4.1. Single-Particle Case: Thermodynamics

Hemmo and Shenker first consider von Neumann’s argument in the single-particle case (see fig. 2). They claim that the argument does not go through here, since S _TD actually remains constant, contrary to our thought experiment’s description. In other words, using thermodynamical considerations, they find that S _VN should not be included in our accounting for S _TD.

Figure 2. Top to bottom, stages 1–7 for the single-particle case, as described by Hemmo and Shenker (Reference Hemmo and Shenker2006).

Here is their argument. Consider the stages where there are entropic changes. In stage 2 when the spin measurement was performed, S _VN increases as before, since it tracks the change of the particle’s spin state from pure to mixed.

Contrariwise, S _TD does not change in stage 5 (isothermal quasi-static compression) or anywhere else (this will be important later). After stage 2, the single particle is in either the $|{\psi }_z^{\uparrow }〉$ state or the $|{\psi }_z^{\downarrow }〉$ state. After stages 3 and 4, with the expansion and separation via a semipermeable wall, there is a particle in one side of the box only and not the other. We make an S _TD-conserving location measurement to figure out which side of the box is empty and which side the particle is at, so as to compress the box against the empty side.Footnote ¹¹ The compression is then performed as before. However, this compression does not decrease S _TD.Footnote ¹² To restore the volume of the ‘gas’ to V, no work needs to be done, since we are compressing against a vacuum. Since there is a change in S _VN in this cycle, but no change in S _TD, the apparent answer, in order to do our entropic accounting, is to ignore, not incorporate, S _VN into S _TD. Hence, S _VN does not correspond to S _TD.

Their analysis is problematic. Although their ultimate point in this analysis—that S _TD fails to corresponds to S _VN—still holds, it does not hold in the way they claim. In fact, the way it fails suggests to us that we should disregard the single-particle case.

For the single-particle case, they claim that “[S _TD] is null throughout the experiment” (Hemmo and Shenker Reference Hemmo and Shenker2006, 162). This then allows them to claim that thermodynamic accounting for S _TD is consistent only if we did not consider S _VN. This then supports their claim that S _VN does not correspond to S _TD since adding S _VN into the thermodynamic accounting actually renders the otherwise consistent calculations inconsistent.

They are right to say that the stage 5 compression (after location measurement) has no thermodynamic effect because we are compressing against a vacuum: no work needs to be done, and so $\Delta S_{\text{TD}}=0$ for stage 5. However, I claim that $\Delta S_{\text{TD}}\ne 0$ for the single-particle case overall, because $\Delta S_{\text{TD}}\ne 0$ in stages 3 and 4 in this context.

As far as I can tell, Hemmo and Shenker did not analyze stages 3 and 4 (i.e., the isothermal expansion and separation) in terms of the single-particle case at all. Rather, they seem to have assumed that $\Delta S_{\text{TD}}=0$ in these stages as with the original case of the macroscopic gas.Footnote ¹³ However, this assumes that there is both a change in entropy of n.R.log 2 because of isothermal expansion and a change in the entropy of mixing of −n.R.log 2 because of separation, as they say so themselves for the original case: “The increase of thermodynamic entropy due to the volume increase $\Delta S=(1/T)\int P\mathrm{dV}$ is exactly compensated by the decrease of thermodynamic mixing entropy $\Delta S=\sum w_k\ln w_k$ (where w_k is the relative frequency of molecules of type k) due to the separation” (Hemmo and Shenker Reference Hemmo and Shenker2006, 157 n. 4, emphasis mine).

In the single-particle case, it makes sense that isothermal expansion should still increase S _TD, since the single-particle ‘gas’ is expanding against a piston and doing work. However, it does not make physical sense to speak of the entropy of mixing here at all, since there is no separation of gases in the single-particle case. The entropy of mixing is explicitly defined for systems where different gases are separated from/mixed with one another via semipermeable walls, but a single particle cannot be separated from/mixed with itself. The quote above makes this conceptual point explicit: by Hemmo and Shenker’s own lights, the relative frequency of a single particle is simply unity (and null for particles of other types), so the entropy of mixing is $1\ln 1=0$. There is no thermodynamic entropy of mixing in the single-particle case.

