Interaction Model in Statistical Mechanics

Abstract

Statistical mechanics considers several models such as Ising model, Potts model, Heisenberg model etc. A rigorous mathematical approach based on the axiomatic foundation of probability would benefit the study and applications of these models. In this paper we use this approach to generalize some of these models into one construction named an interaction model. We introduce a mathematically rigorous definition of the model on an integer lattice that describes a physical system with many particles interacting with an external force and with one another; a random field X_t (t∈Z^v)  models some property of the system such as electric charge, density etc. We introduce a finite model first and then define the thermodynamic limit of the finite models with Gibbs probability measure. The set of values of X_t can be unbounded for more generality. We study properties of the interaction model and show that Ising and Potts models are particular cases of the interaction model.