1. Justification of the Probability Measure in Statistical Mechanics

In an essay published in this journal, Kevin Davey (Reference Davey2008) claims that the justification of the second law of thermodynamics as it is conveyed by the “standard story” of statistical mechanics (SM), roughly speaking, that low-entropy microstates tend to evolve to high-entropy microstates, is “unhelpful at best and wrong at worst” (28). This is a strong claim not only because it leaves the vast empirical success of the standard story mysteriously lacking a principled justification but also because Davey claims to have identified a novel error that is “more basic” than previous criticisms of the standard story, such as those presented by Sklar (Reference Sklar1993) and Earman and Redei (Reference Earman and Redei1996). Moreover, if Davey is correct, all of the research done in justifying the second law through the set of approaches to SM commonly labeled in the literature as “Boltzmannian” statistical mechanics (BSM)—and targeted by Davey’s account—is thoroughly misguided.

How does Davey reach his conclusion? First, he identifies that the standard story, in its attempt to justify the second law, makes essential use of a probability measure. Next, Davey asks, how could such a probability measure be justified? His claim is that the only plausible way to justify a probability measure within the context of the standard story is with a “probability principle” (to be defined later). After this, Davey presents a general argument, illustrated through a thought experiment, for why any probability principle is false. Since some probability principle is supposed to be necessary for justifying the choice of probability measure of the standard story and since any probability principle is false, it follows that the choice of probability measure made in giving the standard story is arbitrary. Hence the standard story fails at justifying the second law: either the probability measure chosen is incorrect, in which case the standard story is “wrong,” or else it is correct but lacks justificatory grounds (i.e., it was a lucky choice), in which case the standard story is “unhelpful.”

Davey’s account, as mentioned above, is restricted to BSM, and, accordingly, this article is also restricted to such approaches. Nevertheless, it is noteworthy to mention that although there exists a plethora of approaches to SM, many of which make use of different types of mathematical formalism, virtually all approaches make use of a probability measure to account for entropic behavior—behavior that abides by the second law of thermodynamics.

My main aim in this article is to criticize what I take to be an unreasonable skepticism applied solely to SM. I will argue that Davey’s argument for rejecting the standard story commits him to a form of skepticism that is more radical than the position he states and that Davey places unreasonable demands on the notion of justification in the physical sciences. Accordingly, I motivate the claim that certain notions of justification in SM should be rejected.Footnote ¹

My account comes in four stages. First, I place Davey’s complaint within the context of BSM (sec. 2). Afterward, I present Davey’s argument that all probability principles are false and identify what I think Davey got right. In the second stage, I show that certain standards of robustness must be placed on Davey’s probability principle in order to warrant his claim that the standard story fails at justifying the second law. I offer a new notion of a probability principle that abides by these standards (sec. 3). Third, I show through an example from classical mechanics that Davey’s argument can be generalized beyond BSM, thus demonstrating that Davey is committed to a stronger position than he states, since it is broader in scope (sec. 4). In the fourth stage, in an effort to identify “what went wrong” with the standard story, I study Davey’s argument and conclude that the standards of justification he sets on BSM are too high. This is so because the failure of justification is neither unique to BSM nor internal to the theory itself (sec. 5).

The main take-home message is not so much that Davey’s account of justification in BSM does not work and thus can safely be ignored. Rather, Davey’s account consists of an important contribution to the scholarly debate on the foundations of SM. Accordingly, it merits a response and serious philosophical consideration. The point of this article is to give such a response.

2. Probability Principles

BSM defines the Boltzmann entropy (which is supposed to mimic the thermodynamic entropy) of a macrostate at time t as , where is the Boltzmann constant, is the Lebesgue measure, and is a region of phase space associated with .Footnote ² For an ideal gas in a box it is maximized at the equilibrium macrostate . The main idea behind BSM is that since , the region of phase space associated with , is overwhelmingly larger than any other , the system is most likely to be found in equilibrium or on its way to equilibrium. In order to give substance to this claim, a connection must be made between regions of phase space and probability. It is here that the traditional statistical postulate enters:

Since is very large, then .Footnote ³

However, Davey intervenes and asks: how could one possibly justify the probability measure used in the above-stated statistical postulate? His answer is that only a probability principle can provide such justification. In this manner the entire justificatory story given by the Boltzmannian depends on the cogency of such a principle. But what is a probability principle?

