III.42 The Ising Model

The Ising model is one of the fundamental models of statistical physics. It was originally designed as a model for the behavior of a ferromagnetic material when it is heated up, but it has since been used to model many other phenomena.

The following is a special case of the model. Let G_n be the set of all pairs of integers with absolute value at most n. A configuration is a way of assigning to each point x in G_n a number σ_x, which equals 1 or -1. The points represent atoms and σ(x) represents whether x has “spin up” or “spin down.” With each configuration σ we associate an “energy” E(σ), which equals - Σ σ_xσ_y, where the sum is taken over all pairs of neighboring points x and y. Thus, the energy is high if many points have different signs from some of their neighbors, and low if G_n is divided into large clusters of points with the same sign.

Each configuration is assigned a probability, which is proportional to e^(-E(σ)/T. Here, T is a positive real number that represents temperature. The probability of a given configuration is therefore higher when it has small energy, so there is a tendency for a typical configuration to have clusters of points with the same sign. However, as the temperature T increases, this clustering effect becomes smaller since the probabilities become more equal.

The two-dimensional Ising model with zero potential is the limit of this model as n tends to infinity For a more detailed discussion of the general model and of the phase transition associated with it, see PROBABILISTIC MODELS OF CRITICAL PHENOMENA [IV.25 §5].