Discounting the entropy of mixing, however, we find that $\Delta S_{\text{TD}}=n.R.\log 2\ne 0$ for stages 3 and 4, and hence for the entire process, contrary to Hemmo and Shenker’s claim. Interestingly, correspondence does fail to obtain between S _TD and S _VN, since $\Delta S_{\text{TD}}+\Delta S_{\text{VN}}=2n.R.\log 2\ne 0$, despite the process being reversible ex hypothesi: incorporating S _VN into thermodynamic accounting violates the Second Law.

However, on this new analysis, we gain some clarity as to why the single-particle case is problematic. While it is true that incorporating S _VN into the thermodynamic accounting violates the Second Law, S _TD accounting by itself violates the Second Law (contrary to Hemmo and Shenker Reference Hemmo and Shenker2006). Even without considering S _VN, $\Delta S_{\text{TD}}\ne 0$ despite the process being reversible. Thermodynamic accounting is inconsistent here no matter what we do, which suggests that the reversible process they described for the single-particle case is thermodynamically unsound: if so, any argument Hemmo and Shenker make in this context may be disregarded.

The upshot: I agree with Hemmo and Shenker that correspondence fails for the single-particle case, but I do not agree about why it fails. It is not because the process they described is already thermodynamically consistent without taking S _VN into account. Rather, it is because the process is already thermodynamically inconsistent anyway.

In recent work, John Norton argued that thermodynamically reversible processes for single-particle systems are impossible in principle, which might explain why the process described by Hemmo and Shenker (Reference Hemmo and Shenker2006) is thermodynamically unsound: it was not justified to assume the process was reversible for a single-particle system. For Norton, a reversible process is “loosely speaking, one whose driving forces are so delicately balanced around equilibrium that only a very slight disturbance to them can lead the process to reverse direction. Because such processes are arbitrarily close to a perfect balance of driving forces, they proceed arbitrarily slowly while their states remain arbitrarily close to equilibrium states” (Reference Norton2017, 135). Norton notes that these thermodynamic equilibrium states are balanced not because there are no fluctuations but because these fluctuations are negligible for macroscopic systems. However, fluctuations relative to single-particle systems are large and generally prevent these systems from being in equilibrium states at any point of the process, rendering reversible processes impossible in the single-particle case (135). If reversible processes are impossible for single-particle systems in general, then it should come as no surprise that the particular single-particle reversible process used by Hemmo and Shenker is likewise thermodynamically unsound, as my analysis above suggests. If so, their claim that correspondence fails in this process is simply beside the point, since this process is not thermodynamic at all.

Since any reversible process cannot be realized for single-particle systems in general, the issue seems not to be with any particular process per se but with the single-particle case simpliciter. To my knowledge, no one before Hemmo and Shenker (Reference Hemmo and Shenker2006) discussed von Neumann’s experiment in terms of a single particle; von Neumann (Reference von Neumann1955), Peres (Reference Peres and Zurek1990, Reference Peres2002), Shenker (Reference Shenker1999), and Henderson (Reference Henderson2003) all explicitly or implicitly assume a large (or infinite) number of particles. This is for good reason. As Hemmo and Shenker acknowledge, and as we have seen: “The case of a single particle is known to be problematic as far as arguments in thermodynamics are concerned” (Reference Hemmo and Shenker2006, 158). Matter in phenomenological thermodynamics is assumed to be continuous.Footnote ¹⁴ A ‘gas’ composed of one particle can be many things, but it is surely not continuous in any commonly accepted sense. In other words, it is just not clear whether the domain of thermodynamics should apply to the single-particle case at all.

