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 Xt (t∈Zv) 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 Xt 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.
Highlights
Statistical mechanics studies models of physical systems with many particles, which interact with an external force and with one another
Kachapova and Kachapov (2016) introduced the concept of interaction model as a generalization of some existing models; there we provided a proof based on this concept that the random field Xt transformed by renormalization group converges to an independent random field with Gaussian distribution
We study properties of the interaction model and show how some well-known models are represented as particular cases of the interaction model
Summary
Statistical mