1. Introduction
The foundations of statistical mechanics (SM) have been receiving much attention from philosophers of science. A glance at the literature, however, reveals that contrary to the case of nonrelativistic quantum mechanics (NRQM), where at least those members of the community who resist the current information-theoretic hype still regard two conceptual difficulties that loom in the foundations of the theory—the measurement problem and the relation of the different interpretations to the special theory of relativity—as open problems but disagree on their solutions, in SM disagreements prevail in much earlier stages on matters of pure definitions: Philosophers of science who show interest in the foundations of SM find it hard to agree even on what the open problems are. The reason for the disagreement is well-known. SM suffers from the lack of a unique mathematical framework. Without such a unique framework, fundamental concepts in the foundations of the theory enjoy multiple formulations, and consequently, the debates in the field remain mostly ineffective: What counts as a problem according to one approach may pass unnoticed from the point of view of another.Footnote 1
The plurality of mathematical frameworks in SM calls for a similar plurality in the discussion of problems in the foundations of the theory.Footnote 2 Yet, for the last decade or so, an astonishing thesis has been put forward in the literature according to which (1) the asymmetry in the initial conditions of our universe is the sole explanans of the puzzle of the thermodynamic asymmetry in time, and (2) once we realize the true nature of this puzzle, causal-dynamical explanations in the foundations of SM become redundant.
The most lucid advocate of this thesis is Huw Price, who has been continually attacking a variety of casual-dynamical approaches in the foundations of SM as giving the wrong answer to the question he has been at pains to promote as the major open question in the field (Price Reference Price1996, Reference Price2002, Reference Price and Callender2003). This discussion aims to rehabilitate causal-dynamical approaches in the foundations of SM by (1) exposing Price's misunderstanding of these approaches and (2) shedding light on the original problem these approaches set forth to solve—a problem that unfortunately receives no attention in Price's celebrated work.
2. The ‘Hard’ Question
In order to appreciate Price's thesis, the following outlook may prove instructive. A general philosophical problem in the philosophy of physics is to explain unobservable phenomena. This problem emerges from the conjunction of a philosophical stance, namely, taking the formalism of our theories seriously as representing our world, and an undisputable fact, namely, the fact that certain phenomena that are allowed to exist according to these formalisms are never (or rarely) observed. Thus, for example, in thermodynamics, which is Price's major concern, the approach of physical systems towards equilibrium is ubiquitous, and yet the time-reversal-invariant character of the underlying dynamical laws that presumably govern thermal phenomena equally allow fluctuations away from equilibrium to occur. The fact that these are rarely reported is commonly known as the puzzle of the thermodynamic arrow in time.
Initially, there exist three possible solutions to this problem that can be distinguished nicely according to the relevant modality of the unobservable phenomena. The first, and easiest, solution puts the blame on our models. It suggests that we should devise new ones in which such phenomena would be forbidden by law. The second puts the blame on us. We are not clever enough, and we lack the appropriate technological capacities to observe such phenomena—our macroscopic observables being so crude that in appropriate time scales, they can be described as evolving autonomously in a single time direction. The third puts the blame on the rarity of the microconditions that generate unobservable phenomena. Although these microconditions are indeed possible, they are also highly improbable, and we are very unlikely to run into them by chance.
It is well-known that the founding fathers of SM shifted back and forth between these three solutions. Maxwell, for one, originally took the second route when he presented his “very observant and neat-fingered being” (Reference Maxwell[1867] 1911, 214) in his letter to Tait in order to emphasize that the second law of thermodynamics is only statistically valid, thus mocking Boltzmann and Clausius, who at that time were following the first route and were unsuccessfully engaged in sterile attempts to derive the second law from Hamiltonian mechanics. The second law, according to Maxwell, could be violated easily if one had intimate knowledge of individual molecules. That Boltzmann retreated from the first route to the third is commonly acknowledged, although the exact point in his career this retreat took place is still a matter of dispute. (For different opinions, see Klein Reference Klein, Cohen and Thirring1973; Gallavotti Reference Gallavotti1999; Von Plato Reference Von Plato1991.)
One (but certainly not the only) problem with Boltzmann's solution—that unobservable phenomena are possible but also highly improbable—is that the theory that accounts for such improbability, namely SM, still employs time-reversible-invariant dynamics. Consequently, nothing forbids entropy from increasing towards the past—a feature of the theory which is falsified by our memories and experience. In order to save the theory from these false retrodictions, Boltzmann introduced the following hypothesis: Our universe started its life in a low entropy state.