A probability principle is any principle of the following general form (where the ellipses in the definition below stand in for a specific probability measure defined for particular cases):

An example of a probability principle used in the standard story and explicitly assumed in the above-stated statistical postulate is the following:

Before presenting my critical reply to Davey’s argument, I would like to identify what I believe Davey got right. His argument portrays a strong objection to the use of any a priori principles, such as the “principle of indifference,” for the justification of the probability measure chosen to describe the state of some system. Davey’s account shows that it is easy to think of situations in which such an a priori principle chooses the wrong probability measure to describe the state of some system, and this point is an insightful one.

However, as an argument that targets the entire set of Boltzmannian approaches to SM, I do not think Davey’s argument works. The following three sections are devoted to establishing this point.

3. Standards

As it stands, the above argument (that the standard story is wrong and unhelpful) is not correct. The reason is that it presupposes that a probability principle is both a necessary and sufficient condition (as opposed to just sufficient) for giving the standard story. But nowhere in Davey’s paper is there an argument to that effect. Rather, there is a burden-shifting plea; what else could account for the standard story?Footnote ⁵ Such an appeal should be charitably accepted if it can be clearly motivated that no other principle will suffice for giving the standard story. In this sense Davey’s probability principle must be robust: it must be the case that when we think hard about justifying the second law, within the context of the standard story, we cannot come up with an alternative to Davey’s probability principle. Otherwise, all that has been shown is that some restricted type of probability principle is both sufficient for giving the standard story and also false. This weaker reading of the argument implies a weaker result—namely, any probability principle (as defined by Davey) is false—but does not warrant his overall conclusion.

In fact, there is another general notion of a probability principle that does abide by the standards of robustness discussed above and is less vulnerable to Davey’s criticism. In the following paragraphs I will present it. But first, notice that for my purposes it is important only that this new notion of a probability principle be a plausible one. My claim—that without modification to Davey’s notion of probability principle, his argument is not correct—is wholly independent of whether this alternative type of probability principle fully succeeds in justifying the second law.

Probability principle∗.—Recall that a probability principle emphasizes the relation between a particular macrostate and a corresponding probability measure. This relationship is central to Davey’s argument. In fact, in rejecting any such principle, Davey stresses this one-to-one relation by claiming that “it is possible for us to know the current macrostate of a system, without having any grounds for believing that any particular probability measure is in fact the correct probability measure with which to describe the microstate of the system. [Consequently,] all Probability Principles are false” (Davey Reference Davey2008, 34).Footnote ⁶ Davey is apparently demanding that we restrict our knowledge to the current macrostate of a system and that we then try to deduce a particular probability measure. But why must we accept such a narrow and taxing demand? It is much more natural to expand our knowledge base to include the typical behavior of the system (as might be studied in a lab) and then to deduce a general set of probability measures for use in the standard story. Since the goal is to give a justification for typical thermodynamic systems (such as coffee cups or gases in boxes studied in labs), and not systems that have interacted with their observer only once through one observed macrostate, I suggest we redefine the general concept of a probability principle to better reflect the aims of the standard story:

Probability principle∗ is more general than, and is inclusive of, Davey’s notion of a probability principle. In this sense, it abides by the standards of robustness discussed above and can support Davey’s claim that the standard story fails at its stated goal if Davey can show that all probability principle∗ candidates are false. More importantly, since probability principle∗ is a plausible candidate for the standard justificatory story given in BSM, Davey’s argument dissolves into its weaker form (mentioned above), which does not warrant his final conclusion that the standard story fails in its intended goal. Of course, Davey is correct that we cannot infer the probabilistic behavior of a system just based on its current macrostate, that is, ignoring the system’s typical behavior and our past experience with such systems. But so what? We cannot get probabilities without putting something probabilistic in. Thus, I will charitably assume that what Davey had in mind for a general notion of a probability principle is probability principle∗.

To be clear, probability principle∗ is objectionable: there is no doubt that many details need to be added, and the ultimate success of the justificatory story remains an open question.Footnote ⁷ What I want to emphasize is that the burden of defending probability principle∗ is not mine. The point is that Davey’s final conclusion is not warranted unless (1) he replaces probability principle with probability principle∗ or (2) shows that the approach I have outlined here fails.

In the next section, I present a kind of reductio ad absurdum, which shows that if Davey’s complaints were correct, then it would be a problem for many other physical sciences, and not just for BSM.

4. Beyond Statistical Mechanics

When Davey’s argument and its corresponding standard of justification are extended beyond BSM to other physical sciences, the results are equally catastrophic. The point is that there is nothing special about his argument that deems it uniquely problematic for BSM. The essential ingredients needed to apply his arguments are the following:

1. I. Identify a system that can be mathematically represented from two different perspectives, one a “higher-level” (or “macroscopic”) representation and the other a “lower-level” (“fundamental” or “microscopic”) representation.

