v})f_{i}(\boldsymbol{v}s_{+}\prime ).\end{eqnarray}$$

Note the overall sign difference between (E 30) and (E 31). Term (a $_{i}$ ) adds to term (a $_{1}$ ), while term (g $_{1}$ ) cancels with term (c $_{1}$ ). One finally obtains

(E 32) $$\begin{eqnarray}X_{e}^{\boldsymbol{p}}=-\left(\unicode[STIX]{x03C0}\,(\overline{n}q^{2})_{e}(\overline{n}q^{2})_{i}\displaystyle \int \text{d}\boldsymbol{v}\displaystyle \int \text{d}\overline{\boldsymbol{v}}\displaystyle \int \frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\frac{\unicode[STIX]{x1D750}_{\boldsymbol{k}}\,\unicode[STIX]{x1D750}_{\boldsymbol{k}}^{\ast }}{|\mathscr{D}(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v})|^{2}}\unicode[STIX]{x1D6FF}\boldsymbol{(}\boldsymbol{k}\boldsymbol{\cdot }(\boldsymbol{v}-\overline{\boldsymbol{v}})\boldsymbol{)}f_{e}(\boldsymbol{v})f_{i}(\overline{\boldsymbol{v}})\!\right)\!\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{B},\end{eqnarray}$$

where $\unicode[STIX]{x0394}\boldsymbol{B}$  is given by (E 14). This is the Balescu–Lenard generalization of the matrix element that defines the hydrodynamic part of the momentum exchange term. Specifically, the positive–definite term in large parentheses is (to within a normalization factor) $-\text{i}(\unicode[STIX]{x1D734}_{\text{C}})_{\boldsymbol{p}}^{\boldsymbol{p}}$ , where $\unicode[STIX]{x1D734}_{\text{C}}$  is the collisional contribution to the frequency matrix defined by (1:2.56b). The Landau form of this term (used by Braginskii) is obtained in the standard way by setting $\mathscr{D}(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v})$ to one and integrating in wavenumber magnitude  $k$ between $k_{\text{D}}$ and $k_{\text{max}}$ . (Classically, $k_{\text{max}}=b_{0}^{-1}$ , where $b_{0}$  is the impact parameter for $90^{\circ }$ scattering. See footnote 88 on page 92 for more discussion about the cutoff.) The final result is in complete agreement with the calculations of Part 1.

Appendix F. Evaluation of term (iv)

Here I provide some details that were omitted by Brey et al. (Reference Brey, Zwanzig and Dorfman1981). Upon inserting the first term of (2.74) into (3.36), one has

(F 1a ) $$\begin{eqnarray}\displaystyle \text{term (iv)} & {\approx} & \displaystyle \displaystyle \int _{0}^{t}\!\text{d}\overline{s}\,\langle Gs_{+}\prime \text{U}(\overline{s})\text{i}\mathscr{L}\boldsymbol{A}s_{+}\prime \rangle _{0}\star \unicode[STIX]{x1D648}^{-1}\star \langle \boldsymbol{A}s_{+}\prime \unicode[STIX]{x1D713}_{\unicode[STIX]{x0394}\boldsymbol{B}}(t-\overline{s})\boldsymbol{A}\text{}s_{+}\prime \text{}^{\text{T}}\rangle _{0}\star \unicode[STIX]{x0394}\boldsymbol{B}(t-\overline{s})\end{eqnarray}$$
(F 1b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle -\displaystyle \int _{0}^{t}\!\text{d}\overline{s}\displaystyle \int _{0}^{t-\overline{s}}\!\text{d}\overline{s}s_{+}\prime \displaystyle \int \text{d}\overline{\boldsymbol{x}}\displaystyle \int \text{d}\overline{\boldsymbol{x}}s_{+}\prime \,\langle Gs_{+}\prime \text{U}(\overline{s})\text{i}\mathscr{L}\boldsymbol{A}s_{+}\prime \rangle _{0}\star \unicode[STIX]{x1D648}^{-1}\nonumber\\ \displaystyle & & \displaystyle \star \,\langle \boldsymbol{A}s_{+}\prime [U(\overline{s}s_{+}\prime )\text{i}\mathscr{L}\text{A}s_{+}\prime ^{\unicode[STIX]{x1D6FD}}(\overline{\unicode[STIX]{x1D707}}s_{+}\prime )]As_{+}\prime ^{\unicode[STIX]{x1D6FE}}(\overline{\unicode[STIX]{x1D707}}s_{+}\prime )\rangle _{0}\unicode[STIX]{x0394}B_{\unicode[STIX]{x1D6FD}}(\overline{\unicode[STIX]{x1D707}}s_{+}\prime ,t-\overline{s}-\overline{s}s_{+}\prime )\unicode[STIX]{x0394}B_{\unicode[STIX]{x1D6FE}}(\overline{\unicode[STIX]{x1D707}},t-\overline{s}).\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
Upon replacing  $\overline{s}s_{+}\prime$ by $\overline{\unicode[STIX]{x1D70F}}\doteq \overline{s}+\overline{s}s_{+}\prime$ , one is led to
(F 2a ) $$\begin{eqnarray}\displaystyle \text{term (iv)} & = & \displaystyle -\displaystyle \int _{0}^{t}\!\text{d}\overline{s}\displaystyle \int _{s}^{t}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\boldsymbol{x}}\displaystyle \int \text{d}\overline{\boldsymbol{x}}s_{+}\prime \,\langle \widehat{G}(\unicode[STIX]{x1D707})\text{U}(\overline{s})\text{i}\mathscr{L}\boldsymbol{A}s_{+}\prime \rangle _{0}\star \unicode[STIX]{x1D648}^{-1}\nonumber\\ \displaystyle & & \displaystyle \star \,\langle \boldsymbol{A}s_{+}\prime [\text{U}(\overline{\unicode[STIX]{x1D70F}}-\overline{s})\text{i}\mathscr{L}As_{+}\prime ^{\unicode[STIX]{x1D6FD}}(\overline{\unicode[STIX]{x1D707}}s_{+}\prime )]As_{+}\prime ^{\unicode[STIX]{x1D6FE}}(\overline{\unicode[STIX]{x1D707}})\rangle _{0}\unicode[STIX]{x0394}B_{\unicode[STIX]{x1D6FD}}(\boldsymbol{r}-\overline{\unicode[STIX]{x1D746}}s_{+}\prime ,t-\overline{\unicode[STIX]{x1D70F}})\unicode[STIX]{x0394}B_{\unicode[STIX]{x1D6FE}}(\boldsymbol{r}-\overline{\unicode[STIX]{x1D746}},t-\overline{s})\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
(F 2b ) $$\begin{eqnarray}\displaystyle & {\approx} & \displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int _{0}^{\overline{\unicode[STIX]{x1D70F}}}\!\text{d}\overline{s}\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,\langle \widehat{G}(\unicode[STIX]{x1D707})\text{U}(\overline{s})\text{i}\mathscr{L}\boldsymbol{A}s_{+}\prime \rangle _{0}\star \unicode[STIX]{x1D648}^{-1}\nonumber\\ \displaystyle & & \displaystyle \star \,\langle \boldsymbol{A}s_{+}\prime [\text{U}(\overline{\unicode[STIX]{x1D70F}}-\overline{s})\widehat{\pmb{\pmb{\mathscr{J}}}}\text{}^{\unicode[STIX]{x1D6FD}}]As_{+}\prime ^{\unicode[STIX]{x1D6FE}}(\overline{\unicode[STIX]{x1D707}})\rangle _{0}\boldsymbol{\cdot }\unicode[STIX]{x1D735}B_{\unicode[STIX]{x1D6FD}}(\boldsymbol{r},t)\overline{\unicode[STIX]{x1D746}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}B_{\unicode[STIX]{x1D6FE}}(\boldsymbol{r},t).\end{eqnarray}$$
This requires evaluation of
(F 3) $$\begin{eqnarray}\boldsymbol{I}\doteq \displaystyle \int _{0}^{\overline{\unicode[STIX]{x1D70F}}}\!\text{d}\overline{s}\,\text{U}_{1}(\overline{s})\text{i}\mathscr{L}_{1}\boldsymbol{A}_{1}^{\prime }\text{U}_{2}(\overline{\unicode[STIX]{x1D70F}}-\overline{s}).\end{eqnarray}$$

(The subscripts distinguish the propagators, which appear under different expectations and thus operate on different variables.) Use the identity (3.24) and integrate the $\text{U}_{1}$  factor by parts:

(F 4) $$\begin{eqnarray}\boldsymbol{I}=(\boldsymbol{A}_{1}^{\prime }-\text{e}^{-\text{Q}_{1}\text{i}\mathscr{L}_{1}\overline{\unicode[STIX]{x1D70F}}}\boldsymbol{A}_{1}^{\prime })\underbrace{\text{U}_{2}(0)}_{1}-\displaystyle \int _{0}^{\overline{\unicode[STIX]{x1D70F}}}\!\text{d}\overline{s}\,(\boldsymbol{A}_{1}^{\prime }-\text{e}^{-\text{Q}_{1}\text{i}\mathscr{L}_{1}\overline{s}}\boldsymbol{A}_{1}^{\prime })\text{U}_{2}(\overline{\unicode[STIX]{x1D70F}}-\overline{s})\text{Q}_{2}\text{i}\mathscr{L}_{2}\text{Q}_{2}.\end{eqnarray}$$

The first $\boldsymbol{A}_{1}^{\prime }$ factor in each parenthesis does not contribute because $\langle \widehat{G}(\unicode[STIX]{x1D707})\boldsymbol{A}s_{+}\prime \rangle =0$ . Use of the identity (3.29) shows that to lowest order in the gradients one can replace $\exp (-\text{Q}\text{i}\mathscr{L}\overline{\unicode[STIX]{x1D70F}})$ by $\exp (-\text{i}\mathscr{L}\overline{\unicode[STIX]{x1D70F}})$ . From the last term in (F 4) arises the quantity

(F 5a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D646}\doteq \langle \boldsymbol{A}s_{+}\prime [\text{e}^{-\text{Q}\text{i}\mathscr{L}\text{Q}(\overline{\unicode[STIX]{x1D70F}}-\overline{s})}\text{Q}\text{i}\mathscr{L}\text{Q}\widehat{\pmb{\pmb{\mathscr{J}}}}\text{}^{\unicode[STIX]{x1D6FD}}]As_{+}\prime ^{\unicode[STIX]{x1D6FE}}\rangle _{0} & = & \displaystyle -\langle [\text{Q}\text{i}\mathscr{L}\text{Q}\text{e}^{\text{Q}\text{i}\mathscr{L}\text{Q}(\overline{\unicode[STIX]{x1D70F}}-\overline{s})}\boldsymbol{A}s_{+}\prime As_{+}\prime ^{\unicode[STIX]{x1D6FE}}]\widehat{\pmb{\pmb{\mathscr{J}}}}\text{}^{\unicode[STIX]{x1D6FD}}\rangle _{0}\end{eqnarray}$$
(F 5b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle -\langle [\text{Q}\text{i}\mathscr{L}\text{e}^{\text{Q}\text{i}\mathscr{L}(\overline{\unicode[STIX]{x1D70F}}-\overline{s})}\text{Q}(\boldsymbol{A}s_{+}\prime As_{+}\prime ^{\unicode[STIX]{x1D6FE}})]\widehat{\pmb{\pmb{\mathscr{J}}}}\text{}^{\unicode[STIX]{x1D6FD}}\rangle _{0}.\end{eqnarray}$$
Because $\text{Q}(\boldsymbol{A}s_{+}\prime \boldsymbol{A}s_{+}\prime )=\boldsymbol{A}s_{+}\prime \boldsymbol{A}s_{+}\prime -\boldsymbol{A}s_{+}\prime \star \unicode[STIX]{x1D648}^{-1}\star \langle \boldsymbol{A}s_{+}\prime \boldsymbol{A}s_{+}\prime \boldsymbol{A}s_{+}\prime \rangle$ and $\mathscr{L}$  is a linear operator, it can be seen that $\unicode[STIX]{x1D646}$  contains at least one power of $\text{i}\mathscr{L}\boldsymbol{A}s_{+}\prime$ and thus is at least of first order in the gradients. The contribution of the integral term in (F 4) is thus negligible, and one obtains
(F 6) $$\begin{eqnarray}\text{term (iv)}\approx -\int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\boldsymbol{x}}\,\langle [\text{e}^{\text{i}\mathscr{L}\overline{\unicode[STIX]{x1D70F}}}\widehat{G}(\unicode[STIX]{x1D707})]\text{P}\widehat{\pmb{\pmb{\mathscr{J}}}}\text{}^{\unicode[STIX]{x1D6FD}}As_{+}\prime ^{\unicode[STIX]{x1D6FE}}(\overline{\unicode[STIX]{x1D707}})\rangle _{0}\boldsymbol{\cdot }\unicode[STIX]{x1D735}B_{\unicode[STIX]{x1D6FD}}(\boldsymbol{r},t)\overline{\unicode[STIX]{x1D746}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}B_{\unicode[STIX]{x1D6FE}}(\boldsymbol{r},t).\end{eqnarray}$$

This is the result quoted by Brey et al.

Appendix G. Balescu–Lenard theory

Various of the calculations in the main text based on multiple-time hierarchies produce equations that after appropriate Markovian approximations lead to some variant of the Balescu–Lenard collision operator. Here I shall review the basic manipulations for an unmagnetized plasma, mostly using the assumption of a statistically homogeneous background state.

G.1 The nonlinear Balescu–Lenard operator

Although the simplest derivation of the Balescu–Lenard operator is accomplished with the Klimontovich formalism (Klimontovich Reference Klimontovich and ter Harr1967), I shall proceed instead from the BBGKY hierarchyFootnote 80 since that was used in the main text. The first member of that hierarchy is exactly

(G 1) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\,f(1)+\cdots =-\boldsymbol{\unicode[STIX]{x2202}}\boldsymbol{\cdot }\pmb{\mathbb{E}}(\overline{\text{}\underline{2}})g(\text{}\underline{1},\overline{\text{}\underline{2}},t).\end{eqnarray}$$

With no further approximations, this equation is time reversible, as is the entire BBGKY hierarchy. The conceptual foundations of the program for obtaining an irreversible equation of the form

(G 2) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\,f+\cdots =-C[\,f],\end{eqnarray}$$

which introduces the nonlinear, irreversible collision operator $\text{C}[\,f]$ as a functional of the one-particle distribution function  $f$ , require deep discussion not reviewed here. The standard procedure is to consider space–time scales much longer than the microscopic correlation scales, which for weakly coupled, unmagnetized plasmas are the Debye length  $\unicode[STIX]{x1D706}_{\text{D}}$ and inverse plasma frequency  $\unicode[STIX]{x1D714}_{\text{p}}^{-1}$ ; formally, one passes to the time-asymptotic limit $t\rightarrow \infty$ . Thus, one must calculate the pair correlation function  $g$ in that limit.

G.1.1 Representation of the collision operator in terms of the pair correlation function

The pair correlation function  $g$ obeys (5.17b ), which for weakly coupled plasma reduces to

(G 3) $$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}+\text{i}\text{L}_{1}+\text{i}\text{L}_{2})g(\text{}\underline{1},\text{}\underline{2},t)=-\unicode[STIX]{x1D750}_{12}\boldsymbol{\cdot }(q_{2}\boldsymbol{\unicode[STIX]{x2202}}_{1}-q_{1}\boldsymbol{\unicode[STIX]{x2202}}_{2})f(\text{}\underline{1},t)f(\text{}\underline{2},t).\end{eqnarray}$$

With Green’s function for the linearized Vlasov equation being defined as in (5.24) and (5.25), the inhomogeneous solutionFootnote 81 is

(G 4) $$\begin{eqnarray}g(\text{}\underline{1},\text{}\underline{2},t)=-\displaystyle \int _{-\infty }^{t}\!\text{d}\overline{t}\,\unicode[STIX]{x1D6EF}(\text{}\underline{1},t;\overline{\text{}\underline{1}},\overline{t})\unicode[STIX]{x1D6EF}(\text{}\underline{2},t;\overline{\text{}\underline{2}},\overline{t})\unicode[STIX]{x1D750}_{\overline{\text{}\underline{1}}\,\overline{\text{}\underline{2}}}\boldsymbol{\cdot }(q_{\overline{2}}\boldsymbol{\unicode[STIX]{x2202}}_{\overline{\text{}\underline{1}}}-q_{\overline{1}}\boldsymbol{\unicode[STIX]{x2202}}_{\overline{\text{}\underline{2}}})f(\overline{\text{}\underline{1}},\overline{t})f(\overline{\text{}\underline{2}},\overline{t}).\end{eqnarray}$$

(Summations/integrations over repeated barred indices are understood.) It is assumed that on the time scale for the formation of a Debye shielding cloud $f$  is essentially stationary; this justifies the Markovian approximation $f(\overline{t})\rightarrow f(t)$ . The time integral can then be extended to  $\infty$ . Treatment of the spatial dependence is more complicated. If one allows for weak spatial gradients in the background state, one must pursue a Wentzel–Kramers–Brillouin (WKB) treatment such as described by Kent & Taylor (Reference Kent and Taylor1969). I shall ignore that possibility and review the standard calculations that assume a homogeneous background. Then use of Parseval’s theorem leads one to the representation

(G 5) $$\begin{eqnarray}\text{C}[\,f]=\boldsymbol{\unicode[STIX]{x2202}}\boldsymbol{\cdot }\displaystyle \int \frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\,\pmb{\mathbb{E}}_{\boldsymbol{k}}^{\ast }(\overline{\text{}\underline{2}})g_{\boldsymbol{ k}}(\text{}\underline{1},\overline{\text{}\underline{2}},\infty ),\end{eqnarray}$$

where

(G 6) $$\begin{eqnarray}g_{\boldsymbol{k}}(\text{}\underline{1},\text{}\underline{2},\infty )=\displaystyle \int \frac{\text{d}\unicode[STIX]{x1D714}}{2\unicode[STIX]{x03C0}}\,g_{\boldsymbol{k},\unicode[STIX]{x1D714}}(\text{}\underline{1},\text{}\underline{2}),\end{eqnarray}$$

with

(G 7) $$\begin{eqnarray}g_{\boldsymbol{k},\unicode[STIX]{x1D714}}(\text{}\underline{1},\text{}\underline{2})\doteq -\unicode[STIX]{x1D6EF}_{\boldsymbol{k},\unicode[STIX]{x1D714}}(\text{}\underline{1};\overline{\text{}\underline{1}})\unicode[STIX]{x1D6EF}_{\boldsymbol{k},\unicode[STIX]{x1D714}}^{\ast }(\text{}\underline{2};\overline{\text{}\underline{2}})\unicode[STIX]{x1D750}_{\boldsymbol{ k}}\boldsymbol{\cdot }(q_{\overline{\text{}\underline{2}}}\boldsymbol{\unicode[STIX]{x2202}}_{\overline{1}}-q_{\overline{1}}\boldsymbol{\unicode[STIX]{x2202}}_{\overline{\text{}\underline{2}}})f(\overline{\text{}\underline{1}})f(\overline{\text{}\underline{2}}).\end{eqnarray}$$

The result for the Vlasov response function  $\unicode[STIX]{x1D6EF}$ is given by (E 15)–(E 19). Because one requires only the electric-field operator applied to  $g$ , it is possible to immediately simplify (G 5) by using the fundamental shielding identity (Krommes Reference Krommes2002, § 6.5.1 and references therein; Krommes Reference Krommes2015, equation (3.33))

(G 8) $$\begin{eqnarray}\pmb{\mathbb{E}}\unicode[STIX]{x1D6EF}=\mathscr{D}^{-1}\pmb{\mathbb{E}}r.\end{eqnarray}$$

However, for various purposes it is useful to first find a formula for $g_{\boldsymbol{k}}$  itself.

G.1.2 Calculation of the pair correlation function $g$

To simplify (G 7), use the result (E 15) to find (upon omitting underlines as well as frequency and wavenumber arguments for brevity)

(G 9) $$\begin{eqnarray}\displaystyle g_{\boldsymbol{k},\unicode[STIX]{x1D714}}(1,2) & = & \displaystyle \left(r(1)-r(1)(\boldsymbol{\unicode[STIX]{x2202}}f)_{1}\boldsymbol{\cdot }\frac{1}{\mathscr{D}}\pmb{\mathbb{E}}(\overline{1})r(\overline{1})\right)\left(r^{\ast }(2)-r^{\ast }(2)(\boldsymbol{\unicode[STIX]{x2202}}f)_{2}\boldsymbol{\cdot }\frac{1}{\mathscr{D}^{\ast }}\pmb{\mathbb{E}}^{\ast }(\overline{2})r^{\ast }(\overline{2})\right)\nonumber\\ \displaystyle & & \displaystyle \times \,(q_{2}\boldsymbol{\unicode[STIX]{x2202}}_{1}-q_{1}\boldsymbol{\unicode[STIX]{x2202}}_{2})\boldsymbol{\cdot }\unicode[STIX]{x1D750}_{\boldsymbol{k}}^{\ast }\,f(\overline{1})f(\overline{2}).\end{eqnarray}$$

(The minus sign in (G 7) was absorbed by using $-\unicode[STIX]{x1D750}_{\boldsymbol{k}}=\unicode[STIX]{x1D750}_{\boldsymbol{k}}^{\ast }$ .) Here

(G 10a-c ) $$\begin{eqnarray}r(1)\doteq [-\text{i}(\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}_{1}+\text{i}\unicode[STIX]{x1D716})]^{-1},\quad \pmb{\mathbb{E}}(\overline{1})\doteq \unicode[STIX]{x1D750}_{\boldsymbol{ k}}(\overline{n}q)_{\overline{1}},\quad \text{and}\quad \mathscr{D}\doteq 1+\pmb{\mathbb{E}}r\boldsymbol{\cdot }\boldsymbol{\unicode[STIX]{x2202}}f.\end{eqnarray}$$

Only derivatives in the direction of  $\boldsymbol{k}$ enter, so introduce the scalar quantities

(G 11a-c ) $$\begin{eqnarray}\mathbb{E}_{\boldsymbol{k}}\doteq \widehat{\boldsymbol{k}}\boldsymbol{\cdot }\pmb{\mathbb{E}}_{\boldsymbol{k}},\quad \unicode[STIX]{x1D716}_{k}\doteq \widehat{\boldsymbol{k}}\boldsymbol{\cdot }\unicode[STIX]{x1D750}_{\boldsymbol{k}}=-4\unicode[STIX]{x03C0}\,\text{i}/k,\quad \text{and}\quad \unicode[STIX]{x2202}_{\boldsymbol{k}}\doteq \widehat{\boldsymbol{k}}\boldsymbol{\cdot }\boldsymbol{\unicode[STIX]{x2202}}\equiv \unicode[STIX]{x2202},\end{eqnarray}$$

so that $\mathscr{D}=1+\mathbb{E}r\unicode[STIX]{x2202}f$ . Also define

(G 12) $$\begin{eqnarray}F\doteq \mathbb{E}rqf.\end{eqnarray}$$

Then, upon multiplying out the right-hand side of (G 9) and using the above definitions, one finds

