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 ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}\widetilde{E} & = & \displaystyle -\unicode[STIX]{x1D735}\boldsymbol{\cdot }\widetilde{\boldsymbol{J}}\text{}^{E},\end{eqnarray}$$

$\widetilde{\unicode[STIX]{x1D749}}$

$\widetilde{\boldsymbol{J}}\text{}^{E}$

(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 ) $$\begin{eqnarray}\displaystyle \widetilde{\boldsymbol{J}}\text{}^{E}(\boldsymbol{k}) & \doteq & \displaystyle \mathop{\sum }_{i=1}^{\mathscr{N}}[E_{i}\boldsymbol{v}_{i}+\unicode[STIX]{x0394}\widetilde{\unicode[STIX]{x1D749}}_{i}(\boldsymbol{k})\boldsymbol{\cdot }\boldsymbol{v}_{i}]\text{e}^{-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{i}},\end{eqnarray}$$

(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 ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}\widetilde{E}_{s}(\boldsymbol{r},t) & = & \displaystyle -\unicode[STIX]{x1D735}\boldsymbol{\cdot }\widetilde{\boldsymbol{J}}\text{}_{s}^{E}+\dot{\widetilde{E}}_{\unicode[STIX]{x0394},s},\end{eqnarray}$$

$\boldsymbol{E}(\boldsymbol{r},t)$

$\widehat{\boldsymbol{b}}$

$\unicode[STIX]{x1D714}_{\text{c}s}\doteq (qB/mc)_{s}$

$\widetilde{\unicode[STIX]{x1D749}}_{s}^{\prime }$

$\dot{\widetilde{\boldsymbol{P}}}_{\unicode[STIX]{x0394},s}\text{}s_{+}\prime$

$\dot{\widetilde{E}}_{\unicode[STIX]{x0394},s}$

$\unicode[STIX]{x0394}$

$\unicode[STIX]{x0394}\boldsymbol{u}$

$\unicode[STIX]{x0394}T$

$\unicode[STIX]{x0394}\boldsymbol{u}\,\unicode[STIX]{x0394}\boldsymbol{u}$

$\unicode[STIX]{x0394}\boldsymbol{u}\,\unicode[STIX]{x0394}T$

$\unicode[STIX]{x0394}T^{2}$

$\unicode[STIX]{x0394}\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}T$

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 ) $$\begin{eqnarray}\displaystyle \langle \widetilde{\boldsymbol{A}}\text{}s_{+}\prime _{s,\boldsymbol{B}}(\boldsymbol{r})\widetilde{\boldsymbol{A}}\text{}s_{+}\prime _{s^{\prime },\boldsymbol{B}}(\boldsymbol{r}s_{+}\prime )\rangle _{\boldsymbol{B}} & = & \displaystyle \frac{\unicode[STIX]{x1D6FF}^{2}\ln Z_{\boldsymbol{ B}}}{\unicode[STIX]{x1D6FF}\boldsymbol{B}_{s}(\boldsymbol{r})\unicode[STIX]{x1D6FF}\boldsymbol{B}_{ss_{+}\prime }(\boldsymbol{r}s_{+}\prime )}=\frac{\unicode[STIX]{x1D6FF}\langle \widetilde{\boldsymbol{A}}_{s}(\boldsymbol{r})\rangle _{\boldsymbol{B}}}{\unicode[STIX]{x1D6FF}\boldsymbol{B}_{ss_{+}\prime }(\boldsymbol{r}s_{+}\prime )},\end{eqnarray}$$

$\widetilde{\boldsymbol{A}}\text{}s_{+}\prime _{s,\boldsymbol{B}}\doteq \widetilde{\boldsymbol{A}}_{s}-\langle \widetilde{\boldsymbol{A}}_{s}\rangle _{\boldsymbol{B}}$

$\mathscr{N}$

$F(\unicode[STIX]{x1D6E4},t)$

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 ) $$\begin{eqnarray}\displaystyle \left(\frac{\unicode[STIX]{x2202}\ln Z_{0}}{\unicode[STIX]{x2202}[-\unicode[STIX]{x1D6FD}_{s}(\boldsymbol{r},t)]}\right)_{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}} & = & \displaystyle \langle \widetilde{\mathscr{E}}_{0}\rangle =\mathscr{V}\left(e_{0s}(\boldsymbol{r},t)-\frac{1}{2}m_{s}\overline{n}_{s}u_{s}^{2}(\boldsymbol{r},t)\right).\end{eqnarray}$$