As Myrvold (Reference Myrvold2011) notes, Maxwell also made a similar claim with regard to phenomenological thermodynamics in general; it does not and should not hold in the single-particle case. On his view, the laws of phenomenological thermodynamics, notably the Second Law, must be continually violated on small scales:

The Second Law, and hence phenomenological thermodynamics, should not be expected to hold true universally in small-scale cases, especially not in the single-particle case. Von Neumann and everyone else in the debate should have recognized this point. Why, then, should it matter that the thought experiment succeeds or fails in this case? Phenomenological thermodynamics does not apply to single-particle cases. There is thus no profit in trying to establish correspondence between S _VN and S _TD in this case. Indeed, if we took seriously Maxwell’s claim that the Second Law fails at small scales, a failure of thermodynamic entropic accounting might even be expected; it does not rule out the possible thermodynamic nature of S _VN even though the sum of S _VN and S _TD might be inconsistent with the Second Law. In short, it is not clear why the single-particle case is relevant to the discussion at hand.

Hemmo and Shenker’s reasoning is untenable, because they fail to respect the context of phenomenological thermodynamics by bringing it into a context in which it is not expected to hold. Instead, it seems more appropriate that the single-particle case is precisely beyond the purview of classical thermodynamics, requiring an analogue that only corresponds to classical thermodynamics at the appropriate scales and limits. We may then take S _VN to be the analogue of S _TD in this case, only approximating S _TD as the system in question approaches the context suitable for traditional thermodynamic analysis. If so, we may see von Neumann as merely demonstrating that S _VN corresponds, not at all domains but in the domain where thermodynamics is taken to hold, to S _TD.

4.2. Single-Particle Case Redux: Statistical Mechanics and Information

Given the foregoing discussion, Hemmo and Shenker might insist that S _VN fails to correspond to S _TD even when taking into account a more relevant domain for single particles—statistical mechanics. After directly arguing that S _VN does not correspond to S _TD (Hemmo and Shenker Reference Hemmo and Shenker2006, 162–65), they further argue that S _VN does not correspond to information entropy (more on this below) in the single-particle case. Prima facie, this should seem irrelevant to von Neumann’s argument, which was to establish the correspondence of the thermodynamic S _TD and quantum S _VN: Why should information entropy’s failure to correspond with S _VN be a worry at all?

Here is one plausible worry, on a charitable reading. If information entropy corresponds to S _TD, and Hemmo and Shenker show that S _VN fails to correspond to information entropy, then we might conclude, indirectly, that S _VN does not correspond to S _TD after all.Footnote ¹⁵ This argument assumes that information entropy does correspond to S _TD, an assumption Hemmo and Shenker seem to hold as well: this is in line with the so-called subjectivist view of statistical mechanics (notably, see Jaynes Reference Jaynes1957). Furthermore, my above argument against the misapplication of phenomenological thermodynamics does not seem to apply here, since this argument is being made in the context of statistical mechanics and its particle picture, with no commitment to phenomenological thermodynamics.

However, Hemmo and Shenker do not do much to motivate the linkage between information entropy and S _TD; indeed, in their words, “a linkage between the Shannon information and thermodynamic entropy has not been established” (Reference Hemmo and Shenker2006, 164). Without this link, the failure of correspondence between the information entropy and S _VN appears, at best, irrelevant to the correspondence between S _TD and S _VN. Nevertheless, I will take a charitable view here and assume that there is a correspondence between information entropy and S _TD, for the sake of assessing their argument. Here is a plausible (if arguable) sketch: if one were a subjectivist like Jaynes (Reference Jaynes1957), one might take the Gibbs entropy in statistical mechanics to be a special case of the information entropy. After all, both have the following form:

(5) $\begin{array}{}& -\sum _i{p}_i\ln p_i,\end{array}$

with i being the number of possible states with associated probabilities of occurring p_i and the Gibbs entropy being multiplied by an additional Boltzmann’s constant k.Footnote ¹⁶ We know that statistical mechanics corresponds to phenomenological thermodynamics at the thermodynamic limit, so we can think of the Gibbs entropy, and hence information entropy, as corresponding to S _TD. I take this to be in line with what Hemmo and Shenker have in mind: “to the extent that the Shannon information underwrites the thermodynamic entropy, it does so via statistical mechanics” (Reference Hemmo and Shenker2006, 165). Assuming that the above picture is plausible, a failure of correspondence between S _VN and the information entropy provides evidence against the correspondence between S _VN and S _TD.