(G 13) $$\begin{eqnarray}\displaystyle g_{\boldsymbol{k},\unicode[STIX]{x1D714}}(1,2) & = & \displaystyle r(1)r^{\ast }(2)\left(\vphantom{\left(\frac{1}{2}\right)}\right.\!\!\underbrace{(\unicode[STIX]{x2202}f)_{1}(qf)_{2}}_{(\text{a}_{1})}-\underbrace{(qf)_{1}(\unicode[STIX]{x2202}f)_{2}}_{(\text{b}_{1})}\nonumber\\ \displaystyle & & \displaystyle -\,(\unicode[STIX]{x2202}f)_{1}\mathscr{D}^{-1}(\underbrace{\mathscr{D}}_{(\text{a}_{2})}-1)(qf)_{2}+\underbrace{(\unicode[STIX]{x2202}f)_{1}\mathscr{D}^{-1}F(\unicode[STIX]{x2202}f)_{2}}_{(\text{c}_{1})}\nonumber\\ \displaystyle & & \displaystyle +\,(\unicode[STIX]{x2202}f)_{2}{\mathscr{D}^{-1}}^{\ast }(\underbrace{\mathscr{D}}_{(\text{b}_{2})}-1)^{\ast }(qf)_{1}-\underbrace{(\unicode[STIX]{x2202}f)_{2}{\mathscr{D}^{-1}}^{\ast }F^{\ast }(\unicode[STIX]{x2202}f)_{1}}_{(\text{d}_{1})}\nonumber\\ \displaystyle & & \displaystyle +\frac{(\unicode[STIX]{x2202}f)_{1}(\unicode[STIX]{x2202}f)_{2}}{|\mathscr{D}|^{2}}[(\underbrace{\mathscr{D}}_{(\text{d}_{2})}-1)F^{\ast }-(\underbrace{\mathscr{D}}_{(\text{c}_{2})}-1)^{\ast }F]\!\!\left.\vphantom{\left(\frac{1}{2}\right)}\right)\unicode[STIX]{x1D716}_{k}^{\ast }.\end{eqnarray}$$

Due to the cancellations indicated by the underbraces (for example, terms (a $_{1}$ ) and (a $_{2}$ ) cancel), this reduces toFootnote 82

(G 14) $$\begin{eqnarray}g_{\boldsymbol{k},\unicode[STIX]{x1D714}}(1,2)=r(1)r^{\ast }(2)\left(\frac{(\unicode[STIX]{x2202}f)_{1}(qf)_{2}}{\mathscr{D}}-\frac{(\unicode[STIX]{x2202}f)_{2}(qf)_{1}}{\mathscr{D}^{\ast }}+\frac{(\unicode[STIX]{x2202}f)_{1}(\unicode[STIX]{x2202}f)_{2}}{|\mathscr{D}|^{2}}2\text{i}Fs_{+}\prime \prime \right)\unicode[STIX]{x1D716}_{k}^{\ast },\end{eqnarray}$$

where $r(1)r^{\ast }(2)=[(\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}_{1}+\text{i}\unicode[STIX]{x1D716})(\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}_{2}-\text{i}\unicode[STIX]{x1D716})]^{-1}$ , double prime denotes imaginary part, and

(G 15) $$\begin{eqnarray}Fs_{+}\prime \prime =-\text{i}\unicode[STIX]{x03C0}\unicode[STIX]{x1D716}_{k}\mathop{\sum }_{\overline{s}}(\overline{n}q^{2})_{\overline{s}}\displaystyle \int \text{d}\overline{\boldsymbol{v}}\,\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{v}})f_{\overline{s}}(\overline{\boldsymbol{v}}).\end{eqnarray}$$

The formula (G 15) must be integrated over frequency according to (G 6). Upon noting that $r_{\unicode[STIX]{x1D714}}(1)$ and  $\mathscr{D}(\unicode[STIX]{x1D714})$ are analytic in the upper half-plane, the first term of (G 14) can be integrated by residues by closing the contour in the upper half-plane, where $r_{\unicode[STIX]{x1D714}}^{\ast }(2)$ has a simple pole at $\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}_{2}+\text{i}\unicode[STIX]{x1D716}$ . Similarly, the second term can be integrated by closing in the lower half-plane. One finds

(G 16) $$\begin{eqnarray}\displaystyle g_{\boldsymbol{k}}(1,2) & = & \displaystyle \frac{1}{(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}_{1}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}_{2}-2\text{i}\unicode[STIX]{x1D716})}\left(\frac{(\unicode[STIX]{x2202}f)_{1}(qf)_{2}}{\mathscr{D}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}_{2})}+\frac{(\unicode[STIX]{x2202}f)_{2}(qf)_{1}}{\mathscr{D}^{\ast }(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}_{1})}\right)\unicode[STIX]{x1D716}_{k}^{\ast }\nonumber\\ \displaystyle & & \displaystyle +\,2\text{i}\displaystyle \int \frac{\text{d}\unicode[STIX]{x1D714}}{2\unicode[STIX]{x03C0}}\,r_{\unicode[STIX]{x1D714}}(1)r_{\unicode[STIX]{x1D714}}^{\ast }(2)(\unicode[STIX]{x2202}f)_{1}\,(\unicode[STIX]{x2202}f)_{2}\frac{Fs_{+}\prime \prime (\unicode[STIX]{x1D714})}{|\mathscr{D}(\unicode[STIX]{x1D714})|^{2}}\unicode[STIX]{x1D716}_{k}^{\ast }.\end{eqnarray}$$

The presence of the delta function in (G 15) for $Fs_{+}\prime \prime \equiv \text{Im}\,F$ allows one to perform the frequency integration in the last term of (G 16), as I shall do later. However, a different approach is useful for the reduction to thermal equilibrium, which I shall discuss in the next section.

G.1.3 Reduction of the pair correlation function to thermal equilibrium

A limit that is both physically relevant and also useful for checking signs is the case of thermal equilibrium. Here one has $f=f_{\text{M}}$ and the result $\boldsymbol{\unicode[STIX]{x2202}}f_{\text{M}}=-(q/T)\boldsymbol{v}f_{\text{M}}$ . A simple manipulation of the formula for $\mathscr{D}(\unicode[STIX]{x1D714})$ then leads to

(G 17) $$\begin{eqnarray}F(\unicode[STIX]{x1D714})=-(k/\unicode[STIX]{x1D714})T[\mathscr{D}(\unicode[STIX]{x1D714})-\mathscr{D}_{0}],\end{eqnarray}$$

where $\mathscr{D}_{0}\doteq 1+k_{\text{D}}^{2}/k^{2}$ is the static dielectric function. Upon taking the imaginary part of (G 17) to obtain $Fs_{+}\prime \prime =-(k/\unicode[STIX]{x1D714})T\mathscr{D}s_{+}\prime \prime$ (this relation expresses the equilibrium balance between emission and absorption of fluctuations by the discrete particles), one finds that the last term of (G 16) can be written as

(G 18) $$\begin{eqnarray}I\doteq 4\unicode[STIX]{x03C0}\,\text{i}T\displaystyle \int \frac{\text{d}\unicode[STIX]{x1D714}}{2\unicode[STIX]{x03C0}\,\unicode[STIX]{x1D714}}\,r_{\unicode[STIX]{x1D714}}(1)r_{\unicode[STIX]{x1D714}}^{\ast }(2)\left(\frac{1}{\mathscr{D}(\unicode[STIX]{x1D714})}-\frac{1}{\mathscr{D}^{\ast }(\unicode[STIX]{x1D714})}\right)(\unicode[STIX]{x2202}f)_{1}\,(\unicode[STIX]{x2202}f)_{2}.\end{eqnarray}$$

Note that the integrand of this integral has no singularity at $\unicode[STIX]{x1D714}=0$ . Thus, one may employ the standard trickFootnote 83 of deforming the contour to include a semicircular arc around the origin and closing at  $\infty$ .Footnote 84 For the term in  $\mathscr{D}^{-1}$ , the simplest contour is shown in figure 2. For the term in  $(\mathscr{D}^{\ast })^{-1}$ , the reflection of that contour into the lower half-plane is appropriate. The contributions from the residues at $\unicode[STIX]{x1D714}=\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}_{1}$ and $\unicode[STIX]{x1D714}=\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}_{2}$ cancel the first two terms of (G 16). The contributions from the arc around the origin then leadFootnote 85 to the final, well-known resultFootnote 86

(G 19) $$\begin{eqnarray}g_{\boldsymbol{k}}^{(\text{eq})}(1,2)=-\frac{q_{1}q_{2}}{T}\frac{4\unicode[STIX]{x03C0}}{k^{2}\mathscr{D}_{0}(\boldsymbol{k})}.\end{eqnarray}$$

G.1.4 Final formula for the Balescu–Lenard operator

Formula (G 14) can be used to obtain the Balescu–Lenard collision operator according to (G 5). One has

(G 20) $$\begin{eqnarray}\mathbb{E}^{\ast }(\overline{2})g(1,\overline{2})=r(1)\left(\frac{(\unicode[STIX]{x2202}f)_{1}}{\mathscr{D}}F^{\ast }-\frac{(\mathscr{D}^{\ast }-1)(qf)_{1}}{\mathscr{D}^{\ast }}+\frac{(\unicode[STIX]{x2202}f)_{1}(\mathscr{D}^{\ast }-1)}{|\mathscr{D}|^{2}}(F-F^{\ast })\right)\unicode[STIX]{x1D716}_{k}^{\ast }.\end{eqnarray}$$

Upon rearranging this expression and integrating over  $\unicode[STIX]{x1D714}$ , one finds

(G 21) $$\begin{eqnarray}\displaystyle \displaystyle \int \frac{\text{d}\unicode[STIX]{x1D714}}{2\unicode[STIX]{x03C0}}\,\mathbb{E}^{\ast }(\overline{2})g_{\boldsymbol{ k},\unicode[STIX]{x1D714}}(1,\overline{2}) & = & \displaystyle \displaystyle \int \frac{\text{d}\unicode[STIX]{x1D714}}{2\unicode[STIX]{x03C0}}\,r_{\unicode[STIX]{x1D714}}(1)\left[\vphantom{\left(\frac{1}{2}\right)}\right.\!\!(\unicode[STIX]{x2202}f)_{1}\left(\vphantom{\left(\frac{1}{2}\right)}\right.\!\underbrace{\frac{F^{\ast }(\unicode[STIX]{x1D714})}{\mathscr{D}(\unicode[STIX]{x1D714})}}_{(\text{a}_{1})}+\frac{\mathscr{D}^{\ast }(\unicode[STIX]{x1D714})}{|\mathscr{D}(\unicode[STIX]{x1D714})|^{2}}[\underbrace{F(\unicode[STIX]{x1D714})}_{0}-\underbrace{F^{\ast }(\unicode[STIX]{x1D714})}_{(\text{a}_{2})}]\!\!\!\left.\vphantom{\left(\frac{1}{2}\right)}\right)\nonumber\\ \displaystyle & & \displaystyle -\underbrace{\left(1-\frac{1}{\mathscr{D}^{\ast }(\unicode[STIX]{x1D714})}\right)(qf)_{1}}_{\text{(c)}}-\underbrace{\frac{(\unicode[STIX]{x2202}f)_{1}}{|\mathscr{D}(\unicode[STIX]{x1D714})|^{2}}2\text{i}Fs_{+}\prime \prime (\unicode[STIX]{x1D714})}_{\text{(d)}}\!\!\left.\vphantom{\left(\frac{1}{2}\right)}\right]\unicode[STIX]{x1D716}_{k}^{\ast }.\end{eqnarray}$$

Terms (a $_{1}$ ) and (a $_{2}$ ) cancel. The underbraced term that vanishes does so by Cauchy’s theorem because its sole frequency dependence arises from the product $r_{\unicode[STIX]{x1D714}}(1)\mathscr{D}^{-1}(\unicode[STIX]{x1D714})F(\unicode[STIX]{x1D714})$ , which is analytic in the upper half-plane and falls off sufficiently rapidly at  $\infty$ . Term (c) can be evaluated by residues, and term (d) can be reduced by integrating over the delta function in (G 15). Only the real part survives integration over  $\boldsymbol{k}$ . With $\boldsymbol{v}_{1}\rightarrow \boldsymbol{v}$ , the final result is $\text{C}_{s}[\,f]=\sum _{\overline{s}}\text{C}_{s\overline{s}}^{\text{BL}}[\,f]$ , where

(G 22) $$\begin{eqnarray}\displaystyle \text{C}_{s\overline{s}}^{\text{BL}}[\,f] & \doteq & \displaystyle \unicode[STIX]{x03C0}\,(\overline{n}m)_{s}^{-1}(\overline{n}q^{2})_{s}(\overline{n}q^{2})_{\overline{s}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\boldsymbol{v}}\boldsymbol{\cdot }\displaystyle \int \text{d}\overline{\boldsymbol{v}}\displaystyle \int \frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\,\frac{\unicode[STIX]{x1D750}_{\boldsymbol{k}}\,\unicode[STIX]{x1D750}_{\boldsymbol{k}}^{\ast }}{|\mathscr{D}(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v})|^{2}}\unicode[STIX]{x1D6FF}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}-\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{v}})\nonumber\\ \displaystyle & & \displaystyle \boldsymbol{\cdot }\,\left(\frac{1}{m_{\overline{s}}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\overline{\boldsymbol{v}}}-\frac{1}{m_{s}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\boldsymbol{v}}\right)f_{s}(\boldsymbol{v})f_{\overline{s}}(\overline{\boldsymbol{v}}),\end{eqnarray}$$

which is the Balescu–Lenard operator. Its most important feature is that the natural interacting entities are shielded test particlesFootnote 87   ( $\unicode[STIX]{x1D750}_{\boldsymbol{k}}/\mathscr{D}(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v})$ ) rather than bare particles ( $\unicode[STIX]{x1D750}_{\boldsymbol{k}}$ ). The first term describes polarization drag, while the second term describes velocity-space diffusion. For future use, note that with my convention for the sign of  $\text{C}$ polarization drag enters with a plus sign while diffusion enters with a minus sign.

Figure 2. Contour of integration used for evaluating the term involving $\mathscr{D}^{-1}$ in (G 18).

For practical calculations, it is customary to approximate the effect of dielectric shielding by setting $\mathscr{D}\rightarrow 1$ and inserting a cutoff at a small wavenumber magnitude of the order of the Debye wavenumber  $k_{\text{D}}$ . This results in the Landau collision operator

(G 23) $$\begin{eqnarray}\displaystyle \text{C}_{s\overline{s}}^{\text{L}}[\,f] & \doteq & \displaystyle 2\unicode[STIX]{x03C0}\,(\overline{n}m)_{s}^{-1}(\overline{n}q^{2})_{s}(\overline{n}q^{2})_{\overline{s}}\ln \unicode[STIX]{x1D6EC}_{s\overline{s}}\nonumber\\ \displaystyle & & \displaystyle \times \,\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\boldsymbol{v}}\boldsymbol{\cdot }\displaystyle \int \text{d}\overline{\boldsymbol{v}}\,\unicode[STIX]{x1D650}(\boldsymbol{v}-\overline{\boldsymbol{v}})\boldsymbol{\cdot }\left(\frac{1}{m_{\overline{s}}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\overline{\boldsymbol{v}}}-\frac{1}{m_{s}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\boldsymbol{v}}\right)f_{s}(\boldsymbol{v})f_{\overline{s}}(\overline{\boldsymbol{v}}),\end{eqnarray}$$

where $\unicode[STIX]{x1D6EC}_{s\overline{s}}\doteq \unicode[STIX]{x1D706}_{\text{D}}/b_{\text{min},s\overline{s}}$ ( $b_{\text{min}}$  being classically the impact parameter for $90^{\circ }$ scatteringFootnote 88 ) and $\unicode[STIX]{x1D650}(\boldsymbol{v})\doteq (\unicode[STIX]{x1D644}-\hat{\boldsymbol{v}}\,\hat{\boldsymbol{v}})/|\boldsymbol{v}|$ . Because this operator is bilinear in form, it is often represented as $\text{C}^{\text{L}}[\,f]=\text{C}^{\text{L}}[\,f,\,\overline{f}]$ . The Balescu–Lenard operator has a more complicated functional dependence, since $\mathscr{D}=\mathscr{D}[\,f]$ . A non-standard notation is to write

(G 24) $$\begin{eqnarray}\text{C}^{\text{BL}}[\,f]\equiv \text{C}^{\text{BL}}[f;f,\overline{f}],\end{eqnarray}$$

thus defining an operator $\text{C}[a;b,\overline{c}]$ that is bilinear in its last two slots; the first slot handles the functional dependence of the dielectric properties. The linearization of that operator, discussed in the next section, is then

(G 25) $$\begin{eqnarray}\unicode[STIX]{x0394}\text{C}[\,f,\unicode[STIX]{x0394}f]=\displaystyle \int \text{d}\text{}\underline{1}\,\frac{\unicode[STIX]{x1D6FF}\text{C}[a;f,\overline{f}]}{\unicode[STIX]{x1D6FF}a(\text{}\underline{1})}\bigg|_{a=f}\unicode[STIX]{x0394}f(\text{}\underline{1})+\text{C}[f;f,\overline{\unicode[STIX]{x0394}f}]+\text{C}[f;\unicode[STIX]{x0394}f,\overline{f}]\end{eqnarray}$$

or explicitly

(G 26) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}\text{C}^{\text{BL}}[\,f,\unicode[STIX]{x1D6E4}] & = & \displaystyle \unicode[STIX]{x03C0}\,(\overline{n}m)_{s}^{-1}(\overline{n}q^{2})_{s}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\boldsymbol{v}}\boldsymbol{\cdot }\mathop{\sum }_{\overline{s}}(\overline{n}q^{2})_{\overline{s}}\displaystyle \!\int \!\text{d}\overline{\boldsymbol{v}}\displaystyle \!\int \!\frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\frac{\unicode[STIX]{x1D750}_{\boldsymbol{k}}\,\unicode[STIX]{x1D750}_{\boldsymbol{k}}^{\ast }}{|\mathscr{D}(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v})|^{2}}\unicode[STIX]{x1D6FF}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}-\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{v}})\nonumber\\ \displaystyle & & \displaystyle \times \!\left[-2\text{Re}\,\left(\frac{\unicode[STIX]{x0394}\mathscr{D}(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v})}{\mathscr{D}(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v})}\right)\left(\frac{1}{m_{\overline{s}}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\overline{\boldsymbol{v}}}-\frac{1}{m_{s}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\boldsymbol{v}}\right)f_{s}(\boldsymbol{v})f_{\overline{s}}(\overline{\boldsymbol{v}})\right.\nonumber\\ \displaystyle & & \displaystyle \quad ~+\left.\!\left(\frac{1}{m_{\overline{s}}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\overline{\boldsymbol{v}}}-\frac{1}{m_{s}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\boldsymbol{v}}\right)[f_{s}(\boldsymbol{v})\unicode[STIX]{x1D6E4}_{\overline{s}}(\overline{\boldsymbol{v}})+\unicode[STIX]{x1D6E4}_{s}(\boldsymbol{v})f_{\overline{s}}(\overline{\boldsymbol{v}})]\right],\end{eqnarray}$$

where

(G 27) $$\begin{eqnarray}\unicode[STIX]{x0394}\mathscr{D}[\unicode[STIX]{x1D6E4}]\doteq \mathbb{E}r\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}.\end{eqnarray}$$

For perturbations around a Maxwellian equilibrium, the $\unicode[STIX]{x0394}\mathscr{D}/\mathscr{D}$ term vanishes for the same, detailed-balance reason that the nonlinear Balescu–Lenard operator annihilates a Maxwellian distribution. In § G.3, we shall encounter the second-order operator $\text{C}^{\text{BL}}[f;\unicode[STIX]{x1D6E4}s_{+}\prime ,\overline{\unicode[STIX]{x1D6E4}}s_{+}\prime \prime ]+\text{C}^{\text{BL}}[f;\unicode[STIX]{x1D6E4}s_{+}\prime \prime ,\overline{\unicode[STIX]{x1D6E4}}s_{+}\prime ]$ for certain correlation functions  $\unicode[STIX]{x1D6E4}s_{+}\prime$ and  $\unicode[STIX]{x1D6E4}s_{+}\prime \prime$ ; that operator plays a crucial role in second-order Chapman–Enskog theory and the Burnett transport coefficients.

G.2 The linearized Balescu–Lenard operator

The two-time hierarchy equations for both two- and three-point correlation functions lead to terms of the form

(G 28) $$\begin{eqnarray}\displaystyle \widehat{\text{D}}[\,f,\unicode[STIX]{x1D6E4}] & \doteq & \displaystyle -\boldsymbol{\unicode[STIX]{x2202}}_{1}\boldsymbol{\cdot }\displaystyle \int \frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\displaystyle \int \frac{\text{d}\unicode[STIX]{x1D714}}{2\unicode[STIX]{x03C0}}\,\pmb{\mathbb{E}}_{\boldsymbol{k}}^{\ast }(\overline{2})\unicode[STIX]{x1D6EF}_{\boldsymbol{ k},\unicode[STIX]{x1D714}}(1;\overline{1})\unicode[STIX]{x1D6EF}_{\boldsymbol{k},\unicode[STIX]{x1D714}}^{\ast }(\overline{2};\overline{2}s_{+}\prime )\nonumber\\ \displaystyle & & \displaystyle \times \,\{\!\unicode[STIX]{x1D750}_{\boldsymbol{k}}\boldsymbol{\cdot }(q_{\overline{2}s_{+}\prime }\boldsymbol{\unicode[STIX]{x2202}}_{\overline{1}}-q_{\overline{1}}\boldsymbol{\unicode[STIX]{x2202}}_{\overline{2}s_{+}\prime })[\unicode[STIX]{x1D6E4}(\overline{1})f(\overline{2}s_{+}\prime )+\unicode[STIX]{x1D6E4}(\overline{2}s_{+}\prime )f(\overline{1})]\nonumber\\ \displaystyle & & \displaystyle +\,\boldsymbol{\unicode[STIX]{x2202}}_{\bar{1}}\unicode[STIX]{x1D6E4}(\overline{1})\boldsymbol{\cdot }[\pmb{\mathbb{E}}_{\boldsymbol{k}}^{\ast }(\widehat{1})g_{\boldsymbol{k}}(\overline{2}s_{+}\prime ,\widehat{1})]^{\ast }+\boldsymbol{\unicode[STIX]{x2202}}_{\overline{2}s_{+}\prime }\unicode[STIX]{x1D6E4}(\overline{2}s_{+}\prime )\boldsymbol{\cdot }[\pmb{\mathbb{E}}_{\boldsymbol{k}}^{\ast }(\widehat{2})g_{\boldsymbol{k}}(\overline{1},\widehat{2})]\!\}\end{eqnarray}$$

for some  $\unicode[STIX]{x1D6E4}$ , e.g. $\unicode[STIX]{x1D6E4}(1)=C_{+;\boldsymbol{k}s_{+}\prime =\mathbf{0}}^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )$ or $\unicode[STIX]{x1D6E4}(1)=C_{+;\boldsymbol{k}s_{+}\prime =\mathbf{0},\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )$ . I shall show that $\widehat{\text{D}}$  is the linearized Balescu–Lenard operator: $\widehat{\text{D}}[\,f,\unicode[STIX]{x1D6E4}]=\unicode[STIX]{x0394}\text{C}^{\text{BL}}[\,f,\unicode[STIX]{x1D6E4}]$ ; see formula (G 26).

