2.1. Equilibrium

We want to ask what it means to describe a system in equilibrium with a heat bath using the canonical distribution. It will be useful to begin by asking what it means to be in equilibrium with a heat bath. (At this point, we take the concept of a heat bath as unproblematic.)

The concept of equilibrium has a disparate set of meanings, and we need to fix on one. Sometimes when we talk about a system being in equilibrium, we mean that the system is currently in a particular macrostate—typically, one of the macrostates described by the formalism of classical equilibrium thermodynamics. I call this a static notion of equilibrium, insofar as it is used to determine whether a system is in equilibrium at a given time. This is not the only way in which we can think of equilibrium. In fact, when we talk about a gas being in equilibrium with a heat bath, we generally do not have a static conception of equilibrium in mind. The gas will exchange energy with the heat bath in such a way that it will occasionally find itself out of static equilibrium. So the following rough definition will suffice for our purposes: a system is in equilibrium with a heat bath if, as it evolves over time, it spends most of its time in the macrostate (or macrostates) associated with static equilibrium, only occasionally fluctuating into nonequilibrium macrostates. This is a temporally extended notion of equilibrium, insofar as saying that a system is in equilibrium with a heat bath in this sense is now to make a claim about the behavior of the system over time.

Because heat baths are supposed to be infinite, it may well turn out that any finite system placed in contact with a heat bath will spend much of its life in states of static equilibrium, if considered over a sufficiently large timescale. Thus, it may well turn out that any finite system that remains in contact with a heat bath indefinitely will be in equilibrium with the heat bath in the sense just defined. This should cause no confusion, so long as it is understood that there are different concepts of equilibrium, and a system in one sort of equilibrium need not be in another type of equilibrium.

2.2. Traditional Interpretations

Let us return to the canonical distribution. According to Gibbs, the state of a system in equilibrium with a heat bath at temperature T is described by the canonical distribution:

where X is a point in phase space, H is the Hamiltonian, Ƶ is the partition function, and $\beta =1/ kT$, where $k$ is Boltzmann's constant. For simplicity, we suppose that the Hamiltonian is time independent and that our system is constrained to lie in some finite volume of physical space. But what does it mean to say of a system that it may be described by the canonical distribution (1)? I shall take it for granted that describing systems in equilibrium with heat baths using the canonical distribution involves making some sort of probabilistic or statistical claim about such systems. But precisely what sort of probabilistic or statistical claim could this be?

One approach is to think of the canonical distribution as giving us epistemic probabilities. In its simplest form, the claim here is that when we know a system is in equilibrium with a heat bath, it is rational to assign the degree of belief $\rho ( X) \vert dX\vert $ to the proposition that the system is in a small region dX at time t, where ρ is the canonical distribution (1). But a claim like this is surely implausible. I might have a justified belief that the number of particles in a gas lies in a certain range (say, between $1.1\times 10^{23}$ and $1.2\times 10^{23}$ particles). In ordinary situations, for each integer in this range I will assign a very low probability to that integer being the exact number of particles in the system. The canonical distribution, however, will assign probability 1 to the claim that the gas has exactly N particles, where N is the actual number of particles described in the Hamiltonian. In this case, the canonical distribution gives a probability of 1 where my epistemic probability is almost 0. A more sophisticated proponent of the epistemic interpretation of statistical mechanics might have something to say about this sort of worry, although whether related worries arise is a question I am happy to leave open.Footnote ¹ For the purpose of this article, I will focus instead on the idea that describing a system with the canonical distribution involves making an objective, nonepistemic probabilistic or statistical claim about the system in question.

More specifically, according to the approaches that I do want to consider, using the canonical distribution to describe a system in equilibrium with a heat bath at temperature T involves making two observations: first, that the system is a member of a particular ensemble (or set of systems) and, second, that it is appropriate (in some sense to be specified) to describe this ensemble using the canonical distribution (1). To pursue this approach, we must first construct an ensemble of systems and then explain in what sense it is appropriate to describe this ensemble with the canonical distribution (1).

There are a few ways to carve up possible ways of proceeding. One thing we might do is try to construct our ensemble out of actually occurring systems with the given Hamiltonian H in equilibrium with a heat bath at a given temperature T. We call this an actualist approach.

Given an ensemble constructed in accordance with this requirement, we might then claim that the statistics of the ensemble are (approximately) given by the canonical distribution. More precisely, we might claim that the proportion of systems in the ensemble occupying a fixed small region dX of phase space is (approximately) equal to $\rho ( X) \vert dX\vert $, where ρ is the canonical distribution. We call this an (approximate) identity requirement.