Their argument comes into two parts. Ignoring S _TD for the time being (which does not change throughout the cycle for the single-particle case—see sec. 4.1), they claim that we can consider the stage 5 location measurement to be a decrease in information entropy of ln 2, as a result of learning information about which one of two parts of the box the particle is in. On first glance, this seems to resolve the arithmetic inconsistency in entropic accounting: ln 2 is exactly the increase in S _VN as a result of the spin state changing from a pure state $|{\psi }_x^{\uparrow }〉$ to the mixed state ρ_spin in stage 2. In other words, for both the information and von Neumann entropies’ accounting to be correct (i.e., net change of zero across the cycle), we must consider S _VN as corresponding to information entropy. Now, since information entropy also corresponds, ex hypothesi, to S _TD, we have an indirect argument for the correspondence of S _VN to S _TD.

However, Hemmo and Shenker claim that this argument fails for collapse interpretations, that is, interpretations of quantum mechanics on which a superposed quantum state ontologically collapses into a pure state upon measurement (either precisely or approximately).Footnote ¹⁷ They allow that, on no-collapse interpretations (e.g., Bohmian or many-worlds interpretations), the location measurement in stage 5 does not decrease S _VN, since the state of the system never changes in light of measurements, and so the above argument goes through.

Let us see what they could mean by this claim by following the state of the particle through the cycle. At stage 2, everyone agrees that the state of the system is ρ_spin following the z-spin measurement; S _VN increases by ln 2. At this point, the particle’s position degrees of freedom remain independent from its spin degrees of freedom, as per our ideal gas assumption, although we might assume the particle starts out on the left half of the box, with the mixture of position states ρ_pos(L) with ‘L’ representing the left side. (Consider fig. 1 but with only one particle.) Following the semipermeable wall’s filtering at stages 3 and 4, the location of the particle becomes classically correlated with the spin. Let us say that the semipermeable wall sends $|{\psi }_z^{\uparrow }〉$ particles to the left, represented by ρ_pos(L), and $|{\psi }_z^{\downarrow }〉$ particles to the right, represented by ρ_pos(R). As such, the (mixed) state of the particle is now

(6) $\begin{array}{}& {\rho }_{\mathrm{particle}}=\frac{1}{2}\left(|{\psi }_z^{\uparrow }〉〈{\psi }_z^{\uparrow }|\otimes {\rho }_{\mathrm{pos}}\left(L\right)+|{\psi }_z^{\downarrow }〉〈{\psi }_z^{\downarrow }|\otimes {\rho }_{\mathrm{pos}}\left(R\right)\right)\mathrm{.}\end{array}$

For no-collapse interpretations, Hemmo and Shenker agree that the state of the particle stays the same as above after the location measurement in stage 5. We perform the compression in stage 5 and remove the partition at the end of stage 6, thereby removing the classical correlations between position and spin. No further change in either information entropy or S _VN occurs, and hence the correspondence goes through (Hemmo and Shenker Reference Hemmo and Shenker2006, 164). The spin state remains mixed until unitarily transformed into a pure state and completing the cycle.

For collapse interpretations, they claim that the location measurement decreases S _VN by ln 2, because, on collapse interpretations, the state of the particle upon the measurement, depending on which side the particle is found, becomes

(7) $\begin{array}{}& {\rho }_{\mathrm{particle}}=\{\begin{array}{c}|{\psi }_z^{\uparrow }〉〈{\psi }_z^{\uparrow }|\otimes {\rho }_{\mathrm{pos}}\left(L\right)\\ |{\psi }_z^{\downarrow }〉〈{\psi }_z^{\downarrow }|\otimes {\rho }_{\mathrm{pos}}\left(R\right)\end{array}\mathrm{.}\end{array}$

The spin state of the system here effectively goes from being a mixed state to a pure state as a result of this measurement: S _VN decreases by ln 2. Summing up the entropy changes, there was a decrease of ln 2 in information entropy and a net change of zero for S _VN as a result of the increase in stage 2 and the decrease in stage 5. Overall, then, the change is not zero but −ln 2; our accounting has gone awry, and there is a failure of correspondence between S _VN and information entropy. If this is right, S _VN does not correspond to S _TD.