To begin, note that the construction $\pmb{\mathbb{E}}_{\boldsymbol{k}}^{\ast }(\widehat{\text{}\underline{2}})g_{\boldsymbol{k}}(\overline{\text{}\underline{1}},\widehat{\text{}\underline{2}})$ (the coefficient of the last $\boldsymbol{\unicode[STIX]{x2202}}_{\overline{2}s_{+}\prime }\unicode[STIX]{x1D6E4}$ term) is the same one used in the calculation of the Balescu–Lenard operator in the last section (see the last line of formula (G 21)); the coefficient of $\boldsymbol{\unicode[STIX]{x2202}}_{\bar{1}}\unicode[STIX]{x1D6E4}$ can be obtained from that result by replacing $\overline{1}\rightarrow \overline{2}s_{+}\prime$ and complex conjugating. Cancellations occur between the unshielded part of the $qf$  terms in the last line of (G 28) and the $qf$  terms in the second line of (G 28). Upon omitting most $\boldsymbol{k}$ arguments for brevity, using the notation $\unicode[STIX]{x1D714}_{1}\equiv \boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}_{1}$ , $\mathscr{D}(1)\equiv \mathscr{D}(\boldsymbol{k},\unicode[STIX]{x1D714}_{1})$ , and $r_{\unicode[STIX]{x1D714}}(1)\equiv r_{\unicode[STIX]{x1D714}}(\boldsymbol{v}_{1})\equiv [-\text{i}(\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}_{1}+\text{i}\unicode[STIX]{x1D716})]^{-1}$ , and using $-\unicode[STIX]{x1D750}_{\boldsymbol{k}}=\unicode[STIX]{x1D750}_{\boldsymbol{k}}^{\ast }$ , one thus has

(G 29a ) $$\begin{eqnarray}\displaystyle \widehat{\text{D}}[\,f,\unicode[STIX]{x1D6E4}] & = & \displaystyle \unicode[STIX]{x2202}_{1}\displaystyle \int \frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\displaystyle \int \frac{\text{d}\unicode[STIX]{x1D714}}{2\unicode[STIX]{x03C0}}\,\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D714}}(1;\overline{1})\frac{1}{\mathscr{D}^{\ast }(\unicode[STIX]{x1D714})}\mathbb{E}^{\ast }(\overline{2})r_{\unicode[STIX]{x1D714}}^{\ast }(\overline{2})\nonumber\\ \displaystyle & & \displaystyle \times \,\left[\vphantom{\left(\frac{(qf)_{\overline{1}}}{\mathscr{D}^{\ast }(\overline{1})}-2\text{i}\displaystyle \int \frac{\text{d}\overline{\unicode[STIX]{x1D714}}}{2\unicode[STIX]{x03C0}}\,r_{\overline{\unicode[STIX]{x1D714}}}(\overline{1})(\unicode[STIX]{x2202}f)_{\overline{1}}\frac{Fs_{+}\prime \prime (\overline{\unicode[STIX]{x1D714}})}{|\mathscr{D}(\overline{\unicode[STIX]{x1D714}})|^{2}}\right)}-(q\unicode[STIX]{x1D6E4})_{\overline{1}}(\unicode[STIX]{x2202}f)_{\overline{2}}+(q\unicode[STIX]{x1D6E4})_{\overline{2}}(\unicode[STIX]{x2202}f)_{\overline{1}}\right.\nonumber\\ \displaystyle & & \displaystyle +\,(\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}_{\overline{1}})\left(\frac{(qf)_{\overline{2}}}{\mathscr{D}(\overline{2})}+2\text{i}\displaystyle \int \frac{\text{d}\overline{\unicode[STIX]{x1D714}}}{2\unicode[STIX]{x03C0}}\,r_{\overline{\unicode[STIX]{x1D714}}}^{\ast }(\overline{2})(\unicode[STIX]{x2202}f)_{\overline{2}}\frac{Fs_{+}\prime \prime (\overline{\unicode[STIX]{x1D714}})}{|\mathscr{D}(\overline{\unicode[STIX]{x1D714}})|^{2}}\right)\nonumber\\ \displaystyle & & \displaystyle -\left.(\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4})_{\overline{2}}\left(\frac{(qf)_{\overline{1}}}{\mathscr{D}^{\ast }(\overline{1})}-2\text{i}\displaystyle \int \frac{\text{d}\overline{\unicode[STIX]{x1D714}}}{2\unicode[STIX]{x03C0}}\,r_{\overline{\unicode[STIX]{x1D714}}}(\overline{1})(\unicode[STIX]{x2202}f)_{\overline{1}}\frac{Fs_{+}\prime \prime (\overline{\unicode[STIX]{x1D714}})}{|\mathscr{D}(\overline{\unicode[STIX]{x1D714}})|^{2}}\right)\right]\unicode[STIX]{x1D716}_{k}^{\ast }\end{eqnarray}$$
(G 29b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle \unicode[STIX]{x2202}_{1}\displaystyle \int \frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\displaystyle \int \frac{\text{d}\unicode[STIX]{x1D714}}{2\unicode[STIX]{x03C0}}\,\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D714}}(1;\overline{1})\frac{1}{\mathscr{D}^{\ast }(\unicode[STIX]{x1D714})}\nonumber\\ \displaystyle & & \displaystyle \times \,\left[\vphantom{\left(\frac{(qf)_{\overline{1}}}{\mathscr{D}^{\ast }(\overline{1})}-2\text{i}\displaystyle \int \frac{\text{d}\overline{\unicode[STIX]{x1D714}}}{2\unicode[STIX]{x03C0}}\,r_{\overline{\unicode[STIX]{x1D714}}}(\overline{1})(\unicode[STIX]{x2202}f)_{\overline{1}}\frac{Fs_{+}\prime \prime (\overline{\unicode[STIX]{x1D714}})}{|\mathscr{D}(\overline{\unicode[STIX]{x1D714}})|^{2}}\right)}-(q\unicode[STIX]{x1D6E4})_{\overline{1}}[\mathscr{D}^{\ast }(\unicode[STIX]{x1D714})-1]+(\mathbb{E}r_{\unicode[STIX]{x1D714}}q\unicode[STIX]{x1D6E4})^{\ast }(\unicode[STIX]{x2202}f)_{\overline{1}}\right.\nonumber\\ \displaystyle & & \displaystyle +\,(\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4})_{\overline{1}}\left(\mathbb{E}^{\ast }(\overline{2})r_{\unicode[STIX]{x1D714}}^{\ast }(\overline{2})\frac{(qf)_{\overline{2}}}{\mathscr{D}(\overline{2})}+2\text{i}\displaystyle \int \frac{\text{d}\overline{\unicode[STIX]{x1D714}}}{2\unicode[STIX]{x03C0}}\,[\mathbb{E}(\overline{2})r_{\unicode[STIX]{x1D714}}(\overline{2})r_{\overline{\unicode[STIX]{x1D714}}}(\overline{2})(\unicode[STIX]{x2202}f)_{\overline{2}}]^{\ast }\frac{Fs_{+}\prime \prime (\overline{\unicode[STIX]{x1D714}})}{|\mathscr{D}(\overline{\unicode[STIX]{x1D714}})|^{2}}\right)\nonumber\\ \displaystyle & & \displaystyle -\left.\unicode[STIX]{x0394}\mathscr{D}^{\ast }(\unicode[STIX]{x1D714})\left(\frac{(qf)_{\overline{1}}}{\mathscr{D}^{\ast }(\overline{1})}-2\text{i}\displaystyle \int \frac{\text{d}\overline{\unicode[STIX]{x1D714}}}{2\unicode[STIX]{x03C0}}\,r_{\overline{\unicode[STIX]{x1D714}}}(\overline{1})(\unicode[STIX]{x2202}f)_{\overline{1}}\frac{Fs_{+}\prime \prime (\overline{\unicode[STIX]{x1D714}})}{|\mathscr{D}(\overline{\unicode[STIX]{x1D714}})|^{2}}\right)\right]\unicode[STIX]{x1D716}_{k}^{\ast },\end{eqnarray}$$
where the  $\unicode[STIX]{x1D714}$ in term (d) of (G 21) was replaced by  $\overline{\unicode[STIX]{x1D714}}$ and
(G 30) $$\begin{eqnarray}\unicode[STIX]{x0394}\mathscr{D}(\unicode[STIX]{x1D714})\doteq \mathbb{E}(\overline{2})r_{\unicode[STIX]{x1D714}}(\overline{2})(\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4})_{\overline{2}}.\end{eqnarray}$$

As I noted previously, the $\overline{\unicode[STIX]{x1D714}}$  integral can be performed because

(G 31) $$\begin{eqnarray}Fs_{+}\prime \prime (\overline{\unicode[STIX]{x1D714}})=-\text{i}\unicode[STIX]{x03C0}\,\mathbb{E}_{\overline{s}}(\overline{\boldsymbol{v}})\unicode[STIX]{x1D6FF}(\overline{\unicode[STIX]{x1D714}}-\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{v}})(qf)_{\overline{s}}(\overline{\boldsymbol{v}}).\end{eqnarray}$$

However, I choose to defer that for notational clarity. Upon inserting the formula (cf. (E 20)) $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D714}}(1;\overline{1})A(\overline{1})=r_{\unicode[STIX]{x1D714}}(1)A(1)-r_{\unicode[STIX]{x1D714}}(1)\unicode[STIX]{x2202}_{1}f\mathscr{D}^{-1}(\unicode[STIX]{x1D714})\mathbb{E}(\overline{1})r_{\unicode[STIX]{x1D714}}(\overline{1})A(\overline{1})$ , one finds

(G 32) $$\begin{eqnarray}\displaystyle & & \displaystyle \widehat{\text{D}}[\,f,\unicode[STIX]{x1D6E4}]=\unicode[STIX]{x2202}_{1}\displaystyle \int \frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\displaystyle \int \frac{\text{d}\unicode[STIX]{x1D714}}{2\unicode[STIX]{x03C0}}\,\left\{\vphantom{\left(\frac{1}{2}\right)}\right.\!\!r_{\unicode[STIX]{x1D714}}(1)\left[\vphantom{\left(\frac{1}{2}\right)}\right.\!-(q\unicode[STIX]{x1D6E4})_{1}\left(1-\frac{1}{\mathscr{D}^{\ast }(\unicode[STIX]{x1D714})}\right)+\underbrace{(\mathbb{E}r_{\unicode[STIX]{x1D714}}q\unicode[STIX]{x1D6E4})^{\ast }\frac{(\unicode[STIX]{x2202}f)_{1}}{\mathscr{D}^{\ast }(\unicode[STIX]{x1D714})}}_{(\text{a}_{1})}\!\left.\vphantom{\left(\frac{1}{2}\right)}\right]\nonumber\\ \displaystyle & & \displaystyle \quad +\,r_{\unicode[STIX]{x1D714}}(1)\frac{(\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4})_{1}}{\mathscr{D}^{\ast }(\unicode[STIX]{x1D714})}\left(\mathbb{E}^{\ast }(\overline{2})r_{\unicode[STIX]{x1D714}}^{\ast }(\overline{2})\frac{(qf)_{\overline{2}}}{\mathscr{D}(\overline{2})}+2\text{i}\displaystyle \int \frac{\text{d}\overline{\unicode[STIX]{x1D714}}}{2\unicode[STIX]{x03C0}}\,[\mathbb{E}(\overline{2})r_{\unicode[STIX]{x1D714}}(\overline{2})r_{\overline{\unicode[STIX]{x1D714}}}(\overline{2})(\unicode[STIX]{x2202}f)_{\overline{2}}]^{\ast }\frac{Fs_{+}\prime \prime (\overline{\unicode[STIX]{x1D714}})}{|\mathscr{D}(\overline{\unicode[STIX]{x1D714}})|^{2}}\right)\nonumber\\ \displaystyle & & \displaystyle \quad +\,r_{\unicode[STIX]{x1D714}}(1)(-)\left(\frac{\unicode[STIX]{x0394}\mathscr{D}(\unicode[STIX]{x1D714})}{\mathscr{D}(\unicode[STIX]{x1D714})}\right)^{\ast }\left(\frac{(qf)_{1}}{\mathscr{D}^{\ast }(1)}-2\text{i}\displaystyle \int \frac{\text{d}\overline{\unicode[STIX]{x1D714}}}{2\unicode[STIX]{x03C0}}\,r_{\overline{\unicode[STIX]{x1D714}}}(1)(\unicode[STIX]{x2202}f)_{1}\frac{Fs_{+}\prime \prime (\overline{\unicode[STIX]{x1D714}})}{|\mathscr{D}(\overline{\unicode[STIX]{x1D714}})|^{2}}\right)\nonumber\\ \displaystyle & & \displaystyle \quad -\,r_{\unicode[STIX]{x1D714}}(1)\frac{(\unicode[STIX]{x2202}f)_{1}}{\mathscr{D}(\unicode[STIX]{x1D714})}\left[\vphantom{\left(\frac{1}{2}\right)}\right.\!-(\mathbb{E}r_{\unicode[STIX]{x1D714}}q\unicode[STIX]{x1D6E4})\left(\vphantom{\left(\frac{1}{2}\right)}\right.\!\underbrace{1}_{\text{(b)=0}}-\frac{1}{\mathscr{D}^{\ast }(\unicode[STIX]{x1D714})}\!\left.\vphantom{\left(\frac{1}{2}\right)}\right)+\left(\frac{(\mathbb{E}r_{\unicode[STIX]{x1D714}}q\unicode[STIX]{x1D6E4})}{\mathscr{D}(\unicode[STIX]{x1D714})}\right)^{\ast }[\underbrace{\mathscr{D}(\unicode[STIX]{x1D714})}_{(\text{a}_{2})}-1]\!\left.\vphantom{\left(\frac{1}{2}\right)}\right]\nonumber\\ \displaystyle & & \displaystyle \quad -\,r_{\unicode[STIX]{x1D714}}(1)\frac{(\unicode[STIX]{x2202}f)_{1}}{|\mathscr{D}(\unicode[STIX]{x1D714})|^{2}}\left[\unicode[STIX]{x0394}\mathscr{D}(\unicode[STIX]{x1D714})\left(\mathbb{E}^{\ast }(\overline{2})r_{\unicode[STIX]{x1D714}}^{\ast }(\overline{2})\frac{(qf)_{\overline{2}}}{\mathscr{D}(\overline{2})}\right.\right.\nonumber\\ \displaystyle & & \displaystyle \quad +\left.\left.2\text{i}\displaystyle \int \frac{\text{d}\overline{\unicode[STIX]{x1D714}}}{2\unicode[STIX]{x03C0}}\,[\mathbb{E}(\overline{2})r_{\unicode[STIX]{x1D714}}(\overline{2})r_{\overline{\unicode[STIX]{x1D714}}}(\overline{2})(\unicode[STIX]{x2202}f)_{\overline{2}}]^{\ast }\frac{Fs_{+}\prime \prime (\overline{\unicode[STIX]{x1D714}})}{|\mathscr{D}(\overline{\unicode[STIX]{x1D714}})|^{2}}\right)\right]\nonumber\\ \displaystyle & & \displaystyle \quad -\,r_{\unicode[STIX]{x1D714}}(1)\frac{(\unicode[STIX]{x2202}f)_{1}}{|\mathscr{D}(\unicode[STIX]{x1D714})|^{2}}\left[-\unicode[STIX]{x0394}\mathscr{D}^{\ast }(\unicode[STIX]{x1D714})\left(\mathbb{E}(\overline{2})r_{\unicode[STIX]{x1D714}}(\overline{2})\frac{(qf)_{\overline{2}}}{\mathscr{D}^{\ast }(\overline{2})}\right.\right.\nonumber\\ \displaystyle & & \displaystyle \quad -\left.\left.2\text{i}\displaystyle \int \frac{\text{d}\overline{\unicode[STIX]{x1D714}}}{2\unicode[STIX]{x03C0}}\,\mathbb{E}r_{\unicode[STIX]{x1D714}}(\overline{1})r_{\overline{\unicode[STIX]{x1D714}}}(\overline{1})(\unicode[STIX]{x2202}f)_{\overline{1}}\frac{Fs_{+}\prime \prime (\overline{\unicode[STIX]{x1D714}})}{|\mathscr{D}(\overline{\unicode[STIX]{x1D714}})|^{2}}\right)\right]\vphantom{\left[-(q\unicode[STIX]{x1D6E4})_{1}\left(1-\frac{1}{\mathscr{D}^{\ast }(\unicode[STIX]{x1D714})}\right)+\underbrace{(\mathbb{E}r_{\unicode[STIX]{x1D714}}q\unicode[STIX]{x1D6E4})^{\ast }\frac{(\unicode[STIX]{x2202}f)_{1}}{\mathscr{D}^{\ast }(\unicode[STIX]{x1D714})}}_{(\text{a}_{1})}\right]}\!\!\left.\vphantom{\left(\frac{1}{2}\right)}\right\}\unicode[STIX]{x1D716}_{k}^{\ast }.\end{eqnarray}$$

Terms (a $_{1}$ ) and (a $_{2}$ ) cancel. Term (b) vanishes by analyticity in the upper half of the $\unicode[STIX]{x1D714}$  plane. The sum of the square-bracketed terms (times  $\unicode[STIX]{x1D716}_{k}^{\ast }$ ) in the last two lines can be recognized as twice the imaginary part of the term in $\unicode[STIX]{x0394}\mathscr{D}^{\ast }(\unicode[STIX]{x1D714})$ . I shall now reduce each of the terms. In the algebra, I shall omit the $\unicode[STIX]{x2202}_{1}(2\unicode[STIX]{x03C0})^{-3}\int \text{d}\boldsymbol{k}$ for brevity; however, it is important to remember the presence of the $\boldsymbol{k}$  integration since that enforces the reality of the final expression.

G.2.1 The term in $q\unicode[STIX]{x1D6E4}$

The $\unicode[STIX]{x1D714}$  integration can be performed by closing the contour in the lower half-plane, giving rise to $-(q\unicode[STIX]{x1D6E4})_{1}\{1-[\mathscr{D}^{\ast }(1)]^{-1}\}\unicode[STIX]{x1D716}_{k}^{\ast }$ . The first term vanishes by reality or upon interchanging  $\boldsymbol{k}$ and  $-\boldsymbol{k}$ . The remaining expression can be written as

(G 33) $$\begin{eqnarray}\frac{(q\unicode[STIX]{x1D6E4})_{1}}{|\mathscr{D}(1)|^{2}}[1+\mathbb{E}(\overline{1})r_{1}(\overline{1})(\unicode[STIX]{x2202}f)_{\overline{1}}]\unicode[STIX]{x1D716}_{k}^{\ast }.\end{eqnarray}$$

The first term again vanishes by reality, and only $\text{Re}\,r_{1}(\overline{1})=\unicode[STIX]{x03C0}\,\unicode[STIX]{x1D6FF}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}_{1}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}_{\overline{1}})$ survives. One is led to the $+\unicode[STIX]{x1D6E4}\,\overline{\unicode[STIX]{x2202}f}$ termFootnote 89 in the linearized Balescu–Lenard operator (G 26).

G.2.2 The term in $\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}$

The $\unicode[STIX]{x1D714}$  integration can be performed by closing the contour in the lower half-plane, giving rise to

(G 34) $$\begin{eqnarray}-\frac{(\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4})_{1}}{\mathscr{D}^{\ast }(1)}\left(\vphantom{\left(\frac{1}{2}\right)}\right.\!\underbrace{\mathbb{E}^{\ast }(\overline{2})r_{1}^{\ast }(\overline{2})\frac{(qf)_{\overline{2}}}{\mathscr{D}(\overline{2})}}_{\text{(c)}}+\underbrace{2\text{i}\displaystyle \int \frac{\text{d}\overline{\unicode[STIX]{x1D714}}}{2\unicode[STIX]{x03C0}}\,[\mathbb{E}(\overline{2})r_{\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}_{1}-\text{i}\unicode[STIX]{x1D716}}(\overline{2})r_{\overline{\unicode[STIX]{x1D714}}}(\overline{2})(\unicode[STIX]{x2202}f)_{\overline{2}}]^{\ast }\frac{Fs_{+}\prime \prime (\overline{\unicode[STIX]{x1D714}})}{|\mathscr{D}(\overline{\unicode[STIX]{x1D714}})|^{2}}}_{\text{(d)}}\!\!\left.\vphantom{\left(\frac{1}{2}\right)}\right)\unicode[STIX]{x1D716}_{k}^{\ast }.\end{eqnarray}$$

Now invoke the identity

(G 35) $$\begin{eqnarray}r_{\unicode[STIX]{x1D714}}(\overline{2})r_{\overline{\unicode[STIX]{x1D714}}}(\overline{2})=\frac{r_{\unicode[STIX]{x1D714}}(\overline{2})-r_{\overline{\unicode[STIX]{x1D714}}}(\overline{2})}{\text{i}(\unicode[STIX]{x1D714}-\overline{\unicode[STIX]{x1D714}})}\end{eqnarray}$$

and perform the $\overline{\unicode[STIX]{x1D714}}$  integration using (G 27) to find that term (d) becomes

(G 36) $$\begin{eqnarray}-\frac{(\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4})_{1}}{\mathscr{D}^{\ast }(1)}\mathbb{E}(\overline{1})r_{1}(\overline{1})[\underbrace{\mathscr{D}^{\ast }(1)}_{(\text{d}_{1})}-\underbrace{\mathscr{D}^{\ast }(\overline{1})}_{(\text{d}_{2})}]\frac{(qf)_{\overline{1}}}{|\mathscr{D}(\overline{1})|^{2}}\unicode[STIX]{x1D716}_{k}^{\ast }.\end{eqnarray}$$

Term (d $_{2}$ ) cancels term (c). In term (d $_{1}$ ), the $\mathscr{D}^{\ast }$ quantities cancel and under the $\boldsymbol{k}$  integration only $\text{Re}\,r_{1}(\overline{1})$ contributes. One is led to the $-(\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4})\overline{f}$ term of (G 26).