But the combination of an actualist approach with an approximate identity requirement is not promising at all. Presumably, there are systems that have found themselves in contact with heat baths only a small number of times in the history of the universe. Only a tiny region of their phase space will ever have been instantiated, and perhaps only a tiny region of their phase space will ever be instantiated. For such a system, the approximate identity requirement will typically not hold of any ensemble constructed using the actualist approach.

In response to this, let us try to keep the actualist approach but relax the approximate identity requirement. To motivate our replacement for the approximate identity requirement, consider the case of an ordinary coin that we can toss as much as we like. Take the following hypothesis.

Suppose now that the coin has been thrown 1,000 times and that it comes up heads two times and tails 998 times. In such a case, we would be inclined to say that Hyp has been strongly disconfirmed. (By strong disconfirmation I mean a type of disconfirmation sufficiently severe to render belief in the given hypothesis to be unjustified.) If the coin came up heads 400 times and tails 600 times, we might say that the hypothesis has been disconfirmed but not strongly so. If the coin came up heads 480 times and tails 520 times, we would probably say that Hyp has not been disconfirmed at all. Exactly how statistical confirmation or disconfirmation works will not be important to us. The important point is that claims about the distribution of an ensemble can sometimes be strongly disconfirmed by observation of a sample.

Just as one might have some sort of procedure for tossing coins, or discovering tossed coins, one might also have some sort of procedure for creating or discovering systems with a prespecified Hamiltonian H in contact with a heat bath at temperature T. In general, we call such a procedure a sampling procedure. Note that a possible sampling procedure for tossed coins might be to take note of all coins in the universe that have just been tossed and list them sequentially. (I am supposing that the set of tossed coins at any moment is finite.) Thus, we do not require that a sampling procedure involves the rejection of otherwise eligible data points, nor do we require that it be a procedure that any human being could actually perform. Note that when a sampling procedure is applied over time, it generates an ensemble. At any moment of time, the ensemble generated thus far is assumed to be finite, even though the ensemble generated over all time may well be infinite. Suppose then that we have a sampling procedure for systems with Hamiltonian H in contact with a heat bath at temperature T. Consider now the hypothesis.

This hypothesis is capable of being disconfirmed—and even strongly disconfirmed—by the finite set of systems generated by our sampling procedure thus far. And so, instead of the approximate identity requirement, we impose the following requirement.

The nondisconfirmation requirement can handle the case of uninstantiated or rarely instantiated systems much better, insofar as we typically do not take hypotheses about distributions to be strongly disconfirmed by very small samples.

The nondisconfirmation requirement is very modest. It may well be too modest to be of use. It is therefore interesting to note that even a requirement this modest gets us into trouble. To see how, let us consider some of the possible sampling procedures we might use to generate our ensemble of systems with Hamiltonian H in contact with a heat bath at temperature T.

Suppose that our sampling procedure tells us to include in our ensemble the microstate of any system with Hamiltonian H in contact with a heat bath at temperature T, but only at the moment the system is first placed in contact with the heat bath. The problem here is that many systems are way out of static equilibrium when first placed in contact with a heat bath. Because of this, the method of sampling just described may consistently produce a disproportionate number of systems in microstates that are extraordinarily rare according to the canonical distribution (1). If this is so, I take it that Hyp will be strongly disconfirmed.

Perhaps it is unfair to consider systems only at the moment they are placed in contact with a heat bath. A problem, however, is that if our sampling procedure tells us to take note of all of the microstates of the system whenever it is in contact with a heat bath, the out-of-static-equilibrium microstates may still be disproportionately represented. (Imagine, e.g., a system with a very long relaxation time, such that up until now, such systems in contact with heat baths at a given temperature have spent almost all of their lives in regions of phase space that are extremely rare according to the canonical distribution.) It will also not do to sample the microstate the moment the system reaches static equilibrium, as the canonical distribution predicts energy fluctuations that will not be present if we sample microstates only with fixed energy. The absence of such energy fluctuations will count as strong disconfirmation of Hyp, particularly in cases in which the occurrence of such fluctuations is moderately probable.

The most promising strategy is to sample the microstate of the system in contact with the heat bath only after some prespecified amount of time—some sort of ‘relaxation time’—appropriate to the system at hand. But I am skeptical as to whether this sort of approach can work. Highly contingent facts about the world, including facts about the way in which we choose to prepare systems, have a way of affecting the composition of any such ensemble. For instance, consider a system that, at a given temperature, has multiple macroscopically distinguishable equilibrium states. Any system with potentially variable anisotropic structure, such as a crystal, will do. For a more artificial example, consider a one-dimensional Ising ring with no external magnetic field, that is, a system of N particles with Hamiltonian