However, I think that Hemmo and Shenker are wrong to claim that S _VN decreases following the location measurement for collapse interpretations. As Prunkl (Reference Prunkl2020, 272) notes, there is an inconsistency here. Everyone, including Hemmo and Shenker, agrees that the spin state of the particle is mixed—not pure—after stage 2’s spin measurement, even on collapse interpretations (Reference Hemmo and Shenker2006, 160). In that case, why does the particle’s spin become pure after the location measurement?

I think this results from a confusion over the nature of mixed states. In particular, they seem to have adopted what Hughes (Reference Hughes1992, secs. 5.4, 5.8) calls the “ignorance interpretation” of mixed states, confusing what I call classical and quantum ignorance. They seem to be assuming that mixed states simply represents classical ignorance (i.e., the lack of knowledge about a particular system): a system represented by a mixed state really is in a pure state, but we know not which. This is why the location measurement is supposed to reveal to us the pure state of this system (by revealing which side it is on and hence the correlated spin state) and hence ‘wash away’ our classical ignorance of the real state of the system. Postmeasurement, we know exactly which pure state this system is in, unlike premeasurement; hence, S _VN decreases.

However, as Hughes (Reference Hughes1992, 144–50) argues, this interpretation of mixed states—as representing classical ignorance about which pure state a particular system is in—cannot be the right interpretation of all mixed states. To begin, a mixed state can be decomposed in nonunique ways in general. Here is a simple example: a mixed state representing a mixture of $|{\psi }_z^{\uparrow }〉$ and $|{\psi }_z^{\downarrow }〉$ can also represent a mixture of $|{\psi }_x^{\uparrow }〉$ and $|{\psi }_x^{\downarrow }〉$, and so on. If we insist that a mixed state represent our classical ignorance about the real state of a particular system, then we end up having to say that a system’s state is really both either $|{\psi }_z^{\uparrow }〉$ or $|{\psi }_z^{\downarrow }〉$ and either $|{\psi }_x^{\uparrow }〉$ or $|{\psi }_x^{\downarrow }〉$. Of course, this is impossible given quantum mechanics. The defender of the classical ignorance interpretation might insist that we simply pick one pair of possible pure states but not both at once. In general, however, there is no way to do that nonarbitrarily given some density matrix. Furthermore, this problem only worsens when we consider that there are usually more than just two ways to decompose a density matrix—a principled choice based on the mixed state alone is not feasible. The mixed state cannot be a representation of classical ignorance.

Instead, to paraphrase Hughes (Reference Hughes1992, 144–45), mixed states should be (minimally) interpreted as such. If we prepared in the same way an ensemble of systems, each described with the same mixed state (i.e., a mixture of pure states with certain weights), then the relative frequency of any given measurement outcome from the ensemble is exactly what we would get if the ensemble were composed of various ‘subensembles’ each in one of the pure states in the mixture, with the relative frequency of each subensemble in the ensemble given by the corresponding weights.

In other words, the sort of quantum ignorance relevant in the right interpretation of mixed states is not whether we are ignorant about the real state of this particular system but whether we are ignorant about the measured states of an ensemble of identically prepared systems like this one. If this is right, quantum ignorance cannot be ‘washed away’ upon measurement of a single system unlike the sort of ignorance Hemmo and Shenker were implicitly assuming, and it seems like this quantum ignorance is precisely what remains after the location measurement.