G.2.3 The term in $(\unicode[STIX]{x0394}\mathscr{D}/\mathscr{D})^{\ast }$

The $\unicode[STIX]{x1D714}$  integration can be performed by closing the contour in the lower half-plane, giving rise to

(G 37) $$\begin{eqnarray}-\left(\frac{\unicode[STIX]{x0394}\mathscr{D}(1)}{\mathscr{D}(1)}\right)^{\ast }\left(\vphantom{\left(\frac{1}{2}\right)}\right.\!\underbrace{\frac{(qf)_{1}}{\mathscr{D}^{\ast }(1)}}_{\text{(e)}}-\underbrace{2\text{i}\displaystyle \int \frac{\text{d}\overline{\unicode[STIX]{x1D714}}}{2\unicode[STIX]{x03C0}}\,r_{\overline{\unicode[STIX]{x1D714}}}(1)(\unicode[STIX]{x2202}f)_{1}\frac{Fs_{+}\prime \prime (\overline{\unicode[STIX]{x1D714}})}{|\mathscr{D}(\overline{\unicode[STIX]{x1D714}})|^{2}}}_{(\text{f}_{1})}\!\left.\vphantom{\left(\frac{1}{2}\right)}\right)\unicode[STIX]{x1D716}_{k}^{\ast }.\end{eqnarray}$$

Term (f $_{1}$ ) will combine with a later term. (Only its real part contributes.) Term (e) can be manipulated as follows:

(G 38) $$\begin{eqnarray}-\left(\frac{\unicode[STIX]{x0394}\mathscr{D}(1)}{\mathscr{D}(1)}\right)^{\ast }\frac{(qf)_{1}}{\mathscr{D}^{\ast }(1)}\unicode[STIX]{x1D716}_{k}^{\ast }=-\frac{(qf)_{1}}{|\mathscr{D}(1)|^{2}}\left[\left(\frac{\unicode[STIX]{x0394}\mathscr{D}(1)}{\mathscr{D}(1)}\right)^{\ast }\mathscr{D}(1)\right]\unicode[STIX]{x1D716}_{k}^{\ast }.\end{eqnarray}$$

The term in square brackets can be arranged as

(G 39a ) $$\begin{eqnarray}\displaystyle \left(\frac{\unicode[STIX]{x0394}\mathscr{D}}{\mathscr{D}}\right)^{\ast }\mathscr{D} & = & \displaystyle \left[\left(\frac{\unicode[STIX]{x0394}\mathscr{D}}{\mathscr{D}}\right)^{\ast }+\frac{\unicode[STIX]{x0394}\mathscr{D}}{\mathscr{D}}\right]\mathscr{D}-\unicode[STIX]{x0394}\mathscr{D}\end{eqnarray}$$
(G 39b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle 2\text{Re}\,\left(\frac{\unicode[STIX]{x0394}\mathscr{D}}{\mathscr{D}}\right)(\underbrace{1}_{(\text{e}_{1})}+\underbrace{\mathbb{E}r\unicode[STIX]{x2202}f}_{(\text{e}_{2})})-\underbrace{\unicode[STIX]{x0394}\mathscr{D}}_{(\text{e}_{3})}.\end{eqnarray}$$
Term (e $_{1}$ ) vanishes by reality under the $\boldsymbol{k}$  integration. Only $\text{Re}\,r$ contributes to term (e $_{2}$ ); one is led to the $-f\,\overline{\unicode[STIX]{x2202}f}$ term in (G 26). The contribution of term (e $_{3}$ ) is easily seen to produce the $+f\,\overline{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}}$ term in (G 26).

G.2.4 The terms in $(\unicode[STIX]{x2202}f)(\mathbb{E}\unicode[STIX]{x1D6E4})$

These terms combine to

(G 40) $$\begin{eqnarray}-\displaystyle \int \frac{\text{d}\unicode[STIX]{x1D714}}{2\unicode[STIX]{x03C0}}\,r_{\unicode[STIX]{x1D714}}(1)\frac{(\unicode[STIX]{x2202}f)_{1}}{|\mathscr{D}(\unicode[STIX]{x1D714})|^{2}}2\text{Re}\,[(\mathbb{E}r_{\unicode[STIX]{x1D714}}q\unicode[STIX]{x1D6E4})\unicode[STIX]{x1D716}_{k}^{\ast }].\end{eqnarray}$$

This reduces to the $-(\unicode[STIX]{x2202}f)\overline{\unicode[STIX]{x1D6E4}}$ term of (G 26).

G.2.5 The remaining terms in $\unicode[STIX]{x2202}f$

The last two terms in $(\unicode[STIX]{x2202}f)_{1}$ reduce to

(G 41) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \int \frac{\text{d}\unicode[STIX]{x1D714}}{2\unicode[STIX]{x03C0}}r_{\unicode[STIX]{x1D714}}(1)\frac{(\unicode[STIX]{x2202}f)_{1}}{|\mathscr{D}(\unicode[STIX]{x1D714})|^{2}}\nonumber\\ \displaystyle & & \displaystyle \quad \times \,2\text{Re}\,\left[\vphantom{\left(\frac{1}{2}\right)}\right.\!\!\unicode[STIX]{x0394}\mathscr{D}^{\ast }(\unicode[STIX]{x1D714})\left(\vphantom{\left(\frac{1}{2}\right)}\right.\!\underbrace{\mathbb{E}(\overline{2})r_{\unicode[STIX]{x1D714}}(\overline{2})\frac{(qf)_{\overline{2}}}{\mathscr{D}^{\ast }(\overline{2})}}_{(\text{g}_{1})}-\underbrace{2\text{i}\displaystyle \int \frac{\text{d}\overline{\unicode[STIX]{x1D714}}}{2\unicode[STIX]{x03C0}}\,\mathbb{E}r_{\unicode[STIX]{x1D714}}(\overline{1})r_{\overline{\unicode[STIX]{x1D714}}}(\overline{1})(\unicode[STIX]{x2202}f)_{\overline{1}}\frac{Fs_{+}\prime \prime (\overline{\unicode[STIX]{x1D714}})}{|\mathscr{D}(\overline{\unicode[STIX]{x1D714}})|^{2}}}_{\text{(h)}}\!\left.\vphantom{\left(\frac{1}{2}\right)}\right)\unicode[STIX]{x1D716}_{k}^{\ast }\!\!\left.\vphantom{\left(\frac{1}{2}\right)}\right].\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Thus, only $\text{Re}\,r_{\unicode[STIX]{x1D714}}(1)=\unicode[STIX]{x03C0}\,\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}_{1})$ contributes. Use of the identity (G 35) and the result (G 31) leads to the expression of term (h) as

(G 42) $$\begin{eqnarray}2\text{i}\displaystyle \int \frac{\text{d}\overline{\unicode[STIX]{x1D714}}}{2\unicode[STIX]{x03C0}}\,\mathbb{E}r_{w}(\overline{1})r_{\overline{\unicode[STIX]{x1D714}}}(\overline{1})(\unicode[STIX]{x2202}f)_{1}\frac{Fs_{+}\prime \prime (\overline{\unicode[STIX]{x1D714}})}{|\mathscr{D}(\overline{\unicode[STIX]{x1D714}})|^{2}}=\mathbb{E}(\overline{\boldsymbol{v}})\left(\frac{\mathscr{D}(\unicode[STIX]{x1D714})-\mathscr{D}(\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{v}})}{\text{i}(\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{v}})}\right)\left(\frac{(qf)_{\overline{\boldsymbol{v}}}}{|\mathscr{D}(\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{v}})|^{2}}\right).\end{eqnarray}$$

In the first term in large parentheses, there is no singularity at $\unicode[STIX]{x1D714}=\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{v}}$ . Thus, the value of that term is inessentially changed by replacing $[\text{i}(\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{v}})]^{-1}\rightarrow [\text{i}(\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{v}}+\text{i}\unicode[STIX]{x1D716})]^{-1}=-r_{\unicode[STIX]{x1D714}}(\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{v}})$ . Expression (G 42) thus becomes

(G 43) $$\begin{eqnarray}\text{(h)}=\underbrace{\mathscr{D}(\unicode[STIX]{x1D714})\mathbb{E}(\overline{\boldsymbol{v}})r_{\unicode[STIX]{x1D714}}(\overline{\boldsymbol{v}})\frac{(qf)_{\overline{\boldsymbol{v}}}}{|\mathscr{D}(\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{v}})|^{2}}}_{(\text{f}_{2})}-\underbrace{\mathbb{E}(\overline{\boldsymbol{v}})r_{\unicode[STIX]{x1D714}}(\overline{\boldsymbol{v}})\frac{(qf)_{\overline{\boldsymbol{v}}}}{\mathscr{D}^{\ast }(\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{v}})}}_{(\text{g}_{2})}.\end{eqnarray}$$

Term (g $_{2}$ ) cancels term (g $_{1}$ ). Upon performing the $\unicode[STIX]{x1D714}$  integration, one finds that the entire value of term (f $_{2}$ ) becomes

(G 44) $$\begin{eqnarray}(\text{f}_{2})=\text{Re}\,\left[\left(\frac{\unicode[STIX]{x0394}\mathscr{D}(1)}{\mathscr{D}(1)}\right)\mathbb{E}(\overline{\boldsymbol{v}})r_{1}(\overline{\boldsymbol{v}})\frac{(qf)_{\overline{\boldsymbol{v}}}}{|\mathscr{D}(\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{v}})|^{2}}\unicode[STIX]{x1D716}_{k}^{\ast }\right].\end{eqnarray}$$

This can be seen to combine with term (f $_{1}$ ) in (G 37). Upon using $r_{1}(\overline{\boldsymbol{v}})+r_{\overline{\boldsymbol{v}}}(1)=2\text{Re}\,r_{1}(\overline{\boldsymbol{v}})$ , one readily finds that the sum of terms (f $_{1}$ ) and (f $_{2}$ ) reduces to the $+\unicode[STIX]{x2202}f\,\overline{f}$ term in (G 26).

G.2.6 Summary of the reduction of $\widehat{\text{D}}$

This completes the reduction of the operator  $\widehat{\text{D}}$ . I have shown that $\widehat{\text{D}}[\,f,\unicode[STIX]{x1D6E4}]$ is indeed the linearized Balescu–Lenard operator  $\unicode[STIX]{x0394}\text{C}^{\text{BL}}[\,f,\unicode[STIX]{x1D6E4}]$ . As I discuss in the main text, this operator appears in various of the equations for two-time correlation functions.

G.3 Nonlinear noise terms and the Balescu–Lenard operator

I now turn to the analysis of the contributions from the terms in the last five lines of (5.48) for the source term  $s_{+}^{(2;2)}$ in (5.45) for  $C_{+}^{(1;2)}$ . Equation (5.46) for  $C_{+}^{(2;2)}$ can be solved in terms of the unperturbed response function as usual, so one must evaluate

(G 45) $$\begin{eqnarray}\text{D}[\unicode[STIX]{x1D6E4}_{\boldsymbol{k}s_{+}\prime \prime }]\doteq -\boldsymbol{\unicode[STIX]{x2202}}_{1}\boldsymbol{\cdot }\displaystyle \int _{0}^{\unicode[STIX]{x1D70F}}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\,\pmb{\mathbb{E}}_{\boldsymbol{k}}^{\ast }(\overline{\text{}\underline{2}})\unicode[STIX]{x1D6EF}_{\boldsymbol{ k}}(\text{}\underline{1},\overline{\unicode[STIX]{x1D70F}};\overline{\text{}\underline{1}})\unicode[STIX]{x1D6EF}_{\boldsymbol{k}}^{\ast }(\text{}\underline{2},\overline{\unicode[STIX]{x1D70F}};\overline{\text{}\underline{2}})\unicode[STIX]{x0394}s_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime \prime }^{(2;2)}(\overline{\text{}\underline{1}},\overline{\text{}\underline{2}},\unicode[STIX]{x1D70F}-\overline{\unicode[STIX]{x1D70F}};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime ),\end{eqnarray}$$

where

(G 46) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x0394}s_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime \prime }^{(2;2)}(\text{}\underline{1},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )\doteq \nonumber\\ \displaystyle & & \displaystyle \quad -\,(q_{2}\boldsymbol{\unicode[STIX]{x2202}}_{1}-q_{1}\boldsymbol{\unicode[STIX]{x2202}}_{2})\boldsymbol{\cdot }\unicode[STIX]{x1D750}_{\boldsymbol{k}}[C_{+;\mathbf{0}}^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )C_{+;\boldsymbol{k}s_{+}\prime \prime }^{(1;1)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )+C_{+;\boldsymbol{k}s_{+}\prime \prime }^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )C_{+;\mathbf{0}}^{(1;1)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )]\nonumber\\ \displaystyle & & \displaystyle \quad - [\boldsymbol{\unicode[STIX]{x2202}}_{1}C_{+;\mathbf{0}}^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}_{\boldsymbol{k}}(\overline{\text{}\underline{1}})C_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime \prime }^{(2;1)}(\overline{\text{}\underline{1}},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )\nonumber\\ \displaystyle & & \displaystyle \quad \quad ~+\boldsymbol{\unicode[STIX]{x2202}}_{1}C_{+;\boldsymbol{k}s_{+}\prime \prime }^{(1;1)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}_{\boldsymbol{k}}(\overline{\text{}\underline{1}})C_{+,\boldsymbol{k};\mathbf{0}}^{(2;1)}(\overline{\text{}\underline{1}},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )]\nonumber\\ \displaystyle & & \displaystyle \quad - [\boldsymbol{\unicode[STIX]{x2202}}_{2}C_{+;\mathbf{0}}^{(1;1)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}_{\boldsymbol{k}}^{\ast }(\overline{\text{}\underline{2}})C_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime \prime }^{(2;1)}(\text{}\underline{1},\overline{\text{}\underline{2}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )\nonumber\\ \displaystyle & & \displaystyle \quad \quad ~+\boldsymbol{\unicode[STIX]{x2202}}_{2}C_{+;\boldsymbol{k}s_{+}\prime \prime }^{(1;1)}(\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )\boldsymbol{\cdot }\pmb{\mathbb{E}}_{\boldsymbol{k}}^{\ast }(\overline{\text{}\underline{2}})C_{+,\boldsymbol{k};\mathbf{0}}^{(2;1)}(\text{}\underline{1},\overline{\text{}\underline{2}},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime )].\end{eqnarray}$$

Consider the evaluation of the $\overline{\unicode[STIX]{x1D70F}}$  integral in (G 45), and notice that the $\unicode[STIX]{x1D70F}$  dependence of the first line of (G 46) differs in character from that of the subsequent four lines. The $C_{+}^{(1;1)}$ functions in the first line vary on the collisional time scale, as they are all evaluated at  $\boldsymbol{k}s_{+}\prime$ or  $\boldsymbol{k}s_{+}\prime \prime$ , both of which are small. That is true as well of the $\boldsymbol{\unicode[STIX]{x2202}}C_{+}^{(1;1)}$ terms in the last four lines. However, the $C_{+}^{(2;1)}$ factors in those lines involve  $\boldsymbol{k}$ , which appears under an integral and is a characteristic Debye-scale wavenumber. Thus, they vary on the microscopic autocorrelation time scale, which is the same as that on which the $\unicode[STIX]{x1D6EF}$  functions vary. The consequence is that while a Markovian approximation is immediately valid for the contribution of the first line, the remaining terms require further processing.

Before proceeding with the details, it is helpful to gain an intuitive understanding of what to expect. For a weakly coupled plasma, collisional effects on the one-body distribution function  $f$ are captured by the nonlinear Balescu–Lenard operator $\text{C}^{\text{BL}}[\,f;f,\overline{f}]$ . Upon expanding $f\approx f_{0}+\unicode[STIX]{x1D716}f_{1}+\unicode[STIX]{x1D716}^{2}f_{2}$ and assuming that $f_{0}$  is Maxwellian, one is led at first order to the linearized operator $\widehat{\text{C}}f_{1}\doteq \text{C}[\,f_{0};f_{0},\overline{f}_{1}]+\text{C}[\,f_{0};f_{1},\overline{f}_{0}]$ ; the first-order variation with respect to the first argument (which describes the functional dependence of the dielectric function on the distribution function) vanishes for Maxwellian  $f_{0}$ . At second order, one finds

(G 47) $$\begin{eqnarray}\displaystyle \text{C}[\,f;f,\overline{f}]_{2} & = & \displaystyle \widehat{\text{C}}f_{2}+\text{C}[\,f_{0};f_{1},\overline{f}_{1}]\nonumber\\ \displaystyle & & \displaystyle +\,\displaystyle \int \text{d}\boldsymbol{v}s_{+}\prime \,\left(\frac{\unicode[STIX]{x1D6FF}\text{C}[a;f_{0},\overline{f}_{1}]}{\unicode[STIX]{x1D6FF}a(\boldsymbol{v}s_{+}\prime )}\bigg|_{a=f_{0}}+\frac{\unicode[STIX]{x1D6FF}\text{C}[a;f_{1},\overline{f}_{0}]}{\unicode[STIX]{x1D6FF}a(\boldsymbol{v}s_{+}\prime )}\bigg|_{a=f_{0}}\right)f_{1}(\boldsymbol{v}s_{+}\prime ).\end{eqnarray}$$

(The second variation with respect to the first argument vanishes.) These terms are mirrored in the theory of two-time correlations. The term analogous to  $\widehat{\text{C}}f_{2}$ was derived in § G.2. The term analogous to $\text{C}[\,f_{0};f_{1},\overline{f}_{1}]$ will be shown in the next section to stem from the $C^{(1;1)}C^{(1;1)}$ terms in the second line of (G 46). This leaves the last four lines of (G 46) to capture the effect described by the last line of (G 47), which arises from the first-order variation of  $\widehat{\text{C}}[\,f_{0}]$ (i.e. fluctuations in the dielectric shielding).

G.3.1 The $C_{+}^{(1;1)}C_{+}^{(1;1)}$ contribution

The initial manipulations of the $C_{+}^{(1;1)}C_{+}^{(1;1)}$ terms proceed as in the derivation of the original Balescu–Lenard operator discussed in § G.1. Because the second line of (G 46) is identical in form to the driving term for the pair correlation function  $g$ (see (G 4)), it is easy to see that the contribution from the $\unicode[STIX]{x1D6E4}\!\unicode[STIX]{x1D6E4}$ terms to  $D[\unicode[STIX]{x1D6E4}]$ is

(G 48) $$\begin{eqnarray}\text{C}^{\text{BL}}[f;C_{+;\mathbf{0}}^{(1;1)}(\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ),\overline{C}_{+;\boldsymbol{k}s_{+}\prime \prime }^{(1;1)}(\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime \prime )]+(1s_{+}\prime \Leftrightarrow 1s_{+}\prime \prime ).\end{eqnarray}$$

The crucial role of this term in the theory of triplet correlations will be analysed in § 6.3.2.

G.3.2 The $\mathbb{E}C_{+}^{(2;1)}$ terms

I now briefly discuss the last four lines of (G 46). A term like $C_{+,\boldsymbol{k};\mathbf{0}}^{(2;1)}(\overline{\text{}\underline{1}},\text{}\underline{2},\unicode[STIX]{x1D70F};1s_{+}\prime )$ is evaluated by time-convolving $[\unicode[STIX]{x1D6EF}_{\boldsymbol{k}}(\overline{\text{}\underline{1}},\unicode[STIX]{x1D70F})\unicode[STIX]{x1D6EF}_{\boldsymbol{k}}^{\ast }(\text{}\underline{2},\unicode[STIX]{x1D70F})]$ with a certain, slowly varying source  $S$ . That result must be convolved with another pair of $\unicode[STIX]{x1D6EF}$  functions according to (G 45). The resulting contribution to  $\text{D}[\unicode[STIX]{x1D6E4}_{\boldsymbol{k}s_{+}\prime \prime }]$ drives  $C_{+}^{(1;2)}$ according to (5.45). Ultimately, contributions to transport coefficients are determined from $\int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\,C_{+}^{(1;2)}(\overline{\unicode[STIX]{x1D70F}})$ . Thus, with $\unicode[STIX]{x1D6EF}_{2}\doteq \unicode[STIX]{x1D6EF}\!\unicode[STIX]{x1D6EF}$ , one must consider (schematically)

(G 49) $$\begin{eqnarray}I\doteq \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\,\displaystyle \int _{0}^{\overline{\unicode[STIX]{x1D70F}}}\!\text{d}\widehat{\unicode[STIX]{x1D70F}}\,\unicode[STIX]{x1D6EF}_{2}(\widehat{\unicode[STIX]{x1D70F}})\displaystyle \int _{0}^{\overline{\unicode[STIX]{x1D70F}}-\widehat{\unicode[STIX]{x1D70F}}}\!\text{d}\unicode[STIX]{x1D70F}s_{+}\prime \,\unicode[STIX]{x1D6EF}_{2}(\unicode[STIX]{x1D70F}s_{+}\prime )S(\overline{\unicode[STIX]{x1D70F}}-\widehat{\unicode[STIX]{x1D70F}}-\unicode[STIX]{x1D70F}s_{+}\prime ).\end{eqnarray}$$

Upon interchanging the order of integration of the first and second integrals, then of the second and third ones, one finds that

(G 50) $$\begin{eqnarray}I=\int _{0}^{\infty }\text{d}\widehat{\unicode[STIX]{x1D70F}}\,\unicode[STIX]{x1D6EF}_{2}(\widehat{\unicode[STIX]{x1D70F}})\int _{0}^{\infty }\text{d}\unicode[STIX]{x1D70F}s_{+}\prime \,\unicode[STIX]{x1D6EF}_{2}(\unicode[STIX]{x1D70F}s_{+}\prime )\int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\,S(\overline{\unicode[STIX]{x1D70F}}).\end{eqnarray}$$

Each $\unicode[STIX]{x1D6EF}_{2}$  integral can be expressed in frequency space as $(2\unicode[STIX]{x03C0})^{-1}\int _{-\infty }^{\infty }\!\text{d}\unicode[STIX]{x1D714}\,\unicode[STIX]{x1D6EF}(\unicode[STIX]{x1D714})\unicode[STIX]{x1D6EF}^{\ast }(\unicode[STIX]{x1D714})$ . The second of each pair of $\unicode[STIX]{x1D6EF}$  functions is operated upon by  $\pmb{\mathbb{E}}$ , which allows simplifications according to (G 8).

Thus, evaluation of the contributions of the $\pmb{\mathbb{E}}C_{+}^{(2;1)}$ terms is feasible in principle, as it involves nothing more than multiple instances of the manipulations done in § G.2 to obtain the linearized Balescu–Lenard operator. The algebra is tedious, however, and I shall not pursue it here. Note that effects related to fluctuations in the dielectric properties of the plasma disappear when dielectric shielding is taken into account by a cutoff at the Debye wavenumber  $k_{\text{D}}$ and if fluctuations in  $k_{\text{D}}$ are subsequently ignored. I shall return to this point in § 6.4.

Appendix H. One-sided correlations and nonlinear noise

Equation (5.33) and similar equations contain (i) a left-hand side that includes collisional damping, (ii) an initial condition and (iii) a source term. These are special cases of Dyson equations (Dyson Reference Dyson1949), the general form of which is, for arbitrary linear operator  $\text{L}$ (e.g. $\text{L}\doteq -\text{i}\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}$ ) and independent variables  $\boldsymbol{q}$ ,

(H 1) $$\begin{eqnarray}\displaystyle & & \displaystyle (\unicode[STIX]{x2202}_{t}+\text{i}\text{L})\text{C}(\boldsymbol{q},t,\boldsymbol{q}s_{+}\prime ,ts_{+}\prime )+\displaystyle \int _{0}^{t}\!\text{d}\overline{t}\,\text{d}\overline{\boldsymbol{q}}\,\unicode[STIX]{x1D6F4}(\boldsymbol{q},t;\overline{\boldsymbol{q}},\overline{t})\text{C}(\overline{\boldsymbol{q}},\overline{t},\boldsymbol{q}s_{+}\prime ,ts_{+}\prime )\nonumber\\ \displaystyle & & \displaystyle \quad =\displaystyle \int _{0}^{ts_{+}\prime }\!\text{d}\overline{t}\,\text{d}\overline{\boldsymbol{q}}\,F(\boldsymbol{q},t,\overline{\boldsymbol{q}},\overline{t})R(\boldsymbol{q}s_{+}\prime ,ts_{+}\prime ;\overline{\boldsymbol{q}},\overline{t}),\end{eqnarray}$$

where in the choice of signs and symbols for the fluctuation-related quantities  $\unicode[STIX]{x1D6F4}$ and  $F$ Footnote 90 I have adopted the conventions used in the review of statistical methods by Krommes (Reference Krommes2002). For example, Rose (Reference Rose1979) has proposed his PDIA, which has this form.

