Statistical mechanics, structure of matter Rigorous equilibrium theory Theorem

Gibbs variational principle

Builds on Gibbs measure Gibbs entropy Thermodynamic limit Probability measure

Statement

Let $S$ be a finite set, $\Omega = S^{\Z^d}$ , and let $\Phi = (\Phi_A)$ be a translation-invariant interaction on $\Z^d$ . Assume $\Phi$ is absolutely summable: $\sum_{A \ni 0} \norm{\Phi_A}_\infty < \infty$ . Let $\Lambda_n = \set{-n, \dots, n}^d$ and $H_n(\sigma) = \sum_{A \subseteq \Lambda_n} \Phi_A(\sigma)$ . Let $\beta > 0$ , and let the pressure be the thermodynamic limit $P(\beta) = \lim_{n} \abs{\Lambda_n}^{-1} \ln \sum_{\sigma \in S^{\Lambda_n}} e^{-\beta H_n(\sigma)}$ . For a translation-invariant probability measure $\mu$ on $\Omega$ , define:

the mean entropy $s(\mu) = \lim_n -\abs{\Lambda_n}^{-1} \sum_{\sigma \in S^{\Lambda_n}} \mu(\sigma) \ln \mu(\sigma)$ , where $\mu(\sigma)$ is the probability that the spins in $\Lambda_n$ equal $\sigma$ ;
the mean energy $e(\mu) = \int_\Omega \sum_{A \ni 0} \frac{\Phi_A}{\abs{A}} \, d\mu$ .

Then $s(\mu) - \beta e(\mu) \le P(\beta)$ for every translation-invariant $\mu$ . Equality holds if and only if $\mu$ is a Gibbs measure for $\beta \Phi$ , that is, a solution of the DLR equations.

Remarks

The limits defining $P$ and $s(\mu)$ exist, by subadditivity. Proof idea: in finite volume, $\ln Z_n - \big( S_n(\nu) - \beta \langle H_n \rangle_\nu \big)$ is the Kullback–Leibler divergence of a measure $\nu$ from the Boltzmann distribution, so it is nonnegative. Here $S_n$ is the Gibbs entropy with $k_B = 1$ ; dividing by $\abs{\Lambda_n}$ and letting $n \to \infty$ gives the inequality. The converse, that equality forces the DLR equations, is the harder part, and uses a relative entropy argument conditional on the outside of a box. Example: for $\Phi = 0$ , $P(\beta) = \ln \abs{S}$ , and the only maximizer of $s(\mu)$ is the product of uniform measures, which is the only Gibbs measure.

Going deeper

For a student

The principle is the infinite-volume form of the Gibbs–Bogoliubov inequality, and the pressure is the Legendre transform of the entropy as a function of the energy. It connects to dynamics: for shifts, equilibrium states maximize $h_\mu + \int \varphi \, d\mu$ , and the maximum is the topological pressure. When several translation-invariant measures attain the maximum, the system has coexisting phases. The principle is a standard route to proving properties of Gibbs measures through convex analysis.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Where this leads

1 page builds on this one.

Mean-field theory

Tag	`BKWBKNEAKA`
URI	`m2://BKWBKNEAKA`
Kind	statement, theorem
Writing	W0 stub
MSC	82B05, 82B20, 37D35
Status	This copy of the site has no build data. Run `m2 status BKWBKNEAKA`.
About	Gibbs measure
Uses	Gibbs measure, Gibbs entropy, Thermodynamic limit, Probability measure
Changed	3 November 2026

Gibbs variational principle

Statement

Remarks

Going deeper

For a student

Reading path

Where this leads

Further reading

What this page needs for W1

What links here

Used by