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

Gibbs–Bogoliubov inequality

Builds on Canonical ensemble Jensen's inequality Expected value

Statement

Let $(\Omega, \nu)$ be a $\sigma$ -finite measure space and let $\beta > 0$ . Let $H, H_0 \colon \Omega \to \R$ be measurable with $Z_0 = \int_\Omega e^{-\beta H_0} \, d\nu \in (0, \infty)$ and $Z = \int_\Omega e^{-\beta H} \, d\nu < \infty$ . Let $\langle \cdot \rangle_0$ be the expected value under the canonical ensemble $e^{-\beta H_0} \, d\nu / Z_0$ , and assume $\langle \abs{H - H_0} \rangle_0 < \infty$ . Let $F = -\beta^{-1} \ln Z$ and $F_0 = -\beta^{-1} \ln Z_0$ be the free energies. Then $F \le F_0 + \langle H - H_0 \rangle_0,$ with equality if and only if $H - H_0$ is constant almost everywhere for the measure $e^{-\beta H_0} d\nu$ .

Remarks

Proof idea: $Z / Z_0 = \langle e^{-\beta (H - H_0)} \rangle_0$ , and Jensen's inequality for the convex function $e^{-\beta x}$ gives $Z / Z_0 \ge e^{-\beta \langle H - H_0 \rangle_0}$ . Taking $-\beta^{-1} \ln$ of both sides gives the claim. The same inequality holds for Hermitian matrices, with $Z = \tr e^{-\beta H}$ and $\langle A \rangle_0 = \tr(e^{-\beta H_0} A) / Z_0$ ; this is the Peierls–Bogoliubov inequality. Example: in mean-field theory one takes $H_0$ to be a sum of independent single-site terms with free parameters. Minimizing the right side over these parameters gives the best such upper bound on $F$ . It gives a variational principle: $F = \min_{H_0} \left( F_0 + \langle H - H_0 \rangle_0 \right)$ , attained at $H_0 = H$ .

Going deeper

For a student

This inequality is the rigorous basis of mean-field and variational approximations, such as the Curie–Weiss treatment of the Ising model. It is equivalent to the nonnegativity of the Kullback–Leibler divergence between the trial ensemble and the true canonical ensemble. It is also the finite-volume form of the Gibbs variational principle, which characterizes Gibbs measures as maximizers of entropy minus energy.

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Tag	`WR3RBEH4QQ`
URI	`m2://WR3RBEH4QQ`
Kind	statement, theorem
Writing	W0 stub
MSC	82B05, 82B10, 82B03
Status	This copy of the site has no build data. Run `m2 status WR3RBEH4QQ`.
About	Canonical ensemble
Uses	Canonical ensemble, Jensen's inequality, Expected value
Changed	3 November 2026