I shall discuss some elementary implications of equations with the Dyson form. Note that in (H 1) there is no ordering of  $t$ and  $ts_{+}\prime$ ; $C(t,ts_{+}\prime )$  is two sided in time, not one sided. It is useful to understand the relationship between this equation and the one-sided equations discussed earlier in the paper. To develop intuition, assume that  $\unicode[STIX]{x1D6F4}$ and  $F$ are both positive.Footnote 91 Then if $F$  were omitted, $C$  would decay to zero. That is incompatible with the fact that $C(t,t)$ has a non-zero value in thermal equilibrium (or in steady-state turbulence). Therefore, an $F$  term must always be present. Another way of saying this is that $F$  is necessary in order that conservation laws associated with the nonlinearity are satisfied.

H.1 Equations for one-sided functions

In order to make explicit the non-zero value of  $C(t,t)$ , it is useful to introduce functions that are one sided in time. Define

(H 2a,b ) $$\begin{eqnarray}C_{+}(t,ts_{+}\prime )\doteq H(t-ts_{+}\prime )C(t,ts_{+}\prime )\quad \text{and}\quad C_{-}(t,ts_{+}\prime )\doteq H(ts_{+}\prime -t)C(t,ts_{+}\prime );\end{eqnarray}$$

then $C(t,ts_{+}\prime )=C_{+}(t,ts_{+}\prime )+C_{-}(t,ts_{+}\prime )$ . To focus on the essence of the problem, consider an equation of the general form

(H 3) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}C(t,ts_{+}\prime )+\unicode[STIX]{x1D708}C=s(t,ts_{+}\prime ).\end{eqnarray}$$

This has the same form as the general Dyson equation (H 1) (with non-dissipative terms ignored), with a positive damping coefficient on the left-hand side of the equation for  $C$ (not  $C_{+}$ or  $C_{-}$ ). The equation for  $C_{+}$ is readily found by time differentiating its definition:

(H 4a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}C_{+}(t,ts_{+}\prime ) & = & \displaystyle \unicode[STIX]{x1D6FF}(t-ts_{+}\prime )C(t,t)+H(t-ts_{+}\prime )\unicode[STIX]{x2202}_{t}C\end{eqnarray}$$
(H 4b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle \unicode[STIX]{x1D6FF}(t-ts_{+}\prime )C(t,t)+H(t-ts_{+}\prime )[-\unicode[STIX]{x1D708}C(t,ts_{+}\prime )+s(t,ts_{+}\prime )];\end{eqnarray}$$
thus,
(H 5) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}C_{+}(t,ts_{+}\prime )+\unicode[STIX]{x1D708}C_{+}=\unicode[STIX]{x1D6FF}(t-ts_{+}\prime )C(t,t)+s_{+}(t,ts_{+}\prime ).\end{eqnarray}$$

This has the same form as (5.33). Green’s function for the operator on the left-hand side is

(H 6) $$\begin{eqnarray}R(t;ts_{+}\prime )=H(t-ts_{+}\prime )\unicode[STIX]{x1D6EF}(t,ts_{+}\prime ),\quad \unicode[STIX]{x1D6EF}(t,ts_{+}\prime )\doteq \text{e}^{-\unicode[STIX]{x1D708}(t-ts_{+}\prime )}.\end{eqnarray}$$

Thus, the solution of (H 5) is, with $C_{0}\doteq C(ts_{+}\prime ,ts_{+}\prime )$ ,

(H 7a ) $$\begin{eqnarray}\displaystyle C_{+}(t,ts_{+}\prime ) & = & \displaystyle \int _{-\infty }^{\infty }\!\,\text{d}\overline{t}\,R(t;\overline{t})[\unicode[STIX]{x1D6FF}(\overline{t}-ts_{+}\prime )C_{0}+s_{+}(\overline{t},ts_{+}\prime )]\end{eqnarray}$$
(H 7b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle H(t-ts_{+}\prime )\left(\unicode[STIX]{x1D6EF}(t;ts_{+}\prime )C_{0}+\displaystyle \int _{ts_{+}\prime }^{t}\!\text{d}\overline{t}\,\unicode[STIX]{x1D6EF}(t,\overline{t})s_{+}(\overline{t},ts_{+}\prime )\right).\end{eqnarray}$$
Notably, the contribution from the initial condition decays as $t\rightarrow \infty$ .

In a similar fashion, one finds

(H 8) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}C_{-}(t,ts_{+}\prime )+\unicode[STIX]{x1D708}C_{-}=-\unicode[STIX]{x1D6FF}(t-ts_{+}\prime )C_{0}+s_{-}(t,ts_{+}\prime ).\end{eqnarray}$$

(Note the minus sign in the initial-condition term.) With the ansatz

(H 9) $$\begin{eqnarray}C_{-}(\unicode[STIX]{x1D70F})=H(-\unicode[STIX]{x1D70F})\text{e}^{-\unicode[STIX]{x1D708}\unicode[STIX]{x1D70F}}D(\unicode[STIX]{x1D70F}),\end{eqnarray}$$

one has

(H 10a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}C_{-} & = & \displaystyle -\unicode[STIX]{x1D708}C_{-}-\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})D(0)+H(-\unicode[STIX]{x1D70F})\text{e}^{-\unicode[STIX]{x1D708}\unicode[STIX]{x1D70F}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}D\end{eqnarray}$$
(H 10b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle -\unicode[STIX]{x1D708}C_{-}-\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})C_{0}+H(-\unicode[STIX]{x1D70F})\widehat{s}(\unicode[STIX]{x1D70F}),\end{eqnarray}$$
where $s_{-}(\unicode[STIX]{x1D70F})=H(-\unicode[STIX]{x1D70F})\widehat{s}(\unicode[STIX]{x1D70F})$ and (H 10b ) is a rewrite of (H 8). For consistency between (H 10a ) and (H 10b ), one must therefore satisfy $D(0)=C_{0}$ and
(H 11) $$\begin{eqnarray}D(\unicode[STIX]{x1D70F})=C_{0}+\displaystyle \int _{0}^{\unicode[STIX]{x1D70F}}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\,\text{e}^{\unicode[STIX]{x1D708}\overline{\unicode[STIX]{x1D70F}}}\widehat{s}(\overline{\unicode[STIX]{x1D70F}}).\end{eqnarray}$$

Now if $C(t,ts_{+}\prime )$ is to describe a physical correlation function, it should decay as $\unicode[STIX]{x1D70F}\rightarrow -\infty$ . However, the contribution to (H 9) from the initial-condition term in (H 11) does not do that; rather, it explodes. Therefore, it is clear that $s_{-}$  cannot vanish and must be related in a particular way to  $C_{0}$ .

H.2 Example: the classical Langevin equation

To see how this works for a simple solvable example, consider the classical Langevin equation (Wang & Uhlenbeck Reference Wang and Uhlenbeck1945)

(H 12) $$\begin{eqnarray}\dot{v}+\unicode[STIX]{x1D708}v=\unicode[STIX]{x1D6FF}a(t),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}a$  is centred Gaussian white noise such that $\langle \unicode[STIX]{x1D6FF}a(t)\unicode[STIX]{x1D6FF}a(ts_{+}\prime )\rangle =2D_{v}\unicode[STIX]{x1D6FF}(t-ts_{+}\prime )$ . The two-time velocity-correlation function obeys

(H 13a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}C(t,ts_{+}\prime )+\unicode[STIX]{x1D708}C & = & \displaystyle \langle \unicode[STIX]{x1D6FF}a(t)\unicode[STIX]{x1D6FF}v(ts_{+}\prime )\rangle\end{eqnarray}$$
(H 13b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle \displaystyle \int _{-\infty }^{ts_{+}\prime }\!\text{d}\overline{t}\,\langle \unicode[STIX]{x1D6FF}a(t)\text{e}^{-\unicode[STIX]{x1D708}(t-\overline{t})}\unicode[STIX]{x1D6FF}a(\overline{t})\rangle\end{eqnarray}$$
(H 13c ) $$\begin{eqnarray}\displaystyle & = & \displaystyle 2D_{v}\displaystyle \int _{-\infty }^{ts_{+}\prime }\!\text{d}\overline{t}\,\text{e}^{-\unicode[STIX]{x1D708}(ts_{+}\prime -\overline{t})}\unicode[STIX]{x1D6FF}(t-\overline{t})\end{eqnarray}$$
(H 13d ) $$\begin{eqnarray}\displaystyle & = & \displaystyle \left\{\begin{array}{@{}ll@{}}0\quad & \text{for }t>ts_{+}\prime ,\\ D_{v}\quad & \text{for }t=ts_{+}\prime ,\\ 2D_{v}\text{e}^{-\unicode[STIX]{x1D708}(ts_{+}\prime -t)}\quad & \text{for }t<ts_{+}\prime .\end{array}\right.\end{eqnarray}$$
One therefore concludes that for this model
(H 14a,b ) $$\begin{eqnarray}s_{+}(t,ts_{+}\prime )=0,\quad s_{-}(t,ts_{+}\prime )=2D_{v}H(ts_{+}\prime -t)\unicode[STIX]{x1D6EF}(ts_{+}\prime ,t).\end{eqnarray}$$

The integral in (H 11) can now be done; it is $2D_{v}\displaystyle \int _{0}^{\unicode[STIX]{x1D70F}}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\,\text{e}^{2\unicode[STIX]{x1D708}\overline{\unicode[STIX]{x1D70F}}}=(D_{v}/\unicode[STIX]{x1D708})(\text{e}^{2\unicode[STIX]{x1D708}\unicode[STIX]{x1D70F}}-1)$ . From (H 9), the final solution is therefore

(H 15) $$\begin{eqnarray}C_{-}(\unicode[STIX]{x1D70F})=H(-\unicode[STIX]{x1D70F})\left[\text{e}^{-\unicode[STIX]{x1D708}\unicode[STIX]{x1D70F}}\left(C_{0}-\frac{D_{v}}{\unicode[STIX]{x1D708}}\right)+\frac{D_{v}}{\unicode[STIX]{x1D708}}\text{e}^{\unicode[STIX]{x1D708}\unicode[STIX]{x1D70F}}\right].\end{eqnarray}$$

To ensure convergent behaviour as $\unicode[STIX]{x1D70F}\rightarrow -\infty$ , one must require $C_{0}=D_{v}/\unicode[STIX]{x1D708}$ ; then $C_{-}(\unicode[STIX]{x1D70F})=H(-\unicode[STIX]{x1D70F})\text{e}^{\unicode[STIX]{x1D708}\unicode[STIX]{x1D70F}}C_{0}$ and the two-sided solution is

(H 16) $$\begin{eqnarray}C(\unicode[STIX]{x1D70F})=\text{e}^{-\unicode[STIX]{x1D708}|\unicode[STIX]{x1D70F}|}C_{0}.\end{eqnarray}$$

This is a well-known result for the classical Langevin problem.

This example shows that the two-time source and the one-time ‘initial’ value are not independent for a physical correlation function; one concludes that the noise term  $F$ cannot be neglected. This establishes the conceptual connection between the treatment in the main text, which used the multiple-time BBGKY hierarchy and one-sided functions, and the Dyson equations used by Rose, which do not involve one-sided correlation functions explicitly. When confronted with the complicated hierarchy equations, one might have naively attempted to ignore the initial-condition terms. As hopefully clarified by the above example, that is incorrect, as it would deal inconsistently with the nonlinear noise. Indeed, one sees explicitly in the calculations of § 6.3.2 that a contribution from an initial condition plays a key role in establishing the correct correspondence between the two-time formalism and Chapman–Enskog theory.

H.3 The $s_{+}^{(1;1)}$ term

Now it is possible to discuss the significance of the $s_{+}^{(1;1)}$ term on the right-hand side of (5.33) or (5.35). Its size is set by $C^{(0,3)}=O(\unicode[STIX]{x1D716}_{\text{p}}^{2})$ . It is thus nominally of the same order as the $\widehat{\text{C}}C_{+}^{(1;1)}$ term on the left-hand side of (5.33); it describes a contribution to nonlinear noise. However, the correlation time of  $s_{+}^{(1;1)}$ , being set by  $\unicode[STIX]{x1D6EF}(\unicode[STIX]{x1D70F})$ , is the short Debye-cloud autocorrelation time  $\unicode[STIX]{x1D714}_{\text{p}}^{-1}$ ; therefore, the $\int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\,s_{+}^{(1;1)}$ term in (5.35) is negligible, being $O(\unicode[STIX]{x1D716}_{\text{p}}^{2})$ , relative to the initial-condition term $C^{(0,2)}=O(\unicode[STIX]{x1D716}_{\text{p}})$ . The small $s_{+}^{(1;1)}$ term is analogous to the $s_{+}$  term in the above Langevin calculation. There the term vanishes altogether as a consequence of the assumption of zero autocorrelation time for the random acceleration. The $s_{+}^{(1;1)}$ term would play a role for short times, but it can be neglected in a one-sided equation coarse grained on the kinetic time scale.

Appendix I. Notation

The following list of notation merges symbols used in both Part 1 and Part 2.

I.1 Basic variables and physics symbols

a

$\boldsymbol{A}(\boldsymbol{v})$

The fundamental vector in the $\unicode[STIX]{x1D707}$ -space description: $\boldsymbol{A}\doteq (1,\;\boldsymbol{P}^{\prime \text{T}},\;Ks_{+}\prime )^{\text{T}}$ .

$\widetilde{\boldsymbol{A}}(\boldsymbol{x},t)$

The fundamental random variables in the 𝛤-space description: $\widetilde{\boldsymbol{A}}\doteq (\widetilde{N},\;\widetilde{\boldsymbol{P}}\text{}^{\text{T}},\;\widetilde{E})^{\text{T}}$ .

$\boldsymbol{A}_{\unicode[STIX]{x0394}}$

Momentum and energy exchange terms

$\boldsymbol{a}(\boldsymbol{x},t)$

The fundamental mean densities: $\boldsymbol{a}\doteq \langle \widetilde{\boldsymbol{A}}\rangle$ .

$\unicode[STIX]{x1D6FC}$

Thermal expansion coefficient: $\unicode[STIX]{x1D6FC}\doteq -n^{-1}(\unicode[STIX]{x2202}n/\unicode[STIX]{x2202}T)_{p}$ .

$\unicode[STIX]{x1D6FC}(\boldsymbol{v})$

The normalized kinetic energy: $\unicode[STIX]{x1D6FC}(\boldsymbol{v})\doteq {\textstyle \frac{1}{2}}mv^{2}/T$ .

b

$\boldsymbol{B}$

Variables conjugate to  $\widetilde{\boldsymbol{A}}$

$\boldsymbol{B}^{\text{ext}}$

External magnetic field

$B$

$B\doteq |\boldsymbol{B}^{\text{ext}}|$ .

$\boldsymbol{b}$

Auxiliary vector approximately equal to  $\unicode[STIX]{x0394}\boldsymbol{B}$ , the deviation of the conjugate variables from their values at the reference point; see (2.62).

$\widehat{\boldsymbol{b}}$

Unit vector in the direction of the magnetic field

$b_{0}$

Impact parameter for $90^{\circ }$  scattering: $b_{0}\doteq q_{1}q_{2}/T$ .

$\unicode[STIX]{x1D6FD}$

Inverse temperature: $\unicode[STIX]{x1D6FD}\doteq T^{-1}$ .

$\unicode[STIX]{x1D737}(\boldsymbol{v})$

Subtracted kinetic-energy flux: $\unicode[STIX]{x1D737}(\boldsymbol{v})\doteq [a(\boldsymbol{v})-\frac{5}{2}]\boldsymbol{v}=\unicode[STIX]{x1D6FE}(\boldsymbol{v})\boldsymbol{v}$ .

c

$C(t,ts_{+}\prime )$

Correlation function

$\text{C}[f]$

Nonlinear collision operator

$\widehat{\text{C}}$

Linearized collision operator: $\unicode[STIX]{x2202}_{t}|\unicode[STIX]{x1D712}\!\rangle +\cdots =-\widehat{\text{C}}|\unicode[STIX]{x1D712}\!\rangle$ , where $f\doteq (1+\unicode[STIX]{x1D712})f_{\text{M}}$ .

$\text{C}[f,\overline{f}]$

The bilinear Landau operator. The first slot refers to test particles, the second to field particles.

$\text{C}^{\text{BL}}[f;a,\bar{b}]$

The nonlinear Balescu–Lenard operator. The first slot describes functional dependence of the dielectric function, the second slot operates on test particles and the third slot operates on field particles.

$c_{\text{p}},c_{\text{v}}$

Specific heats at constant pressure and volume

$c_{\text{s}}$

Sound speed: $c_{\text{s}}\doteq (ZT_{e}/m_{i})^{1/2}$ .

$\unicode[STIX]{x1D712}_{\boldsymbol{k},\unicode[STIX]{x1D714}}$

Dielectric susceptibility: $\mathscr{D}=1+\unicode[STIX]{x1D712}$ .

$\unicode[STIX]{x1D712}(\unicode[STIX]{x1D707})$

Correction to the lowest-order one-particle distribution function: $f=(1+\unicode[STIX]{x1D712})f_{0}$ .

d

$D$

Diffusion coefficient

$\mathscr{D}(\boldsymbol{k},\unicode[STIX]{x1D714})$

Dielectric function

$\mathscr{D}_{0}(\boldsymbol{k})$

Static dielectric function: $\mathscr{D}_{0}(\boldsymbol{k})\doteq 1+k_{\text{D}}^{2}/k^{2}$ .

$\mathscr{D}_{\bot }$

Dielectric constant for strongly magnetized plasma: $\mathscr{D}_{\bot }\doteq \unicode[STIX]{x1D714}_{\text{p}i}^{2}/\unicode[STIX]{x1D714}_{\text{c}i}^{2}$ .

$\unicode[STIX]{x0394}$

Ordering parameter: $\unicode[STIX]{x0394}\doteq k_{\Vert }\unicode[STIX]{x1D706}_{\text{mfp}}$ .

$\unicode[STIX]{x0394}\boldsymbol{B}_{\overline{s}s}(\overline{\boldsymbol{r}},\boldsymbol{r},t)$

$\boldsymbol{B}_{\overline{s}}(\overline{\boldsymbol{r}},t)-\boldsymbol{B}_{s}(\boldsymbol{r},t)$

$\unicode[STIX]{x0394}\boldsymbol{B}$

$\unicode[STIX]{x0394}\boldsymbol{B}(\overline{\unicode[STIX]{x1D707}})\equiv \unicode[STIX]{x0394}\boldsymbol{B}_{\overline{s}}(\overline{\boldsymbol{r}},t)\equiv \unicode[STIX]{x0394}\boldsymbol{B}_{\overline{s}\,\overline{s}}(\overline{\boldsymbol{r}},\boldsymbol{r},t)$ .

$\unicode[STIX]{x0394}T$

Interspecies temperature difference: $\unicode[STIX]{x0394}T\doteq T_{e}-T_{i}$ .

$\unicode[STIX]{x0394}\boldsymbol{u}$

Interspecies flow velocity difference: $\unicode[STIX]{x0394}\boldsymbol{u}\doteq \boldsymbol{u}_{e}-\boldsymbol{u}_{i}$ .

$d$

Number of spatial dimensions

$\unicode[STIX]{x1D6FF}$

Ordering parameter: $\unicode[STIX]{x1D6FF}\doteq k_{\bot }\unicode[STIX]{x1D70C}$ .

$\unicode[STIX]{x1D6FF}(x)$

Dirac delta function

$\unicode[STIX]{x1D6FF}_{ij}$

Kronecker delta function

e

$E$

Energy

$\widetilde{E}(\boldsymbol{x},t)$

Microscopic energy density

$\boldsymbol{E}(\boldsymbol{x},t)$

Electric field

$\pmb{\mathbb{E}}$

Electric-field operator

$e$

Electronic charge (positive)

$e(\boldsymbol{x},t)$

Macroscopic energy density. For an ideal gas, $e=\frac{3}{2}nT$ .

$\unicode[STIX]{x1D716}$

Ordering parameter; positive infinitesimal

$\unicode[STIX]{x1D716}_{\text{p}}$

Plasma parameter: $\unicode[STIX]{x1D716}_{\text{p}}\doteq 1/n\unicode[STIX]{x1D706}_{\text{D}}^{3}$ .

$\unicode[STIX]{x1D750}(\boldsymbol{x}_{1},\boldsymbol{x}_{2})$

Interparticle electric field for unit charges: $\unicode[STIX]{x1D750}_{12}\doteq -\unicode[STIX]{x1D735}_{1}|\boldsymbol{x}_{1}-\boldsymbol{x}_{2}|^{-1}$ ; $\unicode[STIX]{x1D750}(\boldsymbol{x})=(2\unicode[STIX]{x03C0})^{-3}\int \text{d}\boldsymbol{k}\,\text{e}^{\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}}\unicode[STIX]{x1D750}_{\boldsymbol{k}}$ .

$\unicode[STIX]{x1D750}_{\boldsymbol{k}}$

$\unicode[STIX]{x1D750}_{\boldsymbol{k}}\doteq -4\unicode[STIX]{x03C0}\,\text{i}\boldsymbol{k}/k^{2}$ .

f

$F(\unicode[STIX]{x1D6E4},t)$

Liouville distribution: $F\equiv P_{\mathscr{N}}$ .

$F_{\boldsymbol{B}}(\unicode[STIX]{x1D6E4};t)$

Local-equilibrium distribution

$F_{0}(\unicode[STIX]{x1D6E4};\boldsymbol{r},t)$

Reference distribution

$f(\unicode[STIX]{x1D707},t)$

One-particle distribution function: $f\doteq \langle \,\widetilde{f}\rangle$ .

$\widetilde{f}(\unicode[STIX]{x1D707},t)$

Klimontovich phase-space distribution function: $\widetilde{f}\doteq \overline{n}^{-1}\sum _{i}\unicode[STIX]{x1D6FF}\boldsymbol{(}\unicode[STIX]{x1D707}-\widetilde{\unicode[STIX]{x1D707}}(t)\boldsymbol{)}$ .

$f_{\text{M}}$

Maxwellian distribution function

$f_{\text{lM}}$

Local Maxwellian

$\boldsymbol{f}$

Random force

$\unicode[STIX]{x1D719}$

Electrostatic potential

g

$\unicode[STIX]{x1D6E4}$

Phase-space coordinates of the many-body system

$g(\text{}\underline{1},\text{}\underline{2},t)$

Pair correlation function

$\unicode[STIX]{x1D6FE}(\boldsymbol{v})$

$\unicode[STIX]{x1D6FC}(\boldsymbol{v})-\frac{5}{2}$

h

$H(\unicode[STIX]{x1D70F})$

Heaviside unit step function

$\text{He}_{n}$

Probabilistic Hermite polynomial

$h_{\text{P}}$

Planck’s constant

$h(\boldsymbol{x},t)$

Enthalpy density. For an ideal gas, $h=\frac{5}{2}nT$ .

