The Curie-Weiss Model

Abstract

In statistical mechanics, a mean-field approximation is often used to approximate a model by a simpler one, whose global behavior can be studied with the help of explicit computations. The information thus extracted can then be used as an indication of the kind of properties that can be expected from the original model. In addition, this approximation turns out to provide quantitatively correct results in sufficiently high dimensions. The Ising model, which will guide us throughout the book, is a classical example of a model with a rich behavior but with no explicit solution in general (the exceptions being the one-dimensional model, see Section 3.3, and the two-dimensional model when h = 0). In this chapter, we consider its mean-field approximation, in the form of the Curie-Weiss model. Although it is an oversimplification of the Ising model, the Curie-Weiss model still displays a phase transition, with distinct behaviors at high and low temperature. It will also serve as an illustration of various techniques and show how the probabilistic behavior is intimately related to the analytic properties of the thermodynamic potentials (free energy and pressure) of the model.