This was roughly Henderson’s (Reference Henderson2003) criticism against Shenker (Reference Shenker1999), which is why it is puzzling that Hemmo and Shenker (Reference Hemmo and Shenker2006) commit the same mistake:

This applies to the location measurement in stage 5 too: measuring the location of the particle in this case does not change the state of the particle from a mixed one to a pure one even on collapse interpretations. First, it seems quite irrelevant whether we adopt a collapse or no-collapse interpretation, because the collapse mechanism applies to superposed pure states, not statistical mixtures. If anything, collapse had already happened in the stage 2 measurement procedure, yet everyone including Hemmo and Shenker (Reference Hemmo and Shenker2006, 160) accept that the system is in a mixed state after stage 2 even for collapse interpretations. More importantly, there remains a probability distribution over the states of the particle as a result of stage 2’s spin measurement, even after the location measurement. Given an ensemble of particles prepared from stages 1 to 5 in the same way, we are still not be able to predict with certainty whether an ensemble of particles would all be measured on the left or right sides of the box (and hence all spin-up or spin-down) as a result of the mixed state resulting from stage 2, only that half of the ensembles will be on the left and the other half will be on the right. Quantum ignorance remains—the system remains in a mixed state even after the location measurement, as

(8) $\begin{array}{}& {\rho }_{\mathrm{particle}}=\frac{1}{2}(|{\psi }_z^{\uparrow }〉〈{\psi }_z^{\uparrow }|\otimes {\rho }_{\mathrm{pos}}(L)+|{\psi }_z^{\downarrow }〉〈{\psi }_z^{\downarrow }|\otimes {\rho }_{\mathrm{pos}}(R\left)\right)\mathrm{.}\end{array}$

This is exactly the state of the system in no-collapse interpretations; that is, quantum ignorance does not discern between collapse and no-collapse interpretations. What has gone away is the classical ignorance that Hemmo and Shenker (mistakenly) assumed was relevant for mixed states, ignorance about this particular system’s state. By measuring the system’s location, we come to learn of the correlations between location measurement and the particle’s spin. This ignorance does not change the mixed state to a pure state: instead, this loss of classical ignorance—gain in information—is represented as a decrease in information entropy just as before, and this information is what we use to perform the compression in stage 5.

As a result, there is no additional decrease in S _VN in stage 5 for collapse interpretations; the entropy accounting lines up after all, as with no-collapse interpretations. The decrease in information entropy does correspond to the increase in S _VN, and so information entropy does correspond to S _VN after all. Hemmo and Shenker’s argument does not establish the failure of correspondence between S _VN and S _TD via the failure of S _VN and information entropy to correspond.

To sum up, their arguments in the single-particle case are either ill motivated and irrelevant to von Neumann and our discussion of correspondence when considered in terms of phenomenological thermodynamics or outright fail when considered in the more relevant domain of (informational approaches to) statistical mechanics. Either way, their argument does not support the failure of correspondence between S _VN and S _TD.Footnote ¹⁸

4.3. Finitely Many Particles

Hemmo and Shenker’s argument in the case of finitely many particles rests on the assumption of equidistribution; that is, that the particles will be equally distributed across the left and right sides of the box after separation by the semipermeable wall. Assuming equidistribution, the increase in S _VN given the spin measurement in stage 2 is N ln 2 (Hemmo and Shenker Reference Hemmo and Shenker2006, 169). Furthermore, the decrease in thermodynamic entropy in the fourth stage is N ln 2 as well. The entropic accounting therefore seems to work out.

However, Hemmo and Shenker press further on the ‘rough’ nature of equidistribution when N is large but finite: they claim that the change in S _VN will only be N ln 2 when N is infinite, since equidistribution only truly holds when $N\to \infty$. In other cases, S _VN will strictly only approximate S _TD, and hence S _VN and S _TD combined will never be exactly zero. Hence, “Von Neumann’s argument goes through as an approximation” (Hemmo and Shenker Reference Hemmo and Shenker2006, 169). However, they claim that this state of affairs suggests, instead, that von Neumann’s argument strictly fails: “since Von Neumann’s argument is meant to establish a conceptual identity between the quantum mechanical entropy and thermodynamic entropy, we think that such an implication is mistaken. … No matter how large N may be, as long as it is finite, the net change of entropy throughout the experiment will not be exactly zero” (169). As I have already discussed in section 4.1, it is not clear to me that von Neumann’s goal really was to establish strict identity (what they call “conceptual identity”), that is, correspondence between S _VN and S _TD in all domains. Rather, it seems to be the establishing of correspondence only in domains where S _TD is taken to hold. If so, their argument here simply misses the point.