$h(\text{}\underline{1},\text{}\underline{2},\text{}\underline{3},t)$

Triplet correlation function

$\unicode[STIX]{x1D6C8}$

Matrix of transport coefficients

$\unicode[STIX]{x1D702},\hspace{2.55527pt}\widehat{\unicode[STIX]{x1D702}}$

Source functions

j

$\boldsymbol{J}$

Generic flux or current

$\widehat{\boldsymbol{J}}$

Subtracted flux: $\widehat{\boldsymbol{J}}\doteq \text{Q}\boldsymbol{J}$ .

k

$K(\boldsymbol{v})$

Kinetic energy: $K\doteq {\textstyle \frac{1}{2}}mv^{2}$ .

$\boldsymbol{k}$

Wavevector

$k_{\text{D}}$

Debye wavenumber: $k_{\text{D}}^{2}\doteq \sum _{s}k_{\text{D}s}^{2}$ , where $k_{\text{D}s}\doteq (4\unicode[STIX]{x03C0}nq^{2}/T)_{s}^{1/2}$ .

$\unicode[STIX]{x1D705}$

Thermal conductivity

$\unicode[STIX]{x1D705}_{T}$

Isothermal compressibility: $\unicode[STIX]{x1D705}_{T}\doteq n^{-1}(\unicode[STIX]{x2202}n/\unicode[STIX]{x2202}p)_{T}$ .

l

$L$

Box size or gradient scale length

$\text{L}_{1}$

Landau operator for particle 1

$\boldsymbol{L}^{2}$

Square of the angular momentum operator

$\mathscr{L}$

Liouville operator

$\unicode[STIX]{x1D6EC}$

Argument of the Coulomb logarithm: $\unicode[STIX]{x1D6EC}\doteq \unicode[STIX]{x1D706}_{\text{D}}/b_{0}$ in the classical limit.

$\unicode[STIX]{x1D706}_{\text{mfp}}$

Mean free path: $\unicode[STIX]{x1D706}_{\text{mfp}}\doteq v_{\text{t}}/\unicode[STIX]{x1D708}$ .

$\unicode[STIX]{x1D706}_{\text{B}}$

de Broglie wavelength: $\unicode[STIX]{x1D706}_{\text{B},s\overline{s}}\doteq h_{\text{P}}/(\unicode[STIX]{x1D707}_{s\overline{s}}|\boldsymbol{v}-\overline{\boldsymbol{v}}|)$ .

$\unicode[STIX]{x1D706}_{\text{D}}$

Debye length: $\unicode[STIX]{x1D706}_{\text{D}}\doteq k_{\text{D}}^{-1}$ .

m

$\widehat{\text{M}}$

Magnetic-field operator: $\widehat{\text{M}}\doteq -\text{i}\unicode[STIX]{x1D714}_{\text{c}}\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D701}$ .

$M_{sss_{+}\prime }$

Total mass: $M_{sss_{+}\prime }\doteq m_{s}+m_{ss_{+}\prime }$ .

$\unicode[STIX]{x1D648}_{}$

Covariance matrix in the reference ensemble: $\unicode[STIX]{x1D648}_{}\doteq \langle \boldsymbol{A}s_{+}\prime \,\boldsymbol{A}s_{+}\prime ^{\text{T}}\rangle _{0}$ .

$m$

Mass

$\unicode[STIX]{x1D662}$

Fourth-rank viscosity tensor

$\unicode[STIX]{x1D707}$

Viscosity; generic observer arguments $\unicode[STIX]{x1D707}\doteq \{\boldsymbol{x},\boldsymbol{v},s\}$ or $\{\boldsymbol{x},s\}$ ; field index; mass ratio $m_{e}/m_{i}$

$\unicode[STIX]{x1D707}_{sss_{+}\prime }$

Reduced mass: $\unicode[STIX]{x1D707}_{sss_{+}\prime }^{-1}=m_{s}^{-1}+m_{ss_{+}\prime }^{-1}$ .

n

$\widetilde{N}(\boldsymbol{x},t)$

Microscopic number density

$\mathscr{N},N$

Total number of particles

$n(\boldsymbol{x},t)$

Macroscopic (averaged) number density

$\overline{n}$

Mean density: $\overline{n}\doteq \mathscr{N}/\mathscr{V}$ .

$\unicode[STIX]{x1D708}$

Collision frequency

$\unicode[STIX]{x1D743}$

$\unicode[STIX]{x1D6EF}(\unicode[STIX]{x1D70F})$

Response function for the linearized Vlasov equation, sans $H(\unicode[STIX]{x1D70F})$ : $R(\unicode[STIX]{x1D70F})=H(\unicode[STIX]{x1D70F})\unicode[STIX]{x1D6EF}(\unicode[STIX]{x1D70F})$ .

$\unicode[STIX]{x1D74E}$

$\unicode[STIX]{x1D734}$

Frequency matrix; vorticity tensor: $\unicode[STIX]{x1D734}\doteq \frac{1}{2}[(\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}-(\unicode[STIX]{x1D735}\boldsymbol{u})]$ .

$\unicode[STIX]{x1D714}$

Fourier transform variable conjugate to  $t$

$\unicode[STIX]{x1D714}_{\text{p}}$

Plasma frequency: $\unicode[STIX]{x1D714}_{\text{p}}^{2}=\sum _{s}\unicode[STIX]{x1D714}_{\text{p}s}^{2}$ , where $\unicode[STIX]{x1D714}_{\text{p}s}\doteq (4\unicode[STIX]{x03C0}nq^{2}/m)_{s}^{1/2}$ .

p

$\text{P}$

Projection operator

$P_{m}^{l}$

Associated Legendre function of the first kind

$\boldsymbol{P}(\boldsymbol{v})$

$\unicode[STIX]{x1D707}$ -space kinetic momentum flux: $\boldsymbol{P}\doteq m\boldsymbol{v}$ .

$\widetilde{\boldsymbol{P}}(\boldsymbol{x},t)$

Microscopic momentum density

$\boldsymbol{p}(\boldsymbol{x},t)$

Macroscopic momentum density

$p(\boldsymbol{x},t)$

Macroscopic pressure. For an ideal gas, $p=nT$ .

$\unicode[STIX]{x1D745}$

Pressureless part of the stress tensor: $\unicode[STIX]{x1D745}\doteq \unicode[STIX]{x1D749}-p\unicode[STIX]{x1D644}$ .

q

$\text{Q}$

Orthogonal projection operator: $\text{Q}\doteq 1-\text{P}$ .

$Q$

Heat generation

$q$

Signed charge

$\boldsymbol{q}$

Heat-flow vector

r

$R_{\{0,1,2\}}$

$\text{R}_{0}(\unicode[STIX]{x1D70F})\doteq \text{e}^{-\text{i}\mathscr{L}\unicode[STIX]{x1D70F}}$ , $\text{R}_{1}(\unicode[STIX]{x1D70F})\doteq \text{e}^{-\text{Q}\text{i}\mathscr{L}\unicode[STIX]{x1D70F}}$ , $\text{R}_{2}(\unicode[STIX]{x1D70F})\doteq \text{e}^{-\text{Q}\text{i}\mathscr{L}\text{Q}\unicode[STIX]{x1D70F}}$ .

$R(t;ts_{+}\prime )$

Causal infinitesimal response function

$\boldsymbol{R}$

Friction force

$r(t;ts_{+}\prime )$

Single-particle causal response function

$\boldsymbol{r}$

Reference position at which the fluid equations are evaluated

$\unicode[STIX]{x1D70C}(\boldsymbol{r},t)$

Charge density. Poisson’s equation is $-\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D719}=4\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}$ .

$\unicode[STIX]{x1D70C}_{s}$

Gyroradius of species  $s$ .

$\unicode[STIX]{x1D70C}_{\text{s}}$

Sound radius: $\unicode[STIX]{x1D70C}_{\text{s}}\doteq c_{\text{s}}/\unicode[STIX]{x1D714}_{\text{c}i}$ .

$\unicode[STIX]{x1D746}$

Spatial difference: $\unicode[STIX]{x1D746}\doteq \boldsymbol{x}-\boldsymbol{x}s_{+}\prime$ .

s

$S$

Number of species

$S_{sss_{+}\prime }$

Coefficient in the Landau operator: $S_{sss_{+}\prime }\doteq (nq^{2})_{s}(nq^{2})_{ss_{+}\prime }\ln \unicode[STIX]{x1D6EC}_{sss_{+}\prime }$ .

$\unicode[STIX]{x1D64E}$

Rate-of-strain tensor: $\unicode[STIX]{x1D64E}\doteq \frac{1}{2}[(\unicode[STIX]{x1D735}\boldsymbol{u})+(\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}]$ .

$s$

Species index (e.g. $s\in \{e,i\}$ ); also an arbitrary time for the evolution of orthogonal perturbations

$s(\boldsymbol{x},t)$

Macroscopic entropy density

t

$T$

Temperature

$t$

Time at which the fluid equations are evaluated (cf. $s$ )

$\unicode[STIX]{x1D70F}$

Time lag: $\unicode[STIX]{x1D70F}\doteq t-ts_{+}\prime$ .

$\unicode[STIX]{x1D70F}_{e},\;\unicode[STIX]{x1D70F}_{i}$

Collision times

$\unicode[STIX]{x1D749}$

Stress tensor: $\unicode[STIX]{x1D749}=p\unicode[STIX]{x1D644}+\unicode[STIX]{x1D745}$ .

u

$\text{U}$

Modified propagator: $\text{U}(\unicode[STIX]{x1D70F})\doteq \text{Q}\text{e}^{-\text{i}\text{Q}\mathscr{L}\text{Q}\unicode[STIX]{x1D70F}}\text{Q}$ .

$U$

Potential energy

$\unicode[STIX]{x1D650}$

Projection operator in the Landau operator: $\unicode[STIX]{x1D650}(\boldsymbol{v})\doteq (\unicode[STIX]{x1D644}-\hat{\boldsymbol{v}}\,\hat{\boldsymbol{v}})/v$ .

$\boldsymbol{u}(\boldsymbol{x},t)$

Fluid velocity

$u^{\text{int}}$

Internal energy density

v

$\mathscr{V},V$

System volume

$v_{\text{t}}$

Thermal velocity: $v_{\text{t}}\doteq (T/m)^{1/2}$ .

$\boldsymbol{v}$

Velocity

w

$W(\unicode[STIX]{x1D6E4},s;\boldsymbol{r},t)$

$W\doteq \ln \boldsymbol{(}F(\unicode[STIX]{x1D6E4},s)/F_{0}(\unicode[STIX]{x1D6E4};\boldsymbol{r},t)\boldsymbol{)}$

$\unicode[STIX]{x1D652}$

$\unicode[STIX]{x1D652}\doteq \unicode[STIX]{x1D735}\boldsymbol{u}+(\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}-\frac{2}{3}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u})\unicode[STIX]{x1D644}$ .

$\boldsymbol{w}(\boldsymbol{x},\boldsymbol{v},t)$

Peculiar velocity: $\boldsymbol{w}(\boldsymbol{x},\boldsymbol{v},t)\doteq \boldsymbol{v}-\boldsymbol{u}(\boldsymbol{x},t)$ .

x

$X^{\unicode[STIX]{x1D6FC}}$

Mean exchange terms

$\boldsymbol{x}$

Generic spatial position (cf. $\boldsymbol{r}$ )

$\unicode[STIX]{x1D6EF}$

Vlasov response function sans $H(\unicode[STIX]{x1D70F})$

y

$Y_{m}^{l}$

Spherical harmonic

$\unicode[STIX]{x1D6F6}_{\unicode[STIX]{x0394}}$

Orthogonal source term for exchange effects

z

$Z$

Partition function; atomic number

$\unicode[STIX]{x1D701}$

Bulk viscosity; gyroangle

I.2 Miscellaneous notation

$\widetilde{A}$

A tilde indicates a random quantity.

$As_{+}\prime$

Fluctuation in the reference ensemble: $As_{+}\prime \doteq \widetilde{A}-\langle A\rangle _{0}$ .

$A^{\ast }$

Complex conjugate of  $A$

$\widehat{A}$

On most symbols, a hat indicates the orthogonal projection: $\widehat{A}\doteq \text{Q}A$ . The symbol  $\widehat{\text{C}}$ is an exception that denotes the linearized collision operator. Another exception is the magnetic-field operator $\widehat{\text{M}}$ .

$A_{+}$

One-sided function

$\unicode[STIX]{x1D63C}^{\text{T}}$

Transpose of  $\unicode[STIX]{x1D63C}$

${\displaystyle\mathop{\,A\,}\limits_{{\sim}}}$

A quantity dependent on a correlation function involving three phase-space points

$A^{\unicode[STIX]{x1D707}},A_{\unicode[STIX]{x1D707}}$

Contravariant and covariant components of  $\boldsymbol{A}$

$A_{(s)}$

Parentheses around an index defeat the Einstein summation convention.

$\unicode[STIX]{x0394}A$

First-order linearization of  $A$

$\mathscr{A}$

Total amount of  $A$ : $\mathscr{A}=\int \text{d}\boldsymbol{r}\,A(\boldsymbol{r})$ .

$A[\unicode[STIX]{x1D713}]$

Square brackets indicate functional dependence.

$\boldsymbol{\unicode[STIX]{x2202}}$

$(q/m)\unicode[STIX]{x2202}/\unicode[STIX]{x2202}_{\boldsymbol{v}}$

$\text{}\underline{1}$

Underlining denotes a set of coordinates sans time.

$\overline{x}$

Typically, an overline denotes an integration variable or summation index.

$\doteq$

Definition

$\star$

Non-local integral in the local-equilibrium distribution: $A\star B\doteq \sum _{\overline{s}}\int \text{d}\overline{\boldsymbol{r}}\,\boldsymbol{A}_{\overline{s}}(\overline{\boldsymbol{r}},t)\boldsymbol{\cdot }\boldsymbol{B}_{\overline{s}}(\overline{\boldsymbol{r}},t)$ .

$\ast$

Convolution

$\boldsymbol{\ast }$

Convolution plus dot product

I.3 Acronyms and abbreviations

B

Burnett

BBGKY

Bogoliubov, Born, Green, Kirkwood, and Yvon

DIA

direct-interaction approximation

MSR

Martin–Siggia–Rose

NS

Navier–Stokes

ODE

ordinary differential equation

PDF

probability density function

PDIA

particle direct-interaction approximation

WKB

Wentzel–Kramers–Brillouin

Footnotes

1 An apparently successful ‘regularized’ theory applicable to neutral fluids has been developed by Struchtrup & Torrolhon (Reference Struchtrup and Torrolhon2003) and Struchtrup (Reference Struchtrup2005).

2 The quotation has been modified to accommodate the author–year style of referencing used in the present article.

3 The regularized theory of Struchtrup & Torrolhon (Reference Struchtrup and Torrolhon2003) does display a Knudsen boundary layer and possesses other desirable qualities.

4 I say ‘consistent’ because in order to be quantitative Catto & Simakov calculate the corrections to the distribution function approximately (inversion of the collision operator is involved), thus obscuring the general structure of the theory. However, if one manipulates the basic equations solved by Catto & Simakov, the equivalence of the formalisms can be demonstrated. That is done in § 6. It can then be seen that the calculations of Catto & Simakov of the parallel viscosity amount to approximate evaluations of some of the formulas of Brey (Reference Brey1983), and that straightforward generalization of those unmagnetized formulas would lead to the magnetic-field effects evaluated by Catto & Simakov.

5 This can already be seen from the linear-response formalism discussed in Part 1. In the treatment of the magnetized Braginskii equations (§ 1:3), the linearized collision operator  $\widehat{\text{C}}$ is merely replaced by $\text{i}\widehat{\text{M}}+\widehat{\text{C}}$ , where $\widehat{\text{M}}$  is the magnetic-field operator. In the present paper, formulas for transport coefficients are written in terms of the Liouville operator  $\mathscr{L}$ . Those formulas remain unchanged if $\mathscr{L}$  is defined to include the Lorentz force. However, approximate evaluation of the general formulas requires additional ordering assumptions. The ones used by Catto & Simakov (Reference Catto and Simakov2004) are described in footnote 59 on page 61.

6 Section and equation numbers that refer to Part 1 or the online Supplement to Part 2 (Krommes Reference Krommes2018c ) are prefaced by ‘1:’ or ‘2-S:’, respectively.

7 This picture cannot possibly be new, but I am unable to cite a reference where it is presented in the present form, including the subsequent discussion of non-Gaussian statistics.

8 Subtracted fluxes are discussed in § 2.6.2 and calculated in appendix  D.

9 Here ‘two-time’ simply means that the correlation function  $C$ depends on two arbitrary times, i.e. $C=C(t,ts_{+}\prime )$ . It does not imply a multiple time scale asymptotic procedure such as is discussed in appendix  1:A.

10 The concepts of $\unicode[STIX]{x1D707}$  space and $\unicode[STIX]{x1D6E4}$ space were introduced in § 1:2.1, where further motivational remarks can also be found.

11 See the previous footnote.

12 The Lagrangian velocity-correlation function is defined by

$$\begin{eqnarray}C_{vv}(\unicode[STIX]{x1D70F})\doteq \langle \unicode[STIX]{x1D6FF}v\boldsymbol{(}\,\widetilde{x}(\unicode[STIX]{x1D70F}),\unicode[STIX]{x1D70F}\boldsymbol{)}\unicode[STIX]{x1D6FF}v\boldsymbol{(}\,\widetilde{x}(0),0\boldsymbol{)}\rangle ,\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}v$ denotes the velocity fluctuation from the mean, $\widetilde{x}(t)$  is the random trajectory of a fluid element (or test particle), and it is assumed that the fluctuations are temporally stationary.

13 This kinetic-theory problem is completely analogous to the problem of Navier–Stokes turbulence, which has been discussed extensively. A highly incomplete list of discussions of statistical closure theory includes Kraichnan (Reference Kraichnan1961, Reference Kraichnan1962), Leslie (Reference Leslie1973), McComb (Reference McComb1990), Krommes (Reference Krommes2002), McComb (Reference McComb2014) and Krommes & Parker (Reference Krommes, Parker, Galperin and Read2018, and references therein).

14 The Gaussian probability density functional has the form

(1.7) $$\begin{eqnarray}P[\unicode[STIX]{x1D713}]=\mathscr{N}\exp \left(-\frac{1}{2}\displaystyle \int _{-\infty }^{\infty }\!\,\text{d}t\displaystyle \int _{-\infty }^{\infty }\!\,\text{d}ts_{+}\prime \,[\unicode[STIX]{x1D713}(t)-\langle \unicode[STIX]{x1D713}\rangle (t)]C^{-1}(t,ts_{+}\prime )[\unicode[STIX]{x1D713}(ts_{+}\prime )-\langle \unicode[STIX]{x1D713}\rangle (ts_{+}\prime )]\right),\end{eqnarray}$$

where $\mathscr{N}$  is a normalization factor. The inverse  $C^{-1}$ is defined by the solution of the equation $\int _{-\infty }^{\infty }\!\,\text{d}\overline{t}\,C(t,\overline{t})C^{-1}(\overline{t},ts_{+}\prime )=\unicode[STIX]{x1D6FF}(t-ts_{+}\prime )$ .

15 The hat on  $\widehat{\unicode[STIX]{x1D702}}$ is used for consistency with the notation of Martin, Siggia & Rose (Reference Martin, Siggia and Rose1973). See Krommes (Reference Krommes2002) for a review of the Martin, Siggia & Rose (MSR) formalism and for further discussion of source functions.

16 The fact that at linear order transport coefficients can be represented in terms of two-time correlation functions is well known from the work of Kubo on linear-response theory. Selected references include Kubo (Reference Kubo1957), Kubo (Reference Kubo, Brittin and Dunham1959) and Kubo (Reference Kubo, Kirczenow and Marro1974).

17 Conventionally, the bilinear Landau collision operator is written as $\text{C}[\,f,f]$ . I shall instead write $\text{C}[\,f,\overline{f}]$ , where the overline indicates dependence on the integration variable  $\overline{\boldsymbol{v}}$ associated with the field particles. Strictly speaking, the overline is unnecessary because the square-bracket notation already indicates general functional dependence. However, the notation will make some subsequent manipulations easier to follow.

18 It is much more conventional to use  $N$ rather than  $\mathscr{N}$ for the total number of particles (and  $V$ rather than  $\mathscr{V}$ for the system volume). The script font is used in this paper in order to be consistent with Brey’s notation for the total (volume-integrated) amount of a quantity. See § 2.2, especially (2.3), for further discussion.

19 Brey et al. do not include the  $\text{i}$ in the definition of the Liouville operator. I prefer to retain it as a reminder that the operator is time reversible (a term such as $\exp (-\text{i}\mathscr{L}\unicode[STIX]{x1D70F})$ suggests oscillations at the microscopic level) and so that the  $\text{i}$ disappears in $\boldsymbol{k}$  space. For example, under Fourier transformation the operator $\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\rightarrow \text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}$ , leading to the streaming operator $\mathscr{L}_{\boldsymbol{v}}=\sum _{i=1}^{\mathscr{N}}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}_{i}$ . Dissipative effects in the many-body system arise from a generalization of the fact that the Maxwellian velocity average of the reversible streaming effect $\exp (-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}\unicode[STIX]{x1D70F})$ decays in  $\unicode[STIX]{x1D70F}$ . See the related discussion of the stochastic-oscillator model in the review of statistical closure theory by Krommes (Reference Krommes2002).

20 Of course, potential-energy interactions cannot be neglected everywhere since the dissipation arising from Coulomb collisions depends on them. The relevant correlations are contained in the Liouville operator.

21 This property follows as a generalization of the basic properties

$$\begin{eqnarray}\text{i}\mathscr{L}\boldsymbol{x}_{i}=\boldsymbol{v}_{i}=\frac{\text{d}\boldsymbol{x}_{i}}{\text{d}t},\quad \text{i}\mathscr{L}\boldsymbol{v}_{i}=\left(\frac{q}{m}\right)_{i}(\boldsymbol{E}_{i}+c^{-1}\boldsymbol{v}_{i}\times \boldsymbol{B}_{i}^{\text{ext}})=\frac{\text{d}\boldsymbol{v}_{i}}{\text{d}t},\end{eqnarray}$$

or for arbitrary phase function  $f(\unicode[STIX]{x1D6E4})$

$$\begin{eqnarray}\text{i}\mathscr{L}f(\unicode[STIX]{x1D6E4})=\frac{\text{d}}{\text{d}t}f\boldsymbol{(}\unicode[STIX]{x1D6E4}(t)\boldsymbol{)}|_{t=0}.\end{eqnarray}$$

An equivalent discussion is given by Piccirelli (Reference Piccirelli1968); see his equations (23) and (24).

22 If the one-particle kinetic equation is written as $\unicode[STIX]{x2202}_{t}\,f+\cdots =-C[\,f]$ , its linearization around a background state  $f_{0}$ (usually the Maxwellian  $f_{\text{M}}$ ) is $\unicode[STIX]{x2202}_{t}\unicode[STIX]{x0394}f+\cdots =-\widehat{\text{C}}\unicode[STIX]{x0394}f$ , where $\widehat{\text{C}}\doteq \unicode[STIX]{x1D6FF}C[\,f]/\unicode[STIX]{x1D6FF}f|_{f=f_{0}}$ (here $\unicode[STIX]{x1D6FF}$  denotes a functional derivative). The linearized Landau collision operator is discussed in some detail in appendix  1:B.