2. II. Demand that the higher-level representation supervene on the lower-level representation so that a lower-level state corresponds to a unique higher-level state but not vice versa.

3. III. Posit some probability measure over a set of lower-level states and derive from it simple higher-level lawlike behavior through the implementation of an appropriate limit (e.g., thermodynamic, hydrodynamic, etc.).

In our case the higher-level representation is thermodynamics, and the higher-level states are the macrostates defined by the thermodynamic variables. The lower-level representation is BSM and the lower-level states are points in a 6N − 1–dimensional phase space—a microstate—representing the positions and momenta of all N particles composing the system. The higher-level law to be derived from the probability measure postulated over the possible microstates of the system is the second law of thermodynamics.

I submit that the representative situation described above is not unique to thermodynamics and BSM but occurs in many areas of science such as in ecosystems, economies, chemical reactions, and so on.Footnote ⁸ For reasons of brevity, I will illustrate the wide applicability of the representative situation through one example from classical mechanics, but I hope it is clear that Davey’s argument can be generalized beyond both BSM and classical mechanics. The consequence is that in Davey’s view, whenever a higher-level law is derived from a probability distribution (justifiable only by a probability principle) over the possible lower-level representations, such a law lacks a principled justification.

The Galton board.—Consider the following situation. A scientist presents us with a “Galton board”—a board with pins distributed symmetrically in the shape of a truncated isosceles triangle, on which balls carry out various trajectories and form an approximately Gaussian distribution at the bottom of the board.Footnote ⁹ On the topmost middle pin sits a ball, held in place. The dynamics of the board are governed by deterministic classical mechanics.Footnote ¹⁰ That is, specifying the exact initial position of the ball at one time singles out a unique corresponding trajectory for the ball. Let us call this exact initial position the “microstate” of the ball. Accordingly, “macrostate” will denote that ball’s inexact initial position, labeled by M, as it is described by statements such as “on the topmost middle pin sits a ball.”

A scientist tells us that at a time t ₀ he will release the ball. We are challenged to infer, just from the macrostate M of the ball, the probability measure describing the possible exact initial conditions, or “microstates,” of the ball. We happen to know that there exists a probability principle that tells us that the correct probability measure to use is the uniform Lebesgue measure over those microstates compatible with the ball being in macrostate M. Moreover, we know that such a probability measure entails a particular higher-level law (or lawlike behavior): the ball’s trajectory will be that of a random walk. That is, the frequency of times the ball will be deflected to the right when it hits a pin will be approximately equal to the number of times the ball is deflected to the left. Thus, given our probability principle, we think that the following belief is justified:

However, suppose the scientist tells us that once he releases the ball from its macroscopic state M, he will make a choice between

and

The scientist will not tell us which option he chose, and knowing that the system is presently in its given macrostate does not help us deduce which choice he will make. The scientist does tell us, however, that 1 hour ago he selected by some unknown procedure a real number with and that he then devised some random process that guaranteed that he will choose option A with probability and option B with probability . For no value of are we justified in believing that is the probability the scientist will choose. So any probability principle, including the probability principle used above, is clearly false. For example, it just might be that 99% of the time our scientist will choose option B. Hence the probability measure that we selected was unjustified and the higher-level law derivable from such a measure is also unjustified.

The take-home message is that Davey’s argument against probability principles and the justification of the higher-level law derived from such principles can be extended beyond BSM to situations governed by other physical sciences, such as classical mechanics, as long as those situations can be represented by both higher- and lower-level representations as specified above in propositions I–III. It is in this sense that Davey’s argument against probability principles commits him to a form of skepticism that is more radical than the position he claims to be stating. What we are left with is either to accept Davey’s argument and conclude that many higher-level laws lack a principled justification or instead to attempt to identify features in the argument representing an unreasonable demand of the notion of justification. In the following section I do the latter, but it is first worthwhile to clarify the above example in light of I–III.

In the case of the Galton board example, the higher- and lower-level representation, or “macro-” and “micro-level” representation, is classical mechanics. The use of “macrostate” and “microstate” here is not standard. The higher-level states, or “macrostates,” are the initial pin positions on which the ball can be placed, corresponding to a plethora of possible trajectories. The lower-level states, or “microstates,” are the possible exact initial conditions of the ball, each corresponding to one unique trajectory. The higher-level law to be derived from the probability measure postulated over the possible lower-level states is that a single ball’s trajectory, with very high probability, will be that of a random walk. It is this law that remains lacking a justification given Davey’s account.

