Statistical mechanics, structure of matter Foundations of equilibrium statistical mechanics Theorem

Equivalence of ensembles

Builds on Microcanonical ensemble Canonical ensemble Thermodynamic limit Legendre transformation

Statement

Let $S$ be a finite set, and let $\Phi = (\Phi_A)$ be a translation-invariant interaction on $\Z^d$ . Here $A$ runs over the finite subsets of $\Z^d$ and $\Phi_A$ is a real function of the spins in $A$ . Assume $\Phi$ is absolutely summable: $\sum_{A \ni 0} \norm{\Phi_A}_\infty < \infty$ . Let $\Lambda_n = \set{-n, \dots, n}^d$ and $H_n(\sigma) = \sum_{A \subseteq \Lambda_n} \Phi_A(\sigma)$ for $\sigma \in S^{\Lambda_n}$ . Then the following limits exist: $p(\beta) = \lim_{n \to \infty} \frac{1}{\abs{\Lambda_n}} \ln \sum_{\sigma \in S^{\Lambda_n}} e^{-\beta H_n(\sigma)}, \qquad \beta \in \R,$ $s(e) = \lim_{\delta \downarrow 0} \lim_{n \to \infty} \frac{1}{\abs{\Lambda_n}} \ln \# \set{\sigma \in S^{\Lambda_n} : \abs{H_n(\sigma) / \abs{\Lambda_n} - e} < \delta}, \qquad e \in \R,$ with $\ln 0 = -\infty$ . The microcanonical entropy density $s$ is concave and upper semicontinuous, and it is related to the canonical pressure $p$ by $p(\beta) = \sup_{e \in \R} \big( s(e) - \beta e \big), \qquad s(e) = \inf_{\beta \in \R} \big( p(\beta) + \beta e \big).$

Remarks

Proof idea: approximate additivity of $\ln$ of counts over disjoint boxes, up to boundary terms, gives the existence of the limits and the concavity of $s$ . The first identity is Laplace's method: the sum defining $Z$ is dominated by the energies that maximize $s(e) - \beta e$ . The second follows from the first by the duality of the Legendre transformation for concave upper semicontinuous functions. In large deviations language, $\ln \abs{S} - s(e)$ is the rate function of the energy per site under the uniform measure, and the first identity is Varadhan's lemma. Example: for independent spins with $\Phi_{\set{x}}(\sigma) = -h \sigma_x$ and $S = \set{-1, +1}$ , one has $p(\beta) = \ln (2 \cosh \beta h)$ , and $s$ is the binary entropy of the fraction of up spins. Equivalence at the level of measures, rather than of thermodynamic functions, needs extra hypotheses and can fail at first-order phase transitions.

Going deeper

For a student

The two thermodynamic functions are Legendre transforms of each other, as in classical thermodynamics. So one may compute in the most convenient ensemble, usually the canonical one, and translate. Where $s$ has a linear piece, a whole interval of energies corresponds to one value of $\beta$ ; this is the signature of a first-order phase transition with phase coexistence. For systems with long-range interactions, such as gravitating particles, $s$ need not be concave and the ensembles can genuinely differ.

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Tag	`8N7VY25V0E`
URI	`m2://8N7VY25V0E`
Kind	statement, theorem
Writing	W0 stub
MSC	82B05, 82B20, 60F10
Status	This copy of the site has no build data. Run `m2 status 8N7VY25V0E`.
About	Microcanonical ensemble, Canonical ensemble
Uses	Microcanonical ensemble, Canonical ensemble, Thermodynamic limit, Legendre transformation
Changed	3 November 2026