Central to statistical mechanics is the notion of a state space. A state space is a space of possible states of the world at a time. All of the possibilities in this space are alike with regards to certain static properties, such as the spatiotemporal dimensions of the system, the number of particles, and the masses of these particles. The individual elements of this space are picked out by certain dynamic properties, the locations and momenta of the particles. In versions of classical statistical mechanics like that proposed by David Albert, one of the statistical mechanical laws is a constraint on the initial entropy of the universe. On such theories the space of classical statistical mechanical worlds (and the state spaces that partition it) will only contain worlds whose initial macroconditions are of a suitably low entropy.
In classical statistical mechanics these static and dynamic properties determine the state of the world at a time. Classical mechanics is deterministic: the state of the world at a time determines the history of the system. Earman, Xia, Norton and others have offered counterexamples to the claim that classical mechanics is deterministic. Two comments are in order, however. First, these cases spell trouble for the standard distinction between dynamic and static properties employed by classical statistical mechanics. For example, in some of these cases new particles will unpredictably zoom in from infinity. Since this leads to a change in the number of particles in the system, it would seem the number of particles cannot be properly understood as a static property. Second, although no proof of this exists, prevailing opinion is that these indeterministic cases form a set of Lebesgue measure zero. So in classical statistical mechanics each point in the state space corresponds to a unique history, i.e., to a possible world. We can therefore take a state space to be a set of possible worlds, and the state space and its subsets to be propositions. The state spaces form a partition of the classical statistical mechanical worlds, dividing the classical statistical mechanical worlds into groups of worlds that share the relevant static properties.
Given a state space, we can provide the classical statistical mechanical probabilities. Let m be the Liouville measure, the Lebesgue measure over the canonical representation of the state space, and let K be a subset of the state space. The classical statistical mechanical probability of A relative to K is m(A∩K)/m(K). Note that statistical mechanical probabilities aren’t defined for all object propositions A and relative propositions K. Given the above formula, two conditions must be satisfied for the chance of A relative to K to be defined. Both m(A ∩ K) and m(K) must be defined, and the ratio of m(A ∩ K) to m(K) must be defined.
Despite the superficial similarity, the statistical mechanical probability of A relative to K is not a conditional probability. If it were, we could define the probability of A ‘simpliciter’ as m(A), and retrieve the formula for the probability of A relative to K using the definition of conditional probability. The reason we can’t do this is that the Liouville measure m is not a probability measure; unlike probability measures, there is no upper bound on the value a Liouville measure can take. We only obtain a probability distribution after we take the ratio of m(A ∩ K) and m(K); since m(A ∩ K) ≤ m(K), the ratio of the two terms will always fall in the range of acceptable values, [0,1].
Now, how should we understand statistical mechanical probabilities? A satisfactory account must preserve their explanatory power and normative force. For example, classical mechanics has solutions where ice cubes grow larger when placed in hot water, as well as solutions where ice cubes melt when placed in hot water. Why is it that we only see ice cubes melt when placed in hot water? Statistical mechanics provides the standard explanation. When we look at systems of cups of hot water with ice cubes in them, we find that according to the Liouville measure the vast majority of them quickly develop into cups of lukewarm water, and only a few develop into cups of even hotter water with larger ice cubes. The explanation for why we always see ice cubes melt, then, is that it’s overwhelmingly likely that they’ll melt instead of grow, given the statistical mechanical probabilities. In addition to explanatory power, we take statistical mechanical probabilities to have normative force: it seems irrational to believe that ice cubes are likely to grow when placed in hot water.
The natural account of statistical mechanical probabilities is to take them to be chances. On this account, statistical mechanical probabilities have the explanatory power they do because they’re chances; they represent lawful, empirical and contingent features of the world. Likewise, statistical mechanical probabilities have normative force because they’re chances, and chances normatively constrain our credences via something like the Principal Principle.
