Constructing Gibbs Measure in a Rigorous Way

Abstract

Equilibrium statistical mechanics studies mathematical models for physical systems with many particles interacting with an external force and with one another. In this paper we describe an interaction model that generalizes several of these models in one model. An infinite model is constructed as the limiting case of finite interaction models, that is as a thermodynamic limit. The key point in constructing a thermodynamic limit is a proof of existence of the limiting probability measure (Gibbs measure). Traditional proofs use DLR formalism and are quite complicated. Here we explain a more transparent and more constructive proof for the case of high temperatures. The paper provides a detailed, step-by-step rigorous construction of a statistical model and corresponding proofs. The paper also includes a version of the central limit theorem for a random field transformed by a renormalization group, in a special case of the interaction model.