Furthermore, as is well known, the particle analogue of thermodynamics, statistical mechanics, becomes equivalent to phenomenological thermodynamics only when $N=\infty$, that is, when N arrives at the thermodynamic limit. As such, to complain that S _VN does not match up to S _TD outside of this domain is to demand the unreasonable, since it is not clear that even statistical mechanics, the bona fide particle analogue of thermodynamics, can satisfy this demand. Since S _VN approximates S _TD the same way statistical mechanical entropies approximate S _TD (and becomes equivalent at $N=\infty$), and physicists generally accept that statistical mechanics corresponds to thermodynamics nevertheless, why should this approximation be particularly problematic for S _VN? I think Hemmo and Shenker take too seriously the notion of conceptual identity involved in von Neumann’s thought experiment to be strict equality, although I suspect a better way to understand von Neumann’s strategy is to understand S _VN as an approximation to S _TD that is more fundamental than S _TD in small N cases but becomes part of the S _TD calculus in domains where S _TD applies.

To have a case against S _VN as a quantum analogue of S _TD in the case of finitely many particles, Hemmo and Shenker must explain what exactly the problem is with approximations in this case, if it has worked out so well for the case of statistical mechanics and thermodynamics. If not, they might just be “taking thermodynamics too seriously” (see Callender Reference Callender2001).

One might say something stronger: unless they can justify why we cannot use approximations at all in science, they do not have a case at all. As they note themselves, S _TD is itself only on average approximately −N ln 2 (Hemmo and Shenker Reference Hemmo and Shenker2006, 169), only being equal to −N ln 2 when $N=\infty$. So, in fact, the approximate quantity of S _VN, ∼N ln 2, exactly matches the approximate quantity of S _TD, ∼−N ln 2, in the case of finitely many particles. Unless there is something wrong with approximations in physics in general, this, then, is in fact a case of S _VN corresponding to S _TD, contrary to their argument.

4.4. Infinitely Many Particles

Hemmo and Shenker consider von Neumann’s argument in the infinite-particles case in two different ways: one as $N\to \infty$ and one as $N=\infty$. As they rightly point out, the two cases are very different for calculations of physical quantities.

Consider stages 2 and 5 in this context. Hemmo and Shenker emphasize that a spin measurement is “a physical operation which takes place in time” (Reference Hemmo and Shenker2006, 170), which constrains what is physically possible. For the case in which $N\to \infty$, stage 2 is to be understood as a succession of physical measurements in which “we measure individual quantities of each of the particles separately and only then count the relative frequencies” (170), before coming up with a density matrix describing this state. In this case, as with the case described in section 4.3, S _VN approaches N ln 2 as $N\to \infty$. Their complaint here consists of two premises: one, that, as with section 4.3, S _VN never reaches N ln 2 unless $N=\infty$ and, two, that since measurements are physical measurements, we can never perform an infinite series of these measurements, and so we can never measure infinite particles. A fortiori the measurable S _VN can never arrive at N ln 2, and so the entropic accounting is again supposed to be inconsistent if we consider both S _VN and S _TD.

However, it is clear that their argument is moot given an understanding of the sort of thermodynamics we are interested in (see sec. 4.3). While it is true that S _VN will never reach N ln 2, recall that S _TD (or, more likely, one of its statistical mechanical analogues, given the domain of finitely many particles merely approaching ∞ rather than $N=\infty$) will likewise never reach N ln 2. In other words, it does not matter that we can never perform an infinite series of these measurements and, hence, never come to know of S _VN at the thermodynamic limit, since we can likewise never have a thermodynamic entropy equivalent to N ln 2 unless we are at the thermodynamic limit. The two entropies, then, in fact correspond in this case.

