A Rigorous Result about Gibbs Measure

Farida Kachapova, Ilias Kachapov

Published 2018-05-27Version 1

Mathematical models in statistical mechanics describe physical systems with many particles interacting with an external force and with one another. Well-known models include the Ising model, Potts model, Heisenberg model, and $XY$ model. Existing literature on these models lacks rigorous mathematical proofs. In this paper we describe a mathematically accurate generalization of these models to one interaction model. An infinite model is constructed as the limiting case of finite models on an integer lattice at high temperature. We give a detailed mathematical proof of existence of a probability measure for the infinite model, that is existence of thermodynamic limit. The proof includes estimates of series of semi-invariants and graph-related estimates.