23 This result would be true if $\widehat{\text{C}}$  were a first-order differential operator with a constant coefficient, such as  $\unicode[STIX]{x2202}_{v}$ . However, $\widehat{\text{C}}$  is, at the very least, a second-order differential operator in velocity. (In fact, it is an integro-differential operator.)

24 Note that if the cells are arranged in a fixed Eulerian tiling, particles can cross cell boundaries and the lowest-order local cell distribution is a grand canonical ensemble. At next order, matching of the parameters at the boundaries of the neighbouring cells induces gradients across each cell that lead to small corrections to the locally Gibbsian statistical distribution. It is those corrections that determine the collisional transport coefficients.

25 If wave-induced transport is to be considered, this discussion must be modified. See the remarks in appendix B.

26 The significance of the $\boldsymbol{B}$  subscript on  $F_{\boldsymbol{B}}$ will become clear later. In this context, $\boldsymbol{B}$  denotes a set of conjugate variables, not the magnetic field.

27 Brey et al. used  $\widetilde{\boldsymbol{G}}$ instead of  $\widetilde{\boldsymbol{P}}$ and  $\boldsymbol{g}$ instead of  $\boldsymbol{p}$ . The boldfaced  $\widetilde{\boldsymbol{P}}$ for microscopic momentum density should be distinguished from a projection operator  $\text{P}$ and a PDF  $P$ ; boldfaced  $\boldsymbol{p}$ (mean momentum density) should be distinguished from the pressure  $p$ .

28 In kinetic theory, it is convenient to define $n$ -particle distribution functions  $f_{n}$ by $f_{n}\doteq \mathscr{V}^{n}P_{n}$ . I shall write $f_{1}\equiv f$ . Also, to make the notation more consistent with that of Brey et al., I shall write $P_{\mathscr{N}}\equiv F$ .

29 The centre-of-mass velocity of the entire collection of particles is proportional to $\sum _{s}\widetilde{\pmb{\pmb{\mathscr{G}}}}_{s}$ , and that can be set to zero with no loss of generality. I make the more restrictive assumption that no species has bulk motion. This does not preclude a non-vanishing local fluid velocity $\boldsymbol{u}_{s}(\boldsymbol{r},t)$ at a particular point in the fluid.

30 Particle energy is not conserved in a finite-volume canonical ensemble. However, it is well known that the relative energy fluctuations scale as $\mathscr{N}^{-1/2}$ , so they are negligible in the thermodynamic limit.

31 Strictly speaking, if the many-particle system is in contact with a heat bath at temperature  $T$ , then the equilibrium temperature of all species should be  $T$ . However, because interspecies temperature relaxation is slow for small mass ratio, I follow the standard assumption of allowing for different temperatures  $T_{s}$ (Braginskii Reference Braginskii and Leontovich1965). In any event, perturbations from a common temperature can clearly be species dependent.

32 In a consistent notation, the microscopic stress tensor would be written in upper case. However, the symbol for Greek Tau is indistinquishable from  $T$ , which is reserved for temperature.

33 In general, each of the terms in the product $n_{s}\boldsymbol{E}$ is random when the system is macroscopically turbulent. However, in this paper fluctuations of those terms are ignored in order to focus on classical transport. Non-local wave-induced transport also follows from more detailed considerations of the microscopic electric forces.

34 See the previous footnote.

35 A clear discussion of the local-equilibrium distribution is given by Piccirelli (Reference Piccirelli1968, § IV), who also cites some historical references. See also footnote 38 on page 25.

36 I also use  $\boldsymbol{B}$ for the magnetic field, but there should be no confusion in context. For the vector field, I write  $\boldsymbol{B}^{\text{ext}}$ , but I drop the superscript for $B=|\boldsymbol{B}^{\text{ext}}|$ and $\widehat{\boldsymbol{b}}\doteq \boldsymbol{B}^{\text{ext}}/B$ .

Why are the non-random conjugate variables  $\boldsymbol{B}$ written in upper case? I yield to the notation used by Brey et al.

37 Brey et al. (Reference Brey, Zwanzig and Dorfman1981) used the asterisk symbol  $\ast$ for this operation, but I prefer to use  $\ast$ to denote convolution. $A(\boldsymbol{r})\ast B(\boldsymbol{r})\doteq \int \text{d}\overline{\boldsymbol{r}}\,A(\boldsymbol{r}-\overline{\boldsymbol{r}})B(\overline{\boldsymbol{r}})$ depends on  $\boldsymbol{r}$ , but $A(\boldsymbol{r})\star B(\boldsymbol{r})$ as defined by (2.30) is independent of  $\boldsymbol{r}$ . The $\star$  operation is a generalization of the familiar operation of summation and/or integration over repeated indices.

38 An extended quotation from Piccirelli (Reference Piccirelli1968) may be useful. In his discussion of projection techniques, he says (using the notation  $D$ instead of $P_{\mathscr{N}}\equiv F$ , and $D_{0}$ instead of  $F_{\boldsymbol{B}}$ ),

Since  …almost closed expressions for the irreversible currents are central to the present work, the following remarks should be made. The expressions are almost closed only because we have assumed that $D(\unicode[STIX]{x1D6E4},t)$ is split into a differentiable time-independent functional  $D_{0}$ of the densities and a remainder and not because of the particular form of  $D_{0}$ . [Several sentences of technical justification are omitted here.]

From this point of view, our choice of  $D_{0}$ would appear to be arbitrary; any of a whole class of choices will yield almost closed expressions for the currents. One should notice concerning this point: First, that a certain arbitrariness over various generalized equilibrium distributions is desirable to reflect differences in experimental circumstances. It will be argued in the concluding section that the final results are invariant to such choices. Second, we have just seen that the use of generalized equilibrium distributions for  $D_{0}$ introduces the conjugate variables and their gradients in a very natural way. As will emerge presently, this yields a structural form of the equations which has a very strong and complete analogy to the classical ones which must be their limiting form. Finally, there is no reason to expect that arbitrary choices of  $D_{0}$ will lead by any route to the classical limit, and in particular, one cannot expect the initial value terms to have the proper behaviour for arbitrary choices of  $D_{0}$ which may not correspond to physically realizable initial preparations $\ldots$ .

39 I deviate from the notation of Brey et al. and call this quantity  $\unicode[STIX]{x0394}\boldsymbol{B}$ instead of  $\boldsymbol{B}_{0}$ in order to emphasize with the $\unicode[STIX]{x0394}$ that $\unicode[STIX]{x0394}\boldsymbol{B}$  will be treated as small.

40 To eliminate the orthogonal projections, one needs to solve equations of the form

$$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D713}(t)+\text{i}\widehat{\text{L}}(t)\unicode[STIX]{x1D713}=s(t).\end{eqnarray}$$

When $\widehat{\text{L}}(t)$  is a scalar function, Green’s function is easily seen to be

$$\begin{eqnarray}G(t;ts_{+}\prime )=H(t-ts_{+}\prime )\exp \left(-\displaystyle \int _{ts_{+}\prime }^{t}\!\text{d}\overline{t}\,\text{i}\widehat{\text{L}}(\overline{t})\right).\end{eqnarray}$$

When $\widehat{\text{L}}(t)$  is instead an operator such that $\widehat{\text{L}}(t_{1})\widehat{\text{L}}(t_{2})-\widehat{\text{L}}(t_{2})\widehat{\text{L}}(t_{1})\neq 0$ , iterative solution shows that Green’s function generalizes to

$$\begin{eqnarray}G(t;ts_{+}\prime )=H(t-ts_{+}\prime )\exp _{+}\left(-\displaystyle \int _{ts_{+}\prime }^{t}\!\text{d}\overline{t}\,\text{i}\widehat{\text{L}}(\overline{t})\right),\end{eqnarray}$$

where the plus subscript denotes time ordering (i.e. in the series expansion of the exponential the operators must be ordered such that their time arguments increase from right to left).

41 One has $\boldsymbol{E}=-\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}$ ; Poisson’s equation is $-\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D719}=4\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}$ . In the one-component case, $\unicode[STIX]{x1D70C}=(nq)_{s}+(\overline{n}q)_{ss_{+}\prime }$ , where $s$  is the active species and $ss_{+}\prime$  represents the neutralizing background. Thus, $\int \text{d}\boldsymbol{r}\,(nq)_{s}\boldsymbol{E}=\int \text{d}\boldsymbol{r}\,[\unicode[STIX]{x1D70C}-(\overline{n}q)_{ss_{+}\prime }](-\unicode[STIX]{x1D735}\unicode[STIX]{x1D719})$ . The $\overline{n}$  term integrates away. From Poisson’s equation, $-\unicode[STIX]{x1D70C}\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}=(4\unicode[STIX]{x03C0})^{-1}(\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D719})(\unicode[STIX]{x1D735}\unicode[STIX]{x1D719})=(8\unicode[STIX]{x03C0})^{-1}\unicode[STIX]{x1D735}|\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}|^{2}$ . This term integrates away as well.

42 This is an expansion in the Knudsen number (defined in § 1:A), not in the Mach number.

43 See the related discussion in footnote 30 on page 22 of Part 1. In detail,

$$\begin{eqnarray}\text{U}(s)\doteq \text{Q}\text{e}^{-\text{Q}\text{i}\mathscr{L}\text{Q}s}\text{Q}=\text{Q}\text{e}^{-\text{Q}\text{i}\mathscr{L}s}\text{Q}=\text{e}^{-\text{Q}\text{i}\mathscr{L}s}\text{Q},\end{eqnarray}$$
$$\begin{eqnarray}\unicode[STIX]{x2202}_{s}\text{U}=-\text{Q}\text{i}\mathscr{L}\text{U}(s)=-(1-\text{P})\text{i}\mathscr{L}\text{U}(s),\quad \text{or}\quad \unicode[STIX]{x2202}_{s}\text{U}+\text{i}\mathscr{L}\text{U}=\text{P}\text{i}\mathscr{L}\text{U}.\end{eqnarray}$$

Solution by means of a Green’s function leads to

$$\begin{eqnarray}\text{U}(s)=\text{e}^{-\text{i}\mathscr{L}s}\text{U}(0)+\displaystyle \int _{0}^{s}\!\text{d}\overline{s}\,\text{e}^{-\text{i}\mathscr{L}(s-\overline{s})}\text{P}\text{i}\mathscr{L}\text{U}(\overline{s}).\end{eqnarray}$$

One has $\text{U}(0)=\text{Q}^{2}=\text{Q}$ . Since one knows from the original definition that $\text{P}\text{U}=0$ , it must be the case that applying  $\text{P}$ to the right-hand side of the last result yields 0. That can be checked term by term upon expanding in  $s$ . Thus, the right-hand side can be multiplied by  $\text{Q}$ with no change in value, whereupon one obtains (3.4).

44 Note that the last explicit term of (3.3) vanishes under the integration by parts, as it is independent of  $\overline{\unicode[STIX]{x1D746}}$ .

45 If the expectation were taken in a stationary, homogeneous ensemble, $M_{0}(\unicode[STIX]{x1D707},\unicode[STIX]{x1D707}s_{+}\prime ,\unicode[STIX]{x1D70F})$ would depend on only the spatial and temporal difference variables: $M_{0}(\unicode[STIX]{x1D707},\unicode[STIX]{x1D707}s_{+}\prime ,\unicode[STIX]{x1D70F})\rightarrow M_{sss_{+}\prime }(\boldsymbol{r}-\boldsymbol{r}s_{+}\prime ,\unicode[STIX]{x1D70F})$ . It could then be Fourier transformed with respect to those variables, giving the function $\widehat{M}_{sss_{+}\prime ,\boldsymbol{k}}(\unicode[STIX]{x1D714})$ . For transport theory, the relevant limit is $k\rightarrow 0$ , where $k\doteq \{\boldsymbol{k},\unicode[STIX]{x1D714}\}$ . In fact, however, the expectation is taken in the reference ensemble, which is weakly dependent on  $\{\boldsymbol{r},t\}\equiv r$ . One must then consider a function

$$\begin{eqnarray}M_{0}(r,\overline{r})\equiv M\left(r-\overline{r}\,\left|\,\frac{1}{2}(r+\overline{r})\right.\right)=\displaystyle \int \frac{\text{d}k}{(2\unicode[STIX]{x03C0})^{4}}\displaystyle \int \frac{\text{d}q}{(2\unicode[STIX]{x03C0})^{4}}\,\widehat{M}_{k,q}\text{e}^{\text{i}k(r-\overline{r})}\text{e}^{\text{i}q(r+\overline{r})/2}.\end{eqnarray}$$

Transport coefficients will be given by

$$\begin{eqnarray}\displaystyle \int \text{d}\overline{r}\,M_{0}(r,\overline{r})=\displaystyle \int \frac{\text{d}q}{(2\unicode[STIX]{x03C0})^{4}}\,\widehat{M}_{q/2,q}\text{e}^{\text{i}qr}\approx \widehat{M}(k=0\mid r),\end{eqnarray}$$

the latter following when the microscopic correlation scales are much smaller than the macroscopic hydrodynamic scales.

46 See appendix H for a discussion of one-sided functions.

47 The linearized Vlasov response function lies at the heart of Rostoker’s important test particle superposition principle; see Krommes (Reference Krommes1976 and references therein).

48 It is easy to verify that (3.24) is true by showing that the time derivatives of both sides are equal. The derivative of the left-hand side is $\text{U}(\unicode[STIX]{x1D70F})\text{i}\mathscr{L}A$ , while the derivative of the right-hand side is $\text{e}^{-\text{Q}\text{i}\mathscr{L}\unicode[STIX]{x1D70F}}\text{Q}\text{i}\mathscr{L}A=\text{Q}\text{e}^{-\text{Q}\text{i}\mathscr{L}\unicode[STIX]{x1D70F}}\text{Q}\text{i}\mathscr{L}A=\text{U}(\unicode[STIX]{x1D70F})\text{i}\mathscr{L}A$ . The integration constant is zero because both sides vanish at $\unicode[STIX]{x1D70F}=0$ . An explicit proof follows by performing the time integration:

$$\begin{eqnarray}\displaystyle \displaystyle \int _{0}^{\unicode[STIX]{x1D70F}}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\,\text{Q}\text{e}^{-\text{Q}\text{i}\mathscr{L}\text{Q}\overline{\unicode[STIX]{x1D70F}}}\text{Q}\text{i}\mathscr{L}A & = & \displaystyle \displaystyle \int _{0}^{\unicode[STIX]{x1D70F}}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\,\text{e}^{-\text{Q}\text{i}\mathscr{L}\overline{\unicode[STIX]{x1D70F}}}\text{Q}\text{i}\mathscr{L}A\nonumber\\ \displaystyle & = & \displaystyle -(\text{e}^{-\text{Q}\text{i}\mathscr{L}\unicode[STIX]{x1D70F}}-1)(\text{Q}\text{i}\mathscr{L})^{-1}\text{Q}\text{i}\mathscr{L}A=A-\text{e}^{-\text{Q}\text{i}\mathscr{L}\unicode[STIX]{x1D70F}}A,\nonumber\end{eqnarray}$$

provided that $\text{Q}\mathscr{L}$ is invertible. That can be argued from the properties of  $\text{Q}$ .

49 The identity (3.25) can be verified by comparing the Taylor expansions of the exponentials and using the fact that $\text{Q}^{2}=\text{Q}$ .

50 For the cases involving projections, multiple terms lead to each of the integrals $K_{20}$ $K_{23}$ . For more specific information, please consult the supplementary details in Krommes (Reference Krommes2018c ).

51 For the last line of (4.11), recall that the trace of the product of a symmetric and an antisymmetric matrix vanishes.

52 $\unicode[STIX]{x1D702}$  plays the role of a source in a particular adjoint of the primitive amplitude equation for  $\widetilde{\unicode[STIX]{x1D713}}$ .

53 Martin et al. actually consider a more complicated, time-ordered generating functional that depends on two sources  $\unicode[STIX]{x1D702}$ and  $\widehat{\unicode[STIX]{x1D702}}$ and is capable of generating both correlation functions and infinitesimal response functions.

54 The Landau operator  $\text{L}$ , which describes self-consistent linearized Vlasov dynamics, should not be confused with the Landau collision operator  $\text{C}^{\text{L}}$ . It is trivial to incorporate an external magnetic field  $\boldsymbol{B}^{\text{ext}}$ into the Landau operator by replacing  $\boldsymbol{E}(1)$ by $\boldsymbol{E}(1)+c^{-1}\boldsymbol{v}_{1}\times \boldsymbol{B}^{\text{ext}}$ .

55 Krommes (Reference Krommes1975) and Krommes & Oberman (Reference Krommes and Oberman1976a ) introduced the second independent function  $\unicode[STIX]{x1D702}s_{+}\prime$ to handle the second discrete time  $ts_{+}\prime$ ; the integrals are over just the phase-space coordinates, and one applies to  $S_{2}$ functional derivatives with respect to functions involving only phase-space arguments. This is to be contrasted with a generating functional of the MSR type, $S[\unicode[STIX]{x1D702}]\doteq \langle \exp \boldsymbol{(}\!\int \text{d}\overline{1}\,\widetilde{f}(\overline{1})\unicode[STIX]{x1D702}(\overline{1})\boldsymbol{)}\rangle$ , from which $\langle \,\widetilde{f}(1s_{+}\prime )\rangle$ can be generated from a functional derivative with respect to  $\unicode[STIX]{x1D702}(1s_{+}\prime )$ , the arguments of which refer to both phase space and time. The MSR approach is clearly the more general. Historically, the two-time generating functional was introduced in order to systematize the linearization procedure discussed after (5.19), which was originally noticed by Williams (Reference Williams1973).

56 The review of Krommes (Reference Krommes2002) contains lengthy discussion of the MSR formalism.

57 Here I have redefined the underline notation such that when spatial or wavenumber arguments are displayed explicitly, $\text{}\underline{1}\equiv \{\boldsymbol{v}_{1},s_{1}\}$ . I have also dropped the hats that signify Fourier transforms in order to avoid confusion with the linearized collision operator  $\widehat{\text{C}}$ . The subscripting convention for a function like $C_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime }^{(2;1)}$ is as follows. The comma simply separates the  $+$ that denotes a one-sided function. The wavenumber arguments before the semicolon are conjugate to the equal-time ( $t$ ) spatial differences referred to  $\boldsymbol{x}_{1}$ (e.g. $\unicode[STIX]{x1D746}\rightarrow \boldsymbol{k}$ ); the ones after the semicolon refer to the arguments at  $ts_{+}\prime$ , again referred to  $\boldsymbol{x}_{1}$ (e.g. $\unicode[STIX]{x1D746}s_{+}\prime \rightarrow \boldsymbol{k}s_{+}\prime$ ). If there is only one argument at time  $t$ , there is no equal-time wavenumber and the slot before the semicolon is left blank (e.g. $C_{+;\boldsymbol{k}s_{+}\prime }^{(1;1)}$ ). If all of the arguments are at equal times, the semicolon is omitted; an example is the initial condition $C_{\boldsymbol{k},\boldsymbol{k}s_{+}\prime }^{(0,3)}(\text{}\underline{1},\text{}\underline{2},\text{}\underline{1}s_{+}\prime ,ts_{+}\prime )$ .

58 A further assumption is that the effect of the magnetic field is negligible during a collision (i.e. that $\unicode[STIX]{x1D714}_{\text{c}}/\unicode[STIX]{x1D714}_{\text{p}}\ll 1$ ).

59 As noted on page 10, Catto & Simakov define the two ordering parameters $\unicode[STIX]{x0394}\doteq \unicode[STIX]{x1D706}_{\text{mfp}}\unicode[STIX]{x1D6FB}_{\Vert }$ and $\unicode[STIX]{x1D6FF}\doteq \unicode[STIX]{x1D70C}\,\unicode[STIX]{x1D6FB}_{\bot }$ . They assume that $\unicode[STIX]{x1D6FF}\sim \unicode[STIX]{x0394}\sim \unicode[STIX]{x1D716}$ .

60 This is equivalent to assuming that $\boldsymbol{u}_{\bot }$  is of the order of the diamagnetic flow $\boldsymbol{u}_{\ast }\doteq \widehat{\boldsymbol{b}}\times \unicode[STIX]{x1D735}p/(mn\unicode[STIX]{x1D714}_{\text{c}})$ , since $u_{\ast }/v_{\text{t}}\sim \unicode[STIX]{x1D70C}\,\unicode[STIX]{x1D6FB}_{\bot }\ln T=O(\unicode[STIX]{x1D6FF})$ . It is also equivalent to the ordering $\boldsymbol{u}\sim \boldsymbol{q}/p$ , where $\boldsymbol{q}=\boldsymbol{q}_{\Vert }+\boldsymbol{q}_{\ast }+\boldsymbol{q}_{\bot }$  is the classical heat flow, since one has $\boldsymbol{q}_{\Vert }/p\sim n(v_{\text{t}}^{2}/\unicode[STIX]{x1D708})\unicode[STIX]{x1D6FB}_{\Vert }T/(nT)\sim v_{\text{t}}\unicode[STIX]{x0394}$ and $\boldsymbol{q}_{\bot }/p\sim n(\unicode[STIX]{x1D70C}^{2}\unicode[STIX]{x1D708})\unicode[STIX]{x1D6FB}_{\bot }T/(nT)\sim v_{\text{t}}\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D708}/\unicode[STIX]{x1D714}_{\text{c}})\sim v_{\text{t}}\unicode[STIX]{x1D6FF}$ .

61 Strictly speaking, it is unnecessary to introduce a second-order spatial scale, although it is convenient to do so. However, it is important to distinguish between the zeroth-order kinetic spatial scale  $\unicode[STIX]{x1D706}_{\text{mfp}}$ and the first-order spatial scale  $L$ that characterizes the gradients. One assumes complete relaxation on the zeroth-order scales, which leads to (6.8a ). Transport coefficients are determined in terms of the autocorrelation scale that dictates the characteristic step size in a random walk. That is the mean free path $\unicode[STIX]{x1D706}_{\text{mfp}}\doteq v_{\text{t}}/\unicode[STIX]{x1D708}$ , built from the thermal velocity. Thermal streaming effects enter at first order in the expansion. In (6.8b ), they involve the first-order gradient, which is on the macroscopic scale.

62 A contribution from the derivative of $C_{\mathbf{0},\boldsymbol{k}s_{+}\prime \prime }^{(0,3)}$ with respect to  $\boldsymbol{k}s_{+}\prime \prime$ can be shown to phase-mix away.

63 Obviously, this consistency check is being done here in the limit of weak coupling. However, such cancellations must occur in general if the formalism is to be consistent.

64 The $\boldsymbol{k}s_{+}\prime \prime$ derivative appears here on the right, operating to the left, in order to retain the proper order of the tensor indices. In the subsequent expressions, I shall not bother with that nicety because the tensor can be shown to be symmetric in its last two indices.

65 Catto & Simakov do not explicitly differentiate a formal collision operator; however, they do differentiate the solution of their first-order Chapman–Enskog equation, which is equivalent.

66 Obviously, this statement is true only in the hydrodynamic limit and when the focus is merely on the calculation of transport coefficients. A general two-time correlation function contains much more information than can be obtained from the one-time kinetic equation.