That in nature the transition from a probable initial state to improbable state does not take place as often as the converse, can be explained by assuming a very improbable initial state of the entire universe surrounding us. (Boltzmann Reference Boltzmann[1895] 1964, 447)
This hypothesis, suggested by Boltzmann and adopted by Schrödinger (Reference Schrödinger1950), Feynman (Reference Feynman1965), and others, is the ‘Past Hypothesis’ (coined as such by Albert Reference Albert2000). If the Past Hypothesis is true, then the most probable history of our universe is one wherein entropy rises because it started off so low. It is this hypothesis that Price takes as a starting point for his thesis:
All the time-asymmetry of observed thermodynamic phenomena resides in an existential or particular fact—roughly, the fact that physical processes in the known universe are constrained by low entropy ‘boundary condition’ in one temporal direction. (Price Reference Price2002, 92)
Boltzmann himself recognized that his hypothesis, while instrumental in correcting wrong retrodictions of the theory, should be regarded as fundamental, i.e., should not be derived from any more fundamental hypotheses:
[The Past Hypothesis] is a reasonable assumption to make, since it enables us to explain the facts of experience, and one should not expect to be able to deduce it from anything more fundamental. (Boltzmann, quoted in Goldstein Reference Goldstein and Bricmont2001, 50)
And yet, Huw Price (Reference Price1996) argues that when one appreciates the above situation, the appropriate question to ask in foundations of SM is no longer “why does entropy rise?” but rather “why was it ever low to begin with?” Price's thesis is that given that the only asymmetry needed to account for the thermodynamic arrow is the asymmetry in the initial conditions of our universe, the explanation of the Past Hypothesis itself turns out to be the true puzzle of the thermodynamic arrow in time:
There are two distinct conceptions of what needs to be explained about the asymmetry of thermodynamics, and it is easy to fail to notice that they are distinct; easy to bark up the wrong tree. In this paper I try to tease these two conceptions apart, and to identify one of them as the problem's vera crux. My aim is to clarify the issue of the proper explanandum in thermodynamics, under the assumption that it is the time-asymmetry which excites our interest, and which drives the inquiry. (Price Reference Price2002, 84)
Those who shrug and question the cogency of this project, claiming, as it were, that it is simply unnecessary to explain why the world is the way it is, encounter the following response:
However, it seems to me that this attitude to the explanation of initial conditions is on shaky ground. Would it take the same view of the need to explain an equivalent condition at any other time? If so, it is perilously close to a kind of global explanatory nihilism, which answers every ‘Why?’ question with the answer that things had to be some way, so why not this way? If not, on the other hand, then the proponent of this ‘no need to explain initial conditions’ view needs to tell us what is special about (what we call) initial conditions. (Price Reference Price2002, 114)
Faithful as I am to the idea of a plurality of problems in the foundation of SM, it is not my intention here to enter into a general methodological discussion on the merit of Price's project. (For a recent criticism, see Callender Reference Callender2004.) In what follows, I address another feature of Price's thesis—the feature according to which one should regard causal-dynamical explanations to the puzzle of the thermodynamic arrow in time as redundant.
3. Wrong Answer?
Price believes that the Past Hypothesis, i.e., the asymmetry in the initial conditions, is the only asymmetry needed to account for the thermodynamic arrow in time; that is, that the Past Hypothesis correctly answers the question that Boltzmann's solution to the problem of unobservable phenomena has left open. Consequently, Price is convinced that efforts should be devoted to explaining the Past Hypothesis itself, rather than to generating superfluous solutions to a problem that has already been solved.
Let us grant Price his starting point and disregard the other two possible routes to the explanation of unobservable phenomena, and let us accept Boltzmann's solution to this problem. In other words, let us accept that the Past Hypothesis is the reason behind the thermodynamic arrow in time. However, Boltzmann's solution to the problem that led to this hypothesis is infected with another distinct problem, which unfortunately receives no attention in Price's work: the problem of justifying the probabilistic considerations that were introduced by the founding fathers of SM into the dynamical models they were constructing for thermal phenomena (Brush Reference Brush1976a, Reference Brush1976b, Reference Brush1983.)