5. Unreasonable Demands on the Notion of Justification

The thought experiment Davey uses to argue that all probability principles are false sets certain standards for the notion of justification in BSM. One can even think of the thought experiment as a sort of test that any probability principle must pass in order to justify a probability measure chosen to describe some system. I will call this the “Davey test.”Footnote ¹¹

The key feature that renders the Davey test impassable is that there exist no grounds for attributing a justified value to , the probability with which the scientist chooses option A. This is so because it is impossible to form any sort of justified belief about the scientist as well as facts about the probability with which the scientist chooses between options A and B. But what exactly is the scientist in the Davey test supposed to correspond to (or represent) in real systems typically studied by BSM? It surely cannot be that the scientist represents an intervening intelligent agent or a Maxwellian-type demon because it would be unreasonable to demand of BSM to make probabilistic statements about the behavior of such agents.Footnote ¹² Instead, it seems fair to assume that the scientist is supposed to represent those procedures that bring about the initial and boundary conditions of a system and their probability of occurrence. In particular, the scientist is allowed to force the system into one of two different initial macroscopic states, option A or B. So there is good reason to think of the scientist as representing those procedures that bring about unknown macroscopic initial conditions and their probabilities. But here’s the rub: With the kind of systems typically studied by BSM, one usually has access to such procedures. This is so because an agent, such as a scientist in a lab, is responsible for creating the initial macroscopic conditions of a system, along with the corresponding probabilities. Thus, the typical behavior of said systems is known. Accordingly, it is then possible to attribute correct probability values to options A and B in Davey’s thought experiment and to pass the Davey test.

However, what are we to do with systems in which initial macroscopic conditions really are unknown—where it is impossible to assign correct probabilities to options A and B unless it is done accidentally? For brevity’s sake, let me call these systems “problematic systems.” Is it the case that in such systems Davey’s argument goes through and the standard justificatory story fails? I think not, for it is unreasonable to demand that BSM make any sort of accurate claims about the conditions that bring about problematic systems. In other words, contra Davey, probabilistic systems do not fall within the scope of BSM’s account of thermodynamic asymmetries.

My reasoning is as follows. Broadly speaking, there are two situations in which problematic systems arise. The procedures that bring about problematic systems are either (a) completely unknown and uncontrolled influences, external to the system being studied, or (b) known but uncontrolled influences, external to the system being studied, which are not able to be correctly modeled by any current science.

For an example of the first, consider the Big Bang. We do not know those procedures that brought about the initial conditions of the universe, and so it is senseless to talk about their probabilities. Yet the failure of justification is not unique to BSM or the second law but is due to the extremely rare epistemic situation we are in.

For an example of the second, consider the scientist in Davey’s thought experiment. We know that the scientist is responsible for attributing probabilities to options A and B, but we lack a science that accurately models and predicts the behavior of the scientist. Presumably, if some human behavior sciences were advanced enough, we might be able to predict the scientist’s behavior in such a manner as to assign justified probabilities to options A and B. In our case, we do not have a science that correctly models the conditions that bring about problematic systems, but this is no fault of BSM. In both cases, the failure of justification is due to reasons external to BSM.

That is to say, we have two options. First, we might have access to the macroscopic initial conditions of a system and their frequency of occurrence, as one would in a lab, for example, in which case Davey’s argument does not go through. Second, we might not have access to such conditions, as is the case for problematic systems that come about by procedures such as a or b, in which case it is unreasonable to demand the standard story make any probabilistic claims about such conditions.

To put things differently, if one could give a Davey-style thought experiment to argue that some probability principle is wrong, which does not make use of a notion of justification corresponding to problematic systems, then Davey would be correct. But I conjecture that if we adopt probability principle∗, this will not be possible. Alternatively, one can bite the bullet and admit that in peculiar situations, corresponding to procedures a and b, the standard story fails and the generalized notion of a probability principle is jeopardized. Still, the majority of systems studied by BSM will not be vulnerable to Davey’s account.

In conclusion, the standard story for the justification of the second law of thermodynamics—a principled justification for the second law—can fail for various reasons. Some of those reasons, for example, include that realistic systems typically studied by BSM do not abide by the dynamical conditions needed to make the standard story work (see Earman and Redei Reference Earman and Redei1996). These reasons are legitimate. Accordingly, further work must be done in order to identify which dynamical conditions are needed to ensure that entropic behavior is obeyed and to show that realistic systems actually do conform to such dynamical constraints.Footnote ¹³ Other reasons for the failure of the standard story include too high a justificatory standard being set on BSM, such as in the case of problematic systems in which the failure of justification is neither unique to BSM nor internal to the theory itself. My suggestion is that such standards of justification be rejected.