Variational principle for lattice pressure

Statement

Let $\Phi$ be a translation-invariant, uniformly absolutely summable interaction on $\mathbb{Z}^d$ with finite spin space $S$, and let $p(\Phi)$ denote the lattice pressure , defined via the thermodynamic limit of the lattice partition function (e.g. along a van Hove sequence).

Then the pressure satisfies the variational formula

where:

• $\mathcal{P}_{\mathrm{t.i.}}(\Omega)$ is the set of translation-invariant probability measures on $\Omega=S^{\mathbb{Z}^d}$,
• $s(\mu)$ is the specific entropy (entropy per site; a mean version of Shannon entropy ),
• $e_\Phi(\mu)$ is the specific energy (energy per site) computed from the interaction energy .

Furthermore, $\mu$ achieves the supremum if and only if $\mu$ is a translation-invariant infinite-volume Gibbs measure for $\Phi$ (equivalently, $\mu$ satisfies the DLR equation and is translation-invariant).

This is the lattice analogue of the Gibbs variational principle .

Key hypotheses

• Interaction: $\Phi$ is translation-invariant and uniformly absolutely summable (finite range is enough).
• Spin space: $S$ is finite (extensions exist under compactness/regularity assumptions).
• Thermodynamic limit: the pressure $p(\Phi)$ exists as the van Hove limit of $(1/|\Lambda|)\log Z_\Lambda$.

Key conclusions

• Pressure as an optimization problem: $p(\Phi)$ is the supremum of (entropy density) minus (energy density) over translation-invariant states.
• Characterization of equilibrium states: the maximizers are precisely the translation-invariant Gibbs (DLR) states.
• Convex-analytic viewpoint: $p(\Phi)$ functions as a “log-partition” density; the variational formula is a Fenchel–Legendre-type duality with entropy (compare Legendre transform and Fenchel conjugate ).

Proof idea / significance (sketch)

A standard route is via relative entropy and the finite-volume Gibbs inequality (a.k.a. Gibbs variational inequality). For any probability law $\rho_\Lambda$ on configurations in $\Lambda$, one compares $\rho_\Lambda$ with the finite-volume Gibbs law and rewrites the inequality in terms of the relative entropy $D(\rho_\Lambda\|\mu_\Lambda^\eta)\ge 0$. This yields a finite-volume bound of the form “$(1/|\Lambda|)\log Z_\Lambda$ is at least entropy density minus energy density,” and taking van Hove limits produces the lower bound $p(\Phi)\ge \sup(\cdot)$. The matching upper bound follows from consistency/subadditivity arguments for entropy and additivity of energy, together with approximation of an infinite-volume $\mu$ by its finite-volume marginals.

The theorem is the main bridge between:

• the “partition function / pressure” side (partition function , pressure ), and
• the “infinite-volume equilibrium state” side (Gibbs measures ).