Beginning with Bernoulli's work in the mid-eighteenth century, a major issue in the foundations of physics was the construction of dynamical models for thermal phenomena. This issue—along with the revival of the atomic hypothesis in the mid-nineteenth century—was the background against which the founding fathers of SM were conducting their research. Clausius and Maxwell were maybe the first to introduce probabilistic considerations into the mechanical models, and Boltzmann soon followed; yet, Maxwell and Boltzmann were ambiguous with respect to the origin of these probabilistic considerations. On one hand, they regarded thermodynamics as only statistically valid and traced the probabilities to the crudeness of the macroscopic description. On the other hand, they realized that without some kind of randomization at the molecular level, there can be no justification for the probabilistic considerations themselves.
Boltzmann's attempts to solve the problem of probability culminated in two well-known projects. The first is the H-Theorem. With it, Boltzmann tried to dynamically justify the choice of Maxwell's velocity distribution that was originally ‘imported’ by Maxwell to his dynamical model from statistics.Footnote 3 The second is the ergodic hypothesis. It offers dynamical considerations as a possible justification for the assumption of equiprobability that underlies the choice of the uniform measure in Boltzmann's post-H-Theorem approach. Although Price (Reference Price1996, 40–43) criticizes the former and completely ignores the latter,Footnote 4 it is hard to overestimate the contribution of these two projects to mathematical physics: Boltzmann's equation from which the H-Theorem is derived is widely applicable today, and the ergodic theory that emerged from the ergodic hypothesis has produced beautiful mathematical results. Both projects, however, are commonly regarded as unsuccessful in meeting their original goal: while the former relies on an unjustified probabilistic assumption, namely, the molecular chaos hypothesis, the latter is satisfied only by highly idealized models. (See, e.g., Farquhar Reference Farquhar1964; Earman and Redei Reference Earman and Redei1996; Gallavotti Reference Gallavotti1999; Emch and Liu Reference Emch and Liu2002.)
Other causal-dynamical approaches followed, aiming to physically underpin the probabilities of SM. One of the approaches that Price takes great pain to disparage is interventionism,Footnote 5 according to which random external perturbations in the system's surroundings are invoked as responsible for the necessary internal randomness of the system itself. Regardless of its so-called ‘solution’ of the puzzle of the thermodynamic arrow in time (criticized by Price), as a solution to the problem of probability, interventionism is indeed a nonstarter: One cannot assume that the laws of nature inside one's system differ from the laws of nature outside it. Consequently, whatever randomness one invokes in the system's surroundings is the very randomness one seeks to underpin.
The lack of physical underpinning of the probabilistic considerations in Boltzmannian SM resulted in recent suggestions to invoke NRQM as a possible surrogate.Footnote 6 Roughly, the idea is to incorporate stochastic transition probabilities that feature in the dynamics of two solutions to the quantum measurement problem as a physical justification for the otherwise unjustified probabilities of SM.Footnote 7
Given the plurality of frameworks in the foundations of SM, it will not come as a surprise that some commentators fail to see that the lack of physical justification to the probabilistic posits of SM is a conceptual problem. In some texts (e.g., Tolman Reference Tolman1938, 63–70), one is advised to regard the assumption of equiprobability as an a priori postulate. In other texts (e.g., Jaynes Reference Jaynes1983), one is referred to the principle of indifference as the reason behind it. Some (e.g., Callender Reference Callender2004), turn the table upside down and regard the success of equilibrium SM as the only justification needed for its methods. Others (e.g., Loewer Reference Loewer2001), urge us to regard the probability measure in SM as yet another law of nature that defies justification. One can argue about the merits of these ‘justifications’, but one thing is clear: They all put aside the ambition to devise mechanical models for thermodynamic phenomena that are devoid of epistemological considerations.
4. Wrong Question!
So far, we have identified at least three distinct questions in the foundations of SM that call for an answer. These, along with Boltzmann's solution to the first, are listed below:
1. Why are nomologically possible processes seldom, if ever, observed? Boltzmann's answer: These are only improbable, and the reason we do not see them lies in the Past Hypothesis.
2. What justifies the probabilistic considerations in SM that led to Boltzmann's answer to Question 1?
3. What justifies the Past Hypothesis?
When matters are put in these terms, it is clear that the causal-dynamical approaches that followed Boltzmann in his quest for answering Question 2 are criticized by Price for giving redundant answers to Question 1! But this is simply a misunderstanding.
Here is Price (Reference Price2002, 90) in a typical passage in which he describes his intellectual opponents:
On one side are what I shall call Causal-General theories. These approaches take the explanandum to be, at least in part, a time-asymmetric generalisation … . What unifies these diverse approaches, in my view, is their sense of the nature of the project. All of them seek a causal-explanatory account of a time-asymmetric gen eralisation about the physical world as we find it.
