Generalized Curie-Weiss Model and Quadratic Pressure in Ergodic Theory

TL;DR

This paper bridges statistical mechanics and ergodic theory by analyzing a generalized Curie-Weiss model, revealing finitely many invariant measures that maximize quadratic free energy and their convergence properties.

Contribution

It introduces a generalized Curie-Weiss model within ergodic theory, identifying invariant measures that maximize quadratic free energy and describing their convergence behavior.

Findings

Finitely many invariant measures maximize quadratic free energy.

All maximizing measures are Dynamical Gibbs Measures.

Probabilistic Gibbs measures converge to conformal measures associated with these Gibbs measures.

Abstract

We explain the Curie Weiss model in Statistical Mechanics within the Ergodic viewpoint. More precisely, we simultaneously define in $\{-1,+1\}^{\mathbb{N}}$, on the one hand a generalized Curie Weiss model within the Statistical Mechanics viewpoint and on the other hand, quadratic free energy and quadratic pressure within the Ergodic Theory viewpoint. We show that there are finitely many invariant measures which maximize the quadratic free energy. They are all Dynamical Gibbs Measures. Moreover, the Probabilistic Gibbs measures for generalized Curie Weiss model converge to a determined combination of the (dynamical) conformal measures associated to these Dynamical Gibbs Measures. The standard Curie Weiss model is a particular case of our generalized Curie Weiss model. An Ergodic viewpoint over the Curie Weiss Potts model is also given.