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

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

2005-04-30

arxiv.org/abs/math-ph/0505001v1

Rev. Math. Phys., 17 (2005) 1209--1239

Physics

Mathematical Physics

30 pages

Scientific paper

10.1142/S0129055X05002510

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.

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