But statistical mechanical probabilities cannot be chances on the Lewisian accounts. First, classical statistical mechanical chances are compatible with classical mechanics, a deterministic theory. But on the Lewisian accounts determinism and chance are incompatible. Second, classical statistical mechanics is time symmetric like the Aharonov, Bergmann and Lebowitz (ABL) theory of quantum mechanics and classical statistical mechanics (generally, the ABL theory assigns chances given pre-measurements, given post- measurements, and given pre and post-measurements), and is incompatible with the Lewisian accounts for similar reasons. Consider two propositions, A and K, where A is the proposition that the temperature of the world at t1 is T1, and K is the proposition that the temperature of the world at t0 and t2 is T0 and T2. Consider the chance of A relative to K. On the Lewisian accounts the arguments of the relevant chance distribution will be the classical statistical mechanical laws and a history up to a time. But a history up to what time? The statistical mechanical laws and this history entail the chance distribution on the Lewisian accounts. The distribution depends on the relative state K, and a history must run up to t2 to entail K, so we need a history up to t2 to obtain the desired distribution. Since the past is no longer chancy, the chance of any proposition entailed by the history up to t2, including A, must be trivial. But the statistical mechanical chance of A is generally not trivial, so the Lewisian account cannot accommodate such chances. Third, the Lewisian restriction of the second argument of chance distributions to histories is too narrow to accommodate statistical mechanical chances. Consider the case just given, where A is a proposition about the temperature of the world at t1 and K a proposition about the temperature of the world at t0 and t2. Consider also a third proposition K′, that the temperature of the world at t0, t1.5 and t2 is T0, T1.5 and T2, respectively. On the Lewisian accounts it looks like the chance of A relative to K and the chance of A relative to K′ will have the same arguments: the statistical mechanical laws and a history up to t2. But for many values of T1.5, statistical mechanics will assign different chances to A relative to K and A relative to K′.
It’s not surprising that the Lewisian account of the arguments of chance distributions is at odds with statistical mechanical chances. It’s natural to take classical statistical mechanics T and the relative state K to be the arguments of statistical mechanical distributions, since T and K alone entail these distributions. But taking T and K to be the arguments conflicts with the Lewisian accounts, since while K can be a history up to a time, often it is not.
So the Lewisian accounts are committed to denying that statistical mechanical probabilities are chances. Instead, they take them to be subjective values of some kind. There’s a long tradition of treating statistical mechanical probabilities this way, taking them to represent the degrees of belief a rational agent should have in a particular state of ignorance. Focusing on classical statistical mechanics, it proceeds along the following lines.
Start with the intuition that some version of the Indifference Principle – the principle that you should have equal credences in possibilities you’re epistemically ‘in- different’ between – should be a constraint on the beliefs of rational beings. There are generally too many possibilities in statistical mechanical cases – an uncountably infinite number – to apply the standard Indifference Principle to. But given the intuition behind indifference, it seems we can adopt a modified version of the Indifference Principle: when faced with a continuum number of possibilities that you’re epistemically indifferent between, your degrees of belief in these possibilities should match the values assigned to them by an appropriately uniform measure. The properties of the Lebesgue measure make it a natural candidate for this measure. Granting this, it seems the statistical mechanical probabilities fall out of principles of rationality: if you only know K about the world, then your credence that the world is in some set of states A should be equal to the proportion (according to the Lebesgue measure) of K states that are A states. Thus it seems we recover the normative force of statistical mechanical probabilities without having to posit chances.
However, this account of statistical mechanical probabilities is untenable. First, the account suffers from a technical problem. The representation of the state space determines the Lebesgue measure of a set of states, and there are an infinite number of ways to represent the state space. So there are an infinite number of ways to ‘uniformly’ assign credences to the space of possibilities. Classical statistical mechanics uses the Lebesgue measure over the canonical representation of the state space, the Liouville measure, but no compelling argument has been given for why this is the right way to represent the space of possibilities when we’re trying to quantify our ignorance. So it doesn’t seem that we can recover statistical mechanical probabilities from intuitions regarding indifference after all.
Second, the kinds of values this account provides can’t play the explanatory role we take statistical mechanical probabilities to play. On this account statistical mechanical probabilities don’t come from the laws. Rather, they’re a priori necessary facts about what it’s rational to believe when in a certain state of ignorance. But if these facts are a priori and necessary, they’re incapable of explaining a posteriori and contingent facts about our world, like why ice cubes usually melt when placed in hot water. Furthermore, as a purely normative principle, the Indifference Principle isn’t the kind of thing that could explain the success of statistical mechanics. Grant that a priori it’s rational to believe that ice cubes will usually melt when placed in hot water: that does nothing to explain why in fact ice cubes do usually melt when placed in hot water.
The indifference account of statistical mechanical probabilities is untenable. The only viable account of statistical mechanical probabilities on offer is that they are chances, and the Lewisian theories of chance are incompatible with statistical mechanical chances. A proposal is in place to correct these views. The proposal is to allow the second argument of chance distributions to be propositions other than histories, and to reject the two additional claims about chance the Lewisian theories make: that the past is no longer chancy, and that determinism and chance are incompatible. The two additional claims of the Lewisian theories stipulate properties of chance distributions that are incompatible with time symmetric and deterministic chances; by rejecting these two additional claims, we eliminate these stipulated incompatibilities. By allowing the second argument to be propositions other than histories, we can incorporate the time symmetric arguments needed for theories like the ABL theory and the more varied arguments needed for statistical mechanical theories.
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