In yet another passage Price (Reference Price and Callender2003, 43) targets the molecular chaos hypothesis which introduces what he calls ‘a second asymmetry’ to the debate.Footnote 8 According to Price, this ‘second’ asymmetry (over and above Boltzmann's Past Hypothesis) is redundant:
Perhaps even more damagingly, this brief excursion into the history of the initial randomness approach suggests that it provides no legitimate basis whatsoever for the nomological second asymmetry required by the two-asymmetry approach. At best, the asymmetry of microcorrelations is simply the first asymmetry redescribed, the asymmetry of macroscopic boundary conditions, characterized in another way.
And here is Price (Reference Price and Callender2003, 33) describing what he thinks one of the causal-dynamical approaches in the foundations of SM come to explain:
Causal approaches seek a dynamical factor responsible for the general tendency of entropy to increase—some factor without which entropy would not increase, at least with the observed regularity. This factor needs to be time-asymmetric, for otherwise entropy would be constrained to be constant (nondecreasing in both directions). One version of such an approach is interventionism, which takes the cause to be provided by influences from the external environment ‘coming at haphazard’ … .
But the ‘Causal General theories’ Price refers to—i.e., those that are based upon Boltzmann's H-Theorem and the molecular chaos hypothesis, those that are based upon interventionism, those that are based upon a quantum-mechanical true or apparent collapse—all acknowledge that the ultimate solution to the puzzle of thermodynamic arrow in time was given by Boltzmann as an answer to Question 1. The problem these approaches are trying to solve is another problem: It is the problem of probability Boltzmann left unsolved; it is Question 2—the physical underpinning of SM probabilities with mechanical models.
Price (Reference Price2002, 97–98) himself acknowledges that the puzzle of the thermodynamic arrow in time is closely related to the attempts to vindicate the atomic hypothesis, but then he simply dismisses these attempts as inappropriate, and attacks the theories that were elaborated in order to fulfil them on the ground that they are nonstarters as answers to Question 1, a question the answer to which has generated Question 3.
Recall, however, that the H-Theorem was originally introduced by Boltzmann to derive Maxwell's distribution—a distribution that at that time was synonymous with probability. Similarly, interventionism was originally introduced by Boltzmann and Maxwell themselves to account for the necessary randomization in the molecular level. Since in both approaches the founding fathers were supplying dynamical grounds for the probabilistic posits of SM, and not as a mechanism for irreversibility, I see no point in criticizing it in the latter context. If criticism should be launched against the H-Theorem or against the classical interventionist school, it is quite different than the one Price alludes to. As we have seen, the molecular chaos hypothesis is instrumental in deriving the Maxwell-Boltzmann distribution from molecular collisions but it itself remains unjustified, and the classical interventionist school presupposes (in its attempt to supplement the ergodic theory) the very randomness it intends to underpin. Both approaches failed because they left open the question they originally set out to answer—not because they added redundant explanations to Boltzmann's solution of the puzzle of the thermodynamic arrow in time.
Price appreciates neither the problem his intellectual opponents aimed to solve, nor the questions to which they sought answers; thus, he regards their answers as irrelevant to a question he himself poses. Here is another attack by Price, this time on Albert's (Reference Albert1994, Reference Albert2000) complaint that human ignorance is called upon in explaining thermodynamic phenomena:
This is correct as far as it goes, I think, but it does not establish that we need objective probabilities. After all, an opponent might agree that if there were to be a causal explanation that the gas stays dispersed, any probabilities it invoked would need to be objective, but go on to deny that such an explanation is needed. Perhaps this is where explanation stops, with brute Humean regularities. (Price Reference Price2002, 105)
But Albert's complaint is directed at the fact that the statistical approach Price champions leaves Question 2 open, in the sense that epistemological, rather than physical, considerations are touted in the so called ‘justification’ of SM probabilities, not against Boltzmann's solution to the puzzle of the thermodynamic arrow in time.Footnote 9
What may strike one as especially odd in this final quote is Price's nonegalitarian attitude towards putative answers to Questions 2 and 3. While it is true that explanation should end with brute facts, the issue at stake is not whether, but rather where, it should end. Given Price's misguided opinion on causal-dynamical approaches in the explanation of unobservable (thermodynamic) phenomena, one begins to wonder what exactly is the criterion for ending an explanation, and why is Price's insistence on raising Question 3 any different in this context than the current honest attempts to answer Question 2. Without such a criterion, both questions should be either answered or left open.Footnote 10