in which each σ _i takes on the value of plus or minus 1, and $\sigma _{N+1}=\sigma _{1}$. At low temperatures, this system organizes itself into large, macroscopic blocks of ones or minus ones. Because the value of the Hamiltonian is unchanged under the transformation $\sigma \rightarrow -\sigma $, for each equilibrium macrostate at a given temperature there is a distinct equilibrium macrostate at the same temperature in which each macroscopic chunk is replaced by one of the opposite sign. At sufficiently low temperatures, these equilibrium macrostates are clearly distinguishable. To make the argument that follows as clean as possible, let us suppose that systems with the given Hamiltonian do not occur naturally but are always manufactured by us. It is perfectly possible for us, having manufactured such a system, to place it in contact with a heat bath at a certain extremely cold temperature T (much lower than that of the environment in which it was produced) if and only if the system is in a particular macrostate—for instance, if and only if the largest macroscopic block consists of ones rather than minus ones.

When placed in contact with the heat bath, such a system is very likely to relax into a particular sort of macrostate—specifically, a macrostate in which it remains the case that the largest macroscopic block consists of ones rather than minus ones. Such a system is very unlikely to fluctuate out of this state if the temperature is very low. As a result, insofar as we sample systems only after this relaxation time, our sample will consist primarily of microstates in a very specific region of phase space—namely, the region of phase space corresponding to this particular sort of macrostate. This ensemble will look extraordinarily unlikely, given the canonical probability distribution. If we place 1,000 such systems in contact with heat baths, and essentially all of them end up with the largest macroscopic block consisting of ones rather than minus ones, it will be as if a fair coin thrown 1,000 times came up heads more or less every time. This is a situation with extraordinarily low probability, and so we would surely have to say that in such a situation Hyp has been strongly disconfirmed.Footnote ²

I think that difficulties like these show that the entire actualist approach is in error. Insofar as our interpretation of the canonical distribution involves identifying an ensemble, I suspect that the ensemble will have to be populated not just with actual physical systems but also with counterfactual systems. This is in agreement with Gibbs, who denied that his ensembles had any sort of objective existence, saying instead that they were creations of the imagination (see Gibbs Reference Gibbs1981, 188). But what counterfactual ensemble should we pick?

The traditional view, championed by Gibbs, was to simply posit an imaginary collection of systems distributed in accordance with the canonical distribution. According to this approach, when one describes a system using the canonical distribution, one is making a claim about an imaginary ensemble of which the particular system is a member. There are many well-known problems with this approach. For instance, the actual system is a member of all sorts of imaginary ensembles—so why privilege the canonical ensemble when it comes to making predictions?Footnote ³ I think the fact that problems like this are so difficult suggests that this way of thinking of the canonical distribution does not get things right and that it makes sense to look for more ‘naturally occurring’ counterfactual ensembles with which to work.

2.3. The Time Ensemble

One proposal seems particularly natural. The rough idea is this—take a system in contact with a heat bath and imagine allowing it to stay in contact with the heat bath indefinitely. To describe this system using the canonical distribution is just to say that the proportion of time the system spends in any small region of phase space dX containing the point X is approximately

In other words, the ensemble we consider is the time ensemble consisting of all the states through which the system would pass were it left in contact with the heat bath indefinitely. This is obviously a counterfactual ensemble, insofar as it involves microstates that may not have been (and may never be) realized.Footnote ⁴ Although we will suggest some slight modifications of this idea shortly, let us focus on the idea in this simpler form first.

The main idea must be formulated with a little care. The most natural way to define a probability density φ on the set of microstates through which our system passes is as follows: focus at first on the set of microstates through which the system passes in a fixed interval of time from $t=0$ to $t=T$. We suppose that initially the system and the heat bath are in some fixed, prespecified state. Let dX be a very small region of phase space—for simplicity, we assume it is a ball centered around a point X of phase space—and let $\tau ( dX) $ be the amount of time that the system spends in the region dX of phase space in the interval of time $[ 0,\, T] $. Our goal is to define a probability distribution φ _T that describes the states of the system in the time interval from zero to T. We would like it to be the case that

and so it is natural to define

where we imagine the radius of the open ball dX going to zero.

The problem is that thus defined, $\phi _{T}( X) =0$ almost everywhere.Footnote ⁵ This is not a useful distribution with which to work. This sort of situation is not unfamiliar. In his Lectures on Gas Theory, Boltzmann (Reference Boltzmann1964) considers the case of a homogeneous gas in a vessel. He defines a function $f( \vec{v},\, t) $—the so-called density of velocities—by requiring $f( \vec{v},\, t) \vert d\vec{v}\vert $ to be the number of particles in a (fixed) region of space whose velocities lie in the region $d\vec{v}$ at time t. Let us suppose we are dealing with a gas consisting of finitely many point particles, and let us suppose that the region $d\vec{v}$ is a sphere centered around a fixed velocity $\vec{v}$. Suppose $N( d\vec{v}) $ is the number of particles in the given spatial region with velocity in the region $d\vec{v}$ (we suppress the argument t for simplicity). Then we would like

and so it is natural to define

But the problem is that for almost all velocities $\vec{v}$, once $d\vec{v}$ is sufficiently small $N( d\vec{v}) =0$, and so $f( \vec{v},\, t) =0$ almost everywhere.

