The Extended Variational Principle for Mean-Field, Classical Spin Systems

TL;DR

This paper explores the extended variational principle (EVP) for classical mean-field spin systems, clarifying its relation to self-consistent equations and extending its applicability beyond disordered models.

Contribution

It simplifies the EVP for non-random, symmetric Hamiltonians and establishes a new duality with mean-field equations, broadening its scope.

Findings

EVP effectively solves classical mean-field models.

Established duality between EVP and mean-field equations.

Generalized EVP to broader classes of models.

Abstract

The purpose of this article is to obtain a better understanding of the extended variational principle (EVP). The EVP is a formula for the thermodynamic pressure of a statistical mechanical system as a limit of a sequence of minimization problems. It was developed for disordered mean-field spin systems, spin systems where the underlying Hamiltonian is itself random, and whose distribution is permutation invariant. We present the EVP in the simpler setting of classical mean-field spin systems, where the Hamiltonian is non-random and symmetric. The EVP essentially solves these models. We compare the EVP with another method for mean-field spin systems: the self-consistent mean-field equations. The two approaches lead to dual convex optimization problems. This is a new connection, and it permits a generalization of the EVP.