Equivalence and Nonequivalence of Ensembles: Thermodynamic, Macrostate, and Measure Levels

Abstract

We present general and rigorous results showing that the microcanonical and canonical ensembles are equivalent at all three levels of description considered in statistical mechanics—namely, thermodynamics, equilibrium macrostates, and microstate measures—whenever the microcanonical entropy is concave as a function of the energy density in the thermodynamic limit. This is proved for any classical many-particle systems for which thermodynamic functions and equilibrium macrostates exist and are defined via large deviation principles, generalizing many previous results obtained for specific classes of systems and observables. Similar results hold for other dual ensembles, such as the canonical and grand-canonical ensembles, in addition to trajectory or path ensembles describing nonequilibrium systems driven in steady states.

Appendices

Appendix 1: Concavity of the Entropy

Let $s{\text {:}}\,\mathbb {R}\rightarrow \mathbb {R}\cup \{-\infty \}$ be a real function with domain ${\text {dom}}\,s,$ and consider the inequality

$$\begin{aligned} s(v)\le s(u)+\beta (v-u),\quad v\in \mathbb {R}. \end{aligned}$$

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The set of all $\beta \in \mathbb {R}$ for which this inequality is satisfied is called the subdifferential set or simply the subdifferential of s at u and is denoted by $\partial s(u).$ ⁸ The interpretation of this inequality is shown in Fig. 1: if it is possible to draw a line passing through the graph of $s(u)$ which is everywhere above s, then $\partial s(u)\ne \emptyset .$ In this case, we also say that s admits a supporting line at u, which is unique if s is differentiable at u. If $\partial s(u)=\emptyset ,$ then s admits no supporting line at u.

It is easy to see geometrically that nonconcave points of s do not admit supporting lines, while concave points have supporting lines, except possibly if they lie on the boundary of ${\text {dom}}\,s;$ see Sect. 24 of [43] or Appendix A of [44]. The reason for possibly excluding boundary points arises because $s(u)$ may have diverging ‘slopes’ where $\partial s(u)$ is not defined, as in the following example adapted from [43, p. 215]:

$$\begin{aligned} s(u)=\left\{ \begin{array}{lll} \sqrt{1-x^{2}} &{} &{} |x|\le 1\\ -\infty &{} &{} \mathrm{otherwise}. \end{array} \right. \end{aligned}$$

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In this case, $s(u)=s^{**}(u)$ for all $u\in {\text {dom}}\,s=[-1,\,1],$ so that s is a concave function, but it has supporting lines only over $(-1,\,1)=\mathrm{int}\ ({\text {dom}}\,s),$ since $s^{\prime }(u)$ diverges as $u\rightarrow \pm 1$ from within its domain. All cases of concave points with no supporting lines are of this type, since it can be proved in $\mathbb {R}$ that

$$\begin{aligned} \mathrm{int}\ ({\text {dom}}\,s)\subseteq {\text {dom}}\,\partial s\subseteq {\text {dom}}\,s; \end{aligned}$$

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see again Sect. 24 of [43] or Appendix A of [44].

With this proviso on boundary points, s is often defined to be strictly concave at u if it admits supporting lines at u that do not touch other points of its graph. If s has a supporting line at u touching other points of its graph, then s is said to be non-strictly concave at s. Finally, if s admits no supporting line at u, then s is said to be nonconcave at u. These definitions are also illustrated in Fig. 1. For generalizations of these definitions to $\mathbb {R}^{d}$ in terms of supporting hyperplanes, see [43] and Appendix A of [44].

Appendix 2: Varadhan’s Theorem and the Laplace Principle

We recall in this section two important results about Laplace approximations of exponential integrals in general spaces. In the following, $\{a_{n}\}_{n=1}^{\infty }$ is an increasing sequence such that $a_{n}\nearrow \infty$ when $n\rightarrow \infty .$ Moreover, $\{P_{n}\}_{n=1}^{\infty }$ is a sequence of probability measures defined on a (Polish) space $\mathcal {X}.$ In this paper, N takes the role of $a_{n}$ and n.

Theorem 13

(Varadhan, 1966 [42]) Assume that $P_{n}(dx)$ satisfies the LDP with speed $a_{n}$ and rate function I on $\mathcal {X}.$ Let F be a continuous function.

• (a) (Bounded case) assume that $\sup _{x} F(x)<\infty .$ Then

  $$\begin{aligned} \lim \limits _{n\rightarrow \infty } \frac{1}{a_{n}} \ln \int \nolimits _{\mathcal {X}} e^{a_{n} F(x)} P_{n}(dx) =\sup \limits _{x\in \mathcal {X}} \{F(x)-I(x)\}<\infty . \end{aligned}$$

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• (b) (Unbounded case) assume that F satisfies

  $$\begin{aligned} \lim \limits _{L\rightarrow \infty } \lim \limits _{n\rightarrow \infty } \frac{1}{a_{n}}\ln \int \nolimits _{\{F\ge L\}} e^{a_{n} F(x)} P_{n}(dx)=-\infty . \end{aligned}$$

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  Then the result of (a) holds and is finite. In particular, if F is bounded above on the support of $P_{n},$ then (a) holds.

For a proof of this result, see the Appendix B of [34] or Theorem 4.3.1 in [38]. For historical notes on this result, see Sect. 3.7 of [37].

Consider now the exponentially tilted probability measure

$$\begin{aligned} P_{n,F}(dx) = \frac{e^{a_{n} F(x)} P_{n}(dx)}{W_{n,F}}, \end{aligned}$$

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where

$$\begin{aligned} W_{n,F}=\int \nolimits _{\mathcal {X}} e^{a_{n}F(x)} P_{n}(dx)=E_{P}\left[ e^{a_{n}F(X)}\right] . \end{aligned}$$

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This is also known as the exponential family or Esscher transform of $P_{n}.$

Theorem 14

(LDP for tilted measures) Assume that $W_{n,F}<\infty .$ Then $P_{n,F}$ satisfies the LDP with speed $a_{n}$ and rate function

$$\begin{aligned} I_{F} (x)=I(x)-F(x)+\lambda _{F}, \end{aligned}$$

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where

$$\begin{aligned} \lambda _{F}=\lim \limits _{n\rightarrow \infty }\frac{1}{a_{n}}\ln W_{n,F}. \end{aligned}$$

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A proof of this result can be found in Theorem 11.7.2 of [34] or by combining Proposition 3.4 and Theorem 9.1 of [35]. A thermodynamic version of this result also appears in Theorem 4.1 of [7].