What of the second case? Here, Hemmo and Shenker concede that “arithmetically Von Neumann’s argument goes through at the infinite limit” (Reference Hemmo and Shenker2006, 172), which makes sense because, as I have insisted so far, von Neumann’s strategy was never to demonstrate the strict identity of S _VN and S _TD, that is, the correspondence of S _VN and S _TD in all domains. Instead, it was to show that S _VN corresponds to S _TD only in the domain where phenomenological thermodynamics hold, in all other cases merely approximating S _TD in large N cases or replacing it altogether (in, e.g., single-particle cases). I maintain that Hemmo and Shenker’s main mistake was to confuse the domain where phenomenological thermodynamics hold with domains where they do not hold.

Hemmo and Shenker complain that “real physical systems are finite. This means that Von Neumann’s argument does not establish a conceptual identity between the Von Neumann entropy and thermodynamic entropy of physical systems. Identities of physical properties mean that the two quantities refer to the same magnitude in the world” (Reference Hemmo and Shenker2006, 172). In line with what I have said in section 4.1, it seems that there was no physically meaningful theoretical term in phenomenological thermodynamics that could refer to some quantity in the single-particle case, which was why von Neumann needed to come up with a new measure of entropy to begin with. Furthermore, extending a concept to a new domain does not require strict identity, as we have seen and understood for a long time in the case of statistical mechanics and phenomenological thermodynamics.

As Peres summarizes: “There should be no doubt that von Neumann’s entropy… is equivalent to the entropy of classical thermodynamics. (This statement must be understood with the same vague meaning as when we say that the quantum notions of energy, momentum, angular momentum, etc., are equivalent to the classical notions bearing the same names)” (Reference Peres2002, 174). ‘Equivalence’ here should not be understood in terms of strict (or conceptual) identity, that is, correspondence at all domains. Rather, we should understand equivalence loosely as correspondence in the suitable domains of application and successful extension of old concepts in these domains to new domains. As Peres noted above, ‘equivalence’ should be understood in the context of discovery, where one is trying to develop new concepts that are analogous to old ones in different domains. For von Neumann, we have a theory (phenomenological thermodynamics) that is well understood but also another theory (quantum mechanics) that we want to understand in light of the former theory. Finding correspondence provides us with ways to extend concepts from the original theory to the new theory: for example, with S _VN we may now define ‘something like’ S _TD, whereas before there was no way to talk about these cases. The same goes for statistical mechanics: by finding a correspondence between, for example, temperature and mean kinetic energy in the thermodynamic limit, we can extend the notion of ‘something like’ temperature beyond its original domain into systems with small numbers of particles, whereas before there was, again, no way to talk about these cases.

I see nothing wrong in these cases in the context of discovery. We should give up a strong and untenable notion of conceptual identity in this context. If so, Hemmo and Shenker’s objection loses much bite.

They further claim that “the fact that the behavior of the two quantities coincides approximately for a very large number of particles is not enough, because in any ensemble of finite gases there are systems in which the identity will not be true. This means that in a real experiment the Von Neumann entropy is not identical with the thermodynamic entropy” (Hemmo and Shenker Reference Hemmo and Shenker2006, 172). This again reveals a confusion between phenomenological and statistical thermodynamics. If they want to talk about particles at all, it seems they must adopt some form of statistical mechanical picture with microscopic variables, given phenomenological thermodynamics’ emphasis on purely macroscopic variables like volume or temperature. Yet, if so, they must recognize that thermodynamic entropy S _TD is in general not strictly identical to statistical mechanical entropy, for example, the Gibbs entropy or information entropy (briefly discussed in sec. 4.2) either. Their complaint about approximate coincidences not being enough for (the relevant sort of) equivalence thus weakens significantly, especially since they must assume some such equivalence (which cannot be strict identity) to even talk about particles within the context of phenomenological thermodynamics to begin with. Furthermore, statistical mechanics is evidently empirically successful in explaining and predicting traditionally thermodynamic phenomena despite this ‘nonequivalence’—it is not clear why this ‘nonequivalence’ should matter if, for all practical purposes, statistical mechanics is the conceptual successor of thermodynamics. Of course, if they could come up with a principled reason why approximations should not be allowed period, while accounting for statistical mechanics’ empirical success in accounting for thermodynamic behavior, then this could change. As of now, I see no such argument forthcoming.