67 I have not done a complete literature survey. For some entry points, see the citations in the quotation by Wong et al. (Reference Wong, MacLennan, Lindenfeld and Dufty1978) reproduced on page 5.

68 Some references in which the stochastic-oscillator model is discussed include Kraichnan (Reference Kraichnan1961), Kraichnan (Reference Kraichnan1964), Krommes (Reference Krommes2002), Krommes & Reiman (Reference Krommes and Reiman2009) and Krommes (Reference Krommes2015).

69 In all of the references cited in the previous footnote, the multiplicatively random coefficient is chosen to be Gaussian. In the present discussion, non-Gaussianity is essential.

70 In the absence of any vector, the only third-rank tensor available from which to construct  $T_{ijk}$ is the Levi-Civita tensor $\unicode[STIX]{x1D716}_{ijk}$ ; however, that is antisymmetric. Useful discussions of symmetry considerations are by Robertson (Reference Robertson1940) and Mathews & Walker (Reference Mathews and Walker1970, § 3–2).

71 A similar ansatz is made in the development of Grad’s 13-moment equations (Grad Reference Grad1958).

72 The form (B 3) agrees with equation (A10) of Brey (Reference Brey1983) since he uses the opposite sign of  $\boldsymbol{k}$ in his definition of the Fourier transform; for example, $\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{ij}\rightarrow -\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{ij}=+\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{ji}$ , which reproduces the exponent in Brey’s term.

73 In principle, the densities in each of the boxes can be macroscopically random if the system is turbulent or if one does coarse graining on the Debye scale. That possibility is ignored in the present work in order to focus on the microscopic fluctuations that give rise to classical transport. It is at this point that one loses effects related to non-local wave transport.

74 The chemical potential is discussed in virtually every introductory book on statistical physics. Its derivation for a monatomic ideal gas often begins with the Sackur–Tetrode entropy formula

$$\begin{eqnarray}\mathscr{S}=k\mathscr{N}\left\{\ln \left[\frac{\mathscr{V}}{\mathscr{N}}\left(\frac{4\unicode[STIX]{x03C0}m\mathscr{E}}{3\mathscr{N}h_{\text{P}}^{2}}\right)^{3/2}\right]+\frac{5}{2}\right\}.\end{eqnarray}$$

One then uses the result $\unicode[STIX]{x1D707}=-T\,\unicode[STIX]{x2202}\mathscr{S}/\unicode[STIX]{x2202}\mathscr{N}$ to find

$$\begin{eqnarray}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}=\ln \left[n\left(\frac{4\unicode[STIX]{x03C0}me}{3h_{\text{P}}^{2}}\right)^{3/2}\right].\end{eqnarray}$$

In the presence of a mean electrostatic potential $\unicode[STIX]{x1D719}(\boldsymbol{r},t)$ , a term  $q_{s}\unicode[STIX]{x1D719}$ must be added to  $\unicode[STIX]{x1D707}$ .

75 Explicitly, as a simple exercise in the use of the projection operator,

$$\begin{eqnarray}\widehat{\boldsymbol{P}}_{s}=\widetilde{\boldsymbol{P}}_{s}-(\langle \widetilde{\boldsymbol{P}}_{s}\rangle _{0}+\widetilde{\boldsymbol{P}}\text{}s_{+}\prime \star \unicode[STIX]{x1D648}_{\boldsymbol{p}\boldsymbol{p}}^{-1}\star \langle \widetilde{\boldsymbol{P}}\text{}s_{+}\prime \,\widetilde{\boldsymbol{P}}_{s}\rangle _{0})=\widetilde{\boldsymbol{P}}_{s}-(\langle \widetilde{\boldsymbol{P}}\rangle _{s}+\widetilde{\boldsymbol{P}}\text{}s_{+}\prime _{s})=\mathbf{0}.\end{eqnarray}$$

76 Symmetry considerations lead to

$$\begin{eqnarray}\displaystyle \boldsymbol{J}_{s}\text{}^{e}(\boldsymbol{r}) & = & \displaystyle \displaystyle \int \frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\,\text{e}^{\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{r}}\nonumber\\ \displaystyle & & \displaystyle \times \,\left\langle \vphantom{\left(\frac{1}{2}\right)}\right.\!\!\mathop{\sum }_{i\in s}\left(\frac{1}{2}m_{s}|\boldsymbol{w}_{i}+\boldsymbol{u}_{s}|^{2}(\boldsymbol{w}_{i}+\boldsymbol{u}_{s})+\widetilde{U}_{i}(\boldsymbol{w}_{i}+\boldsymbol{u}_{s})+\unicode[STIX]{x0394}\widetilde{\unicode[STIX]{x1D70F}}_{i}(\boldsymbol{k})\boldsymbol{\cdot }(\boldsymbol{w}_{i}+\boldsymbol{u}_{s})\right)\text{e}^{-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{i}}\!\!\left.\vphantom{\left(\frac{1}{2}\right)}\right\rangle _{0}\nonumber\\ \displaystyle & = & \displaystyle \left[\vphantom{\left(\frac{1}{2}\right)}\right.\!\underbrace{\left(\frac{1}{2}mnu^{2}+\frac{3}{2}nT+u\right)}_{e}+\underbrace{(nT+\unicode[STIX]{x0394}p)}_{p}\!\left.\vphantom{\left(\frac{1}{2}\right)}\right]_{s}\boldsymbol{u}_{s}=h_{s}\boldsymbol{u}_{s}.\nonumber\end{eqnarray}$$

77 One uses the facts that (i) there are no velocity correlations in the reference ensemble, and (ii)  $\langle w^{2n}\rangle =(2n+1)!!\,v_{\text{t}}^{2n}$ .

78 Both  $f$ and  $g$ are to be calculated in the reference ensemble, so they should be adorned with the subscript 0; however, I shall drop that in the remainder of this appendix.

79 Several different approaches to the calculation of the thermal-equilibrium pair correlation function are described by Montgomery & Tidman (Reference Montgomery and Tidman1964). One route is worked out in § G.1.3.

80 A historically important monograph on plasma kinetic theory that contains closely related manipulations is by Montgomery & Tidman (Reference Montgomery and Tidman1964). See also the modern introduction to that subject by Swanson (Reference Swanson2008).

81 The contribution from the initial condition can be shown to phase-mix away.

82 As a partial sign check, note that in general one has $g_{\boldsymbol{k}}(1,2)=g_{\boldsymbol{k}}^{\ast }(2,1)$ . That symmetry is obeyed by (G 15). (Note that $\unicode[STIX]{x1D750}_{\boldsymbol{k}}^{\ast }=-\unicode[STIX]{x1D750}_{\boldsymbol{k}}$ .)

83 This amounts to an application of the Kramers–Kronig relations; see the discussion by Ichimaru (Reference Ichimaru1973, appendix  C).

84 This trick is commonly used in discussions of the fluctuation–dissipation theorem (see, for example, Birdsall & Langdon (Reference Birdsall and Langdon1985, § 12-3)), where one evaluates the integral $\text{Im}\int \!\text{d}\unicode[STIX]{x1D714}\,[\unicode[STIX]{x1D714}\mathscr{D}(\unicode[STIX]{x1D714})]^{-1}$ . In that case, there is a contribution from the arc at  $\infty$ because $\mathscr{D}(\infty )=1$ . There is no such contribution in (G 18) because of the additional factors of $r_{\unicode[STIX]{x1D714}}r_{\unicode[STIX]{x1D714}}^{\ast }\sim \unicode[STIX]{x1D714}^{-2}$ .

85 One has $r_{0}(1)r_{0}^{\ast }(2)=[k^{2}(\widehat{\boldsymbol{k}}\boldsymbol{\cdot }\boldsymbol{v}_{1})(\widehat{\boldsymbol{k}}\boldsymbol{\cdot }\boldsymbol{v}_{2})]^{-1}$ . The velocities cancel with those arising from $(\unicode[STIX]{x2202}f_{\text{M}})_{1}(\unicode[STIX]{x2202}f_{\text{M}})_{2}$ .

86 As a sign check, note that $g_{k}^{(\text{eq})}<0$ for like-signed particles, signifying repulsion. Note that formula (G 19) holds only for Debye-length scales.

87 Kent & Taylor (Reference Kent and Taylor1969) discuss the generalization of this formula to include the convective amplification of locally unstable modes.

88 The wavenumber integration is logarithmically divergent at large  $k$ (small impact parameter) because large angle scattering is not treated correctly. At low temperatures, that can be rectified by asymptotically matching to the Boltzmann operator (Frieman & Book Reference Frieman and Book1963) or, more commonly, by inserting a cutoff at $k_{\text{max}}=b_{0}^{-1}$ , where $b_{0,s\overline{s}}\doteq q_{s}q_{\overline{s}}/T$ . ( $b_{0}$  is the impact parameter for $90^{\circ }$  scattering; the distance of closest approach is  $2b_{0}$ . In a more precise definition, $T$  is replaced by $\unicode[STIX]{x1D707}v_{\text{rel}}^{2}$ , where $\unicode[STIX]{x1D707}$  is the reduced mass and $v_{\text{rel}}$  is the relative velocity.) The Landau operator is accurate only to terms of $O(1)$ relative to $\ln \unicode[STIX]{x1D6EC}$ .

When the de Broglie wavelength  $\unicode[STIX]{x1D706}_{\text{B},s\overline{s}}\doteq h_{\text{P}}/(\unicode[STIX]{x1D707}_{s\overline{s}}|\boldsymbol{v}-\overline{\boldsymbol{v}}|)$ is greater than  $b_{0}$ (which is true for sufficiently high temperatures), $b_{0}$  must be replaced by  $\unicode[STIX]{x1D706}_{\text{B}}$ . The argument is summarized, and original references are given, by Krommes (Reference Krommes2018a ).

89 When quoting this and similar results, I include an explicit plus or minus sign so that one can quickly check against the expected terms in the linearized Balescu–Lenard operator (G 26). Those are

$$\begin{eqnarray}-2\text{Re}\,\left(\frac{\unicode[STIX]{x0394}\mathscr{D}}{\mathscr{D}}\right)(f\,\overline{\unicode[STIX]{x2202}f}-\unicode[STIX]{x2202}f\,\overline{f})+f\,\overline{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}}-\unicode[STIX]{x2202}f\,\overline{\unicode[STIX]{x1D6E4}}+\overline{\unicode[STIX]{x2202}f}\,\unicode[STIX]{x1D6E4}-\overline{f}\,\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}.\end{eqnarray}$$

When eyeballing signs, it is helpful to remember that $\unicode[STIX]{x1D716}\,\unicode[STIX]{x1D716}^{\ast }>0$ and $\text{Re}\,r>0$ .

90 The noise function  $F$ used here is unrelated to the $\mathscr{N}$ -particle Liouville distribution.

91 For turbulence, that is indeed true for  $F$ ; the nonlinear contribution to  $\unicode[STIX]{x1D6F4}$ is not always positive as a function of wavenumber, but it is typically so.

References

Alder, B. J. & Wainwright, T. E. 1970 Decay of the velocity autocorrelation function. Phys. Rev. A 1, 1821.Google Scholar
Balescu, R. 1975 Equilibrium and Nonequilibrium Statistical Mechanics. Wiley (reprinted by Krieger Publishing Co., 1991).Google Scholar
Birdsall, C. K. & Langdon, A. B. 1985 Plasma Physics via Computer Simulation. McGraw-Hill.Google Scholar
Braginskii, S. I. 1965 Transport processes in a plasma. In Reviews of Plasma Physics (ed. Leontovich, M. N.), vol. 1, pp. 205311. Consultants Bureau.Google Scholar
Brey, J. J. 1983 Long time behavior of the Burnett transport coefficients. J. Chem. Phys. 79, 45854598.Google Scholar
Brey, J. J., Zwanzig, R. & Dorfman, J. R. 1981 Nonlinear transport equations in statistical mechanics. Physica A 109, 425444.Google Scholar
Burnett, D. 1935 The distribution of velocities in a slightly non-uniform gas. Proc. Lond. Math. Soc. 39, 385430.Google Scholar
Burnett, D. 1936 The distribution of molecular velocities and the mean motion in a non-uniform gas. Proc. Lond. Math. Soc. 40, 382435.Google Scholar
Catto, P. J. & Simakov, A. N. 2004 A drift ordered short mean free path description for magnetized plasma allowing strong spatial anisotropy. Phys. Plasmas 11, 90102.Google Scholar
Catto, P. J. & Simakov, A. N. 2005 A new, explicitly collisional contribution to the gyroviscosity and the radial electric field in a collisional tokamak. Phys. Plasmas 12, 114503 (4 pages); 13, 129901(E).Google Scholar
Dawson, J. & Nakayama, T. 1967 Derivations of hierarchies for $N$ -particle systems and Vlasov systems by means of the functional calculus. J. Math. Phys. 8, 553560.Google Scholar
Dyson, F. J. 1949 The radiation theories of Tomonaga, Schwinger, and Feynman. Phys. Rev. 75, 486502.Google Scholar
Ernst, M. H., Cichocki, B., Dorfman, J. R., Sharma, J. & van Beijeren, H. 1978 Kinetic theory of nonlinear viscous flow in two and three dimensions. J. Stat. Phys. 18, 237270.Google Scholar
Ernst, M. H. & Dorfman, J. R. 1972 Nonanalytic dispersion relations in classical fluids: I. The hard-sphere gas. Physica (Utrecht) 61, 157181.Google Scholar
Frieman, E. A. & Book, D. L. 1963 Convergent classical kinetic equation for a plasma. Phys. Fluids 6, 17001706.Google Scholar
García-Colín, L. S., Velasco, R. M. & Uribe, F. J. 2008 Beyond the Navier–Stokes equations: Burnett hydrodynamics. Phys. Rep. 465, 149189.Google Scholar
Gorban, A. N. & Karlin, I. 2014 Hilbert’s 6th problem: exact and approximate hydroynamic manifold for kinetic equations. Bull. Am. Math. Soc. 51 (2), 186246.Google Scholar
Grad, H. 1958 Principles of the kinetic theory of gases. In Handbuch der Physik, vol. XII, pp. 205294. Springer.Google Scholar
Ichimaru, S. 1973 Basic Principles of Plasma Physics – A Statistical Approach. Benjamin.Google Scholar
Kawasaki, K. & Gunton, J. D. 1973 Theory of nonlinear transport processes: nonlinear shear viscosity and normal stress effects. Phys. Rev. A 8, 20482064.Google Scholar
Kent, A. & Taylor, J. B. 1969 Plasma fluctuations and convective modes. Phys. Fluids 12, 209212.Google Scholar
Klimontovich, Y. L. 1967 The Statistical Theory of Non-equilibrium Processes in a Plasma (ed. ter Harr, D.). MIT Press (translated by H. S. H. Massey and O. M. Blunn).Google Scholar
Kraichnan, R. H. 1961 Dynamics of nonlinear stochastic systems. J. Math. Phys. 2, 124148; 3, 205(E).Google Scholar
Kraichnan, R. H. 1962 The closure problem of turbulence theory. Proc. Symp. Appl. Math. 13, 199225.Google Scholar
Kraichnan, R. H. 1964 Kolmogorov’s hypotheses and Eulerian turbulence theory. Phys. Fluids 7, 17231734.Google Scholar
Krommes, J. A.1975 On renormalized kinetic theories of anomalous transport due to hydrodynamic fluctuations in strongly magnetized plasma. PhD thesis, Princeton University.Google Scholar
Krommes, J. A. 1976 Two new proofs of the test particle superposition principle of plasma kinetic theory. Phys. Fluids 19, 649655.Google Scholar
Krommes, J. A. 1984 Statistical descriptions and plasma physics. In Handbook of Plasma Physics (ed. Galeev, A. A. & Sudan, R. N.), vol. 2, chap. 5.5, pp. 183268. North-Holland.Google Scholar
Krommes, J. A. 2002 Fundamental statistical theories of plasma turbulence in magnetic fields. Phys. Rep. 360, 1352.Google Scholar
Krommes, J. A. 2007 Nonequilibrium gyrokinetic fluctuation theory and sampling noise in gyrokinetic particle-in-cell simulations. Phys. Plasmas 14, 090501 (26 pages).Google Scholar
Krommes, J. A. 2015 A tutorial introduction to the statistical theory of turbulent plasmas, a half century after Kadomtsev’s Plasma Turbulence and the resonance-broadening theory of Dupree and Weinstock. J. Plasma Phys. 81, 205810601 (80 pages).Google Scholar
Krommes, J. A.2018a An introduction to the physics of the Coulomb logarithm, with emphasis on quantum-mechanical effects. J. Plasma Phys., arXiv:1806.04990 (8 pages).Google Scholar
Krommes, J. A. 2018b Projection-operator methods for classical transport in magnetized plasmas. Part 1. Linear response, the Braginskii equations and fluctuating hydrodynamics. J. Plasma Phys. 84, 925840401 (80 pages).Google Scholar
Krommes, J. A. 2018c Projection-operator methods for classical transport in magnetized plasmas. II. Nonlinear response and the Burnett equations – Supplement. J. Plasma Phys. doi:10.1017/S0022377818000892.Google Scholar
Krommes, J. A. & Oberman, C. 1976a Anomalous transport due to long-lived fluctuations in plasma. Part I. A general formalism for two-time fluctuations. J. Plasma Phys. 16, 193227.Google Scholar
Krommes, J. A. & Oberman, C. 1976b Anomalous transport due to long-lived fluctuations in plasma. Part II. Hydrodynamic contributions to transport in two-dimensional, strongly magnetized systems. J. Plasma Phys. 16, 229260.Google Scholar
Krommes, J. A. & Parker, J. B. 2018 Statistical closures and zonal flows. In Zonal Jets (ed. Galperin, B. & Read, P.), chap. V.1.1. Cambridge University Press (in press).Google Scholar
Krommes, J. A. & Reiman, A. H. 2009 Plasma equilibrium in a magnetic field with stochastic regions. Phys. Plasmas 16, 072308 (26 pages).Google Scholar
Kubo, R. 1957 Statistical–mechanical theory of irreversible processses. I. General theory and simple applications to magnetic and conduction problems. J. Phys. Soc. Japan 12, 570586.Google Scholar
Kubo, R. 1959 Some aspects of the statistical–mechanical theory of irreversible processes. In Lectures in Theoretical Physics (ed. Brittin, W. E. & Dunham, L. G.), vol. I, pp. 120203. Interscience.Google Scholar
Kubo, R. 1974 Response, relaxation, and fluctuations. In Lecture Notes in Physics: Transport Phenomena (ed. Kirczenow, G. & Marro, J.), vol. 31, pp. 24124. Springer.Google Scholar
Leslie, D. C. 1973 Developments in the Theory of Turbulence. Clarendon.Google Scholar
Martin, P. C., Siggia, E. D. & Rose, H. A. 1973 Statistical dynamics of classical systems. Phys. Rev. A 8, 423437.Google Scholar
Mathews, J. & Walker, R. L. 1970 Mathematical Methods of Physics. Benjamin.Google Scholar
McComb, W. D. 1990 The Physics of Fluid Turbulence. Clarendon.Google Scholar
McComb, W. D. 2014 Homogeneous, Isotropic Turbulence. Oxford University Press.Google Scholar
Mikhaǐlovskiǐ, A. B. 1967 Macroscropic description of a collision[al] plasma in a strong magnetic field for stability problems. Zh. Eksp. Teor. Fiz. 52, 943954; [Sov. Phys. JETP 25 623–630 (1967)].Google Scholar
Mikhaǐlovskiǐ, A. B. & Tsypin, V. S. 1971 Transport equations and gradient instabilities in a high pressure collisional plasma. Plasma Phys. 13, 785798.Google Scholar
Mikhaǐlovskiǐ, A. B. & Tsypin, V. S. 1984 Transport equations of plasma in a curvilinear magnetic field. Beitr. Plasmaphys. 24, 335354.Google Scholar
Montgomery, D. C. & Tidman, D. A. 1964 Plasma Kinetic Theory. McGraw-Hill.Google Scholar
Mori, H. 1965 Transport, collective motion, and Brownian motion. Progr. Theoret. Phys. 33, 423455.Google Scholar
Nemerov, H.1967 Projection. The Atlantic Monthly 219 (5), 87.Google Scholar
Piccirelli, R. A. 1968 Theory of the dynamics of simple fluids for large spatial gradients and long memory. Phys. Rev. 175, 7798.Google Scholar
Reichl, L. E. 1998 A Modern Course in Statistical Physics, 2nd edn. Wiley (The section on ‘Beyond the Boltzmann Equation’ was radically shortened in the third and fourth editions).Google Scholar
Robertson, H. P. 1940 The invariant theory of isotropic turbulence. Math. Proc. Camb. Phil. Soc. 36, 209223.Google Scholar
Rose, H. A. 1979 Renormalized kinetic theory of nonequilibrium many-particle classical systems. J. Stat. Phys. 20, 415447.Google Scholar
Rosenbluth, M. N. & Liu, C. S. 1976 Cross-field energy transport by plasma waves. Phys. Fluids 19, 815818.Google Scholar
Rostoker, N. & Rosenbluth, M. N. 1960 Test particles in a completely ionized plasma. Phys. Fluids 3, 114.Google Scholar
Rudnick, J. & Gaspari, G. 2004 Elements of the Random Walk: An Introduction for Advanced Students and Researchers. Cambridge University Press.Google Scholar
Struchtrup, H. 2005 Failures of the Burnett and super-Burnett equations in steady state processes. Contin. Mech. Thermodyn. 17, 4350.Google Scholar
Struchtrup, H. & Torrolhon, M. 2003 Regularization of Grad’s 13 moment equations: derivation and linear analysis. Phys. Fluids 15, 26682680.Google Scholar
Swanson, D. G. 2008 Plasma Kinetic Theory. CRC Press.Google Scholar
Taylor, G. I. 1921 Diffusion by continuous movements. Proc. Lond. Math. Soc. (2) 20, 196212; reprinted in Turbulence: Classic Papers on Statistical Theory (ed. S. K. Friedlander and L. Topper) (Interscience, 1961), pp. 1–17.Google Scholar
Wang, M. C. & Uhlenbeck, G. E. 1945 On the theory of the Brownian motion II. Rev. Mod. Phys. 17, 323342; reprinted in Selected Papers on Noise and Stochastic Processes (ed. N. Wax) (Dover, 1954), pp. 113–132.Google Scholar
Williams, E. A.1973 On the theory of fluctuations in plasma. PhD thesis, Princeton University.Google Scholar
Wong, C. K., MacLennan, J. A., Lindenfeld, M. & Dufty, J. W. 1978 Theory of nonlinear transport in Burnett order. J. Chem. Phys. 68, 15631753.Google Scholar
Zwanzig, R. 2001 Nonequilibrium Statistical Mechanics. Oxford University Press.Google Scholar
Figure 0

Figure 1. One mechanism leading to a Burnett contribution to the momentum flux that is proportional to $|\unicode[STIX]{x1D6FB}T|^{2}$. The thick solid line depicts a temperature profile with constant gradient. The net second-order momentum flux across the $z=0$ plane arises from the unbalanced portion of the first-order viscous forces exerted on the velocity streams arriving from a mean free path $\unicode[STIX]{x1D706}_{\text{mfp}}$ away. See the text for further discussion.

Figure 1

Figure 2. Contour of integration used for evaluating the term involving $\mathscr{D}^{-1}$ in (G 18).

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