Boltzmann's solution was to think of $d\vec{v}$ as a tiny fixed cell of velocity space centered around $\vec{v}$. Because $d\vec{v}$ is fixed, it is not the sort of thing that can be sent to zero. The equation

is then taken to define the value of $f( \vec{v},\, t) $. Boltzmann's idea was that we should choose our cells $d\vec{v}$ in such a way that the resulting distribution of velocities $f( \vec{v},\, t) $ is more or less continuous—so that when we move from a cell of velocity space to an ‘adjacent’ cell, $f( \vec{v},\, t) $ does not change too much. That it is possible to choose cells' $d\vec{v}$ in this way is a substantive assumption Boltzmann was simply happy to make. So long as this assumption was made, the quantity $f( \vec{v},\, t) $ was well defined enough for Boltzmann's purposes.

We mimic this move. We divide phase space into small cells. Let dX be the cell containing X. We then define

The idea here is to choose the cells in such a way that φ _T is more or less continuous—so that when we move from one of these cells to an adjacent one, $\phi _{T}( X) $ does not change too much. That it is possible to do this is a substantive assumption that we simply take for granted in what follows. So long as this assumption is made, the quantity $\phi _{T}( X) $ is well defined enough to work with.

Given all this, there are various ways in which one might formulate the idea that the canonical distribution describes the time ensemble of states through which the system would evolve were it left in contact with the heat bath. For instance, consider what follows.

Here, we suppose that the system and the heat bath are in fixed, prespecified states at $T=0$. (We leave open the possibility that different coarse grainings may be required at different times T.)

All sorts of variations on this main idea are possible. One of the more important variations involves thinking of the interaction between the heat bath and the system indeterministically, in much the same way that we think of the interactions between a Brownian particle and its environment indeterministically. In this case, the system will not trace out a determinate path through phase space in the interval of time $[ 0,\, T] $—instead, we will have to describe what happens to the system in the interval $[ 0,\, T] $ using a probability measure over paths through phase space.

In this indeterministic setting, there are a number of ways in which one might try to capture the idea that describing a system with the canonical distribution involves making a claim about the set of states through which the system evolves. One way follows. Given a small region of dX of phase space, let $E_{T}[ \tau ( dX) ] $ be the expected value of the amount of time the system spends in the region dX in the interval of time $[ 0,\, T] $. (We suppose that the system is in a fixed, prespecified state at $T=0$.) We then define

For such a system in a particular initial state to be describable by the canonical distribution is for it to be the case that for all points X in phase space,

A different approach is to work with the definition of φ _T given in the deterministic setting and insist instead that

Even this later equation admits of variations. My goal here is not to commit to one particular approach but rather to indicate some possible ways of thinking of the canonical distribution in which time ensembles are central.

Even without committing to a specific definition, some advantages of the general approach sketched here are already visible. According to this approach, to say that a system can be described by the canonical distribution is to make a statistical claim about the time ensemble of microstates through which our system passes. Note that one advantage of working with these definitions is that the focus is on time averages right from the start. If time averages are so important—as the huge amount of literature directed toward understanding the relationship between time and ensemble averages suggests—it seems much more natural to just define the canonical ensemble as a time ensemble of sorts. This is not to suggest that by redefining words we can avoid any of the major philosophical problems with equilibrium statistical mechanics. All the old problems still exist, although many of them will need to be formulated in different language. The hope, however, is that better definitions can give us a better perspective on what these problems are and how deep they run.

Also note that according to our definition, to describe a system with the canonical distribution is to make a claim about the (very) long-term behavior of the system. As such, the values of observables calculated using the canonical distribution are to be thought of as time-averaged expectation values. The time-averaged expectation value of an observable can be quite different from the value of the observable at any moment of time. However, supposing we can show that the variance of the observable over time is small, we can at least assert that the observed value will probably be approximately equal to the time-averaged expectation value.Footnote ⁶ The observed value probably being approximately equal to the time-averaged expectation value is compatible with the observed value being radically different from the time-averaged expectation value. But so long as we realize that the canonical distribution gives us only the probable value of an observable, there is no real problem with the fact that the actual value of the observable might be very different, unless we insist on demanding from the canonical ensemble something that it is simply not designed to give.Footnote ⁷