$\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FD}|_{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}}\equiv \unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FD}$

$\boldsymbol{u}$

$\overline{n}$

$\boldsymbol{r}$

$t$

$e_{0}$

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:

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 ) $$\begin{eqnarray}\displaystyle & = & \displaystyle -\mathop{\sum }_{\overline{s}}\dot{\pmb{\pmb{\mathscr{A}}}}_{\unicode[STIX]{x0394},\overline{s}}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{B}_{\overline{s}s}(\boldsymbol{r},t).\end{eqnarray}$$

$\int \text{d}\boldsymbol{r}\,(nq)_{s}\boldsymbol{E}=\mathbf{0}$

(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 ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{s}\text{Q}W+\text{Q}\text{i}\mathscr{L}\text{Q}W+\text{Q}\text{i}\mathscr{L}\text{P}W & = & \displaystyle \text{Q}S_{\unicode[STIX]{x0394}}.\end{eqnarray}$$

(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.

3.1 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 ) $$\begin{eqnarray}\displaystyle & {\approx} & \displaystyle -\displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}\,\boldsymbol{K}_{G}^{\overline{\unicode[STIX]{x1D6FD}}}(\unicode[STIX]{x1D707},\overline{\unicode[STIX]{x1D707}},\overline{\unicode[STIX]{x1D70F}})\boldsymbol{\cdot }\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t)\nonumber\\ \displaystyle & & \displaystyle -\,\displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}\,\boldsymbol{K}_{G}^{\overline{\unicode[STIX]{x1D6FD}}}(\unicode[STIX]{x1D707},\overline{\unicode[STIX]{x1D707}},\overline{\unicode[STIX]{x1D70F}})\overline{\unicode[STIX]{x1D70F}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}[\unicode[STIX]{x2202}_{t}B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t)]^{(1)}\nonumber\\ \displaystyle & & \displaystyle +\,\displaystyle \int _{0}^{\infty }\text{d}\overline{\unicode[STIX]{x1D70F}}\displaystyle \int \text{d}\overline{\unicode[STIX]{x1D707}}\,\boldsymbol{K}_{G}^{\overline{\unicode[STIX]{x1D6FD}}}(\unicode[STIX]{x1D707},\overline{\unicode[STIX]{x1D707}},\overline{\unicode[STIX]{x1D70F}})\overline{\unicode[STIX]{x1D746}}\boldsymbol{ : }\unicode[STIX]{x1D735}\unicode[STIX]{x1D735}B_{\overline{\unicode[STIX]{x1D6FD}}}(\boldsymbol{r},t),\end{eqnarray}$$

$\boldsymbol{K}=_{\text{a}}\boldsymbol{K}+_{\text{b}}\boldsymbol{K}$

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

3.3.2 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 $\text{B}_{e}^{\boldsymbol{p}}$ .

4.1 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 ) $$\begin{eqnarray}\displaystyle C_{+}^{(2;1)}(\text{}\underline{1},\text{}\underline{2};\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ) & = & \displaystyle C_{+}^{(2;1)}(\unicode[STIX]{x1D746},\boldsymbol{v}_{1},\boldsymbol{v}_{2},\unicode[STIX]{x1D70F};\unicode[STIX]{x1D746}s_{+}\prime ,\boldsymbol{v}s_{+}\prime )\nonumber\\ \displaystyle & = & \displaystyle \displaystyle \int \frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\displaystyle \int \frac{\text{d}\boldsymbol{k}s_{+}\prime }{(2\unicode[STIX]{x03C0})^{3}}\,\text{e}^{\text{i}\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x1D746}+\text{i}\boldsymbol{k}s_{+}\prime \boldsymbol{\cdot }\unicode[STIX]{x1D746}s_{+}\prime }\widehat{\text{C}}_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime }^{(2;1)}(\boldsymbol{v}_{1},\boldsymbol{v}_{2},\unicode[STIX]{x1D70F};\boldsymbol{v}s_{+}\prime ).\end{eqnarray}$$

$\widehat{\pmb{\pmb{\mathscr{J}}}}^{E}$

$\boldsymbol{x}-\boldsymbol{x}s_{+}\prime$

$\widehat{\text{C}}_{\boldsymbol{k}s_{+}\prime =\mathbf{0}}$

$\overline{\boldsymbol{x}}$

$\overline{\boldsymbol{x}}$

$\boldsymbol{x}s_{+}\prime \prime$

(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 ) $$\begin{eqnarray}\displaystyle & & \displaystyle \quad =\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}(-\text{i}\boldsymbol{k}s_{+}\prime \prime )}\widehat{\text{C}}_{+;\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\boldsymbol{v},\unicode[STIX]{x1D70F};\boldsymbol{v}s_{+}\prime ,\boldsymbol{v}s_{+}\prime \prime )|_{\boldsymbol{k}s_{+}\prime =\mathbf{0},\,\boldsymbol{k}s_{+}\prime \prime =\mathbf{0}}.\end{eqnarray}$$

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.

5.1 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 ) $$\begin{eqnarray}\displaystyle & = & \displaystyle -\displaystyle \int \text{d}\widehat{\boldsymbol{q}}\,\left\langle \left[\widehat{\boldsymbol{v}}\boldsymbol{\cdot }\widehat{\unicode[STIX]{x1D735}}\widetilde{f}(\widehat{\boldsymbol{q}},t)+\left(\frac{q}{m}\right)_{\widehat{s}}\pmb{\mathbb{E}}\widetilde{f}(\widehat{\boldsymbol{q}},t)\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}\widetilde{f}}{\unicode[STIX]{x2202}\widehat{\boldsymbol{v}}}\right]\text{e}^{\widetilde{\unicode[STIX]{x1D6F7}}_{\unicode[STIX]{x1D702}}}\right\rangle \unicode[STIX]{x1D702}(\widehat{\boldsymbol{q}}).\end{eqnarray}$$

(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}$$

$\unicode[STIX]{x1D750}_{ij}$

$O(\unicode[STIX]{x1D716}_{\text{p}})$

$h$

$k$

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 ) $$\begin{eqnarray}\displaystyle & & \displaystyle \quad =\displaystyle \int \frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\displaystyle \int \frac{\text{d}\boldsymbol{k}s_{+}\prime }{(2\unicode[STIX]{x03C0})^{3}}\displaystyle \int \frac{\text{d}\boldsymbol{k}s_{+}\prime \prime }{(2\unicode[STIX]{x03C0})^{3}}\,\text{e}^{\text{i}\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x1D746}+\text{i}\boldsymbol{k}s_{+}\prime \boldsymbol{\cdot }\unicode[STIX]{x1D746}s_{+}\prime +\text{i}\boldsymbol{k}s_{+}\prime \prime \boldsymbol{\cdot }\unicode[STIX]{x1D746}s_{+}\prime \prime }C_{+,\boldsymbol{k};\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(2;2)}(\text{}\underline{1},\text{}\underline{2},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime ).\qquad\end{eqnarray}$$

$\text{i}\boldsymbol{\unicode[STIX]{x2202}}_{\boldsymbol{k}s_{+}\prime \prime }C_{+;\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\text{}\underline{1},\unicode[STIX]{x1D70F};\text{}\underline{1}s_{+}\prime ,\text{}\underline{1}s_{+}\prime \prime )|_{\boldsymbol{k}s_{+}\prime =\mathbf{0},\,\boldsymbol{k}s_{+}\prime \prime =\mathbf{0}}$

$C_{+;\boldsymbol{k}s_{+}\prime ,\boldsymbol{k}s_{+}\prime \prime }^{(1;2)}(\unicode[STIX]{x1D70F})$

(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).

6.1 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 ) $$\begin{eqnarray}\displaystyle \text{i}\widehat{\text{M}}f_{2}+\widehat{\text{C}}f_{2} & = & \displaystyle \boldsymbol{w}\boldsymbol{\cdot }\unicode[STIX]{x1D735}f_{1}+(mn)^{-1}\unicode[STIX]{x1D735}p\boldsymbol{\cdot }\unicode[STIX]{x2202}_{\boldsymbol{w}}f_{1}\nonumber\\ \displaystyle & & \displaystyle +\,[\unicode[STIX]{x2202}_{t}f_{0}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}f_{0}-\boldsymbol{w}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{u})\boldsymbol{\cdot }\unicode[STIX]{x2202}_{\boldsymbol{w}}f_{0}]-\text{C}[\,f_{1},\overline{f}_{1}].\end{eqnarray}$$

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

$\unicode[STIX]{x1D735}\equiv \unicode[STIX]{x1D735}_{1}$

$t\equiv t_{1}$

$\unicode[STIX]{x2202}_{t_{1}}f_{0}$

$\text{D}f_{0}/\text{D}t$

$\unicode[STIX]{x2202}_{t_{1}}f_{1}$

$t$

$\unicode[STIX]{x2202}_{t}\,f_{0}$

$t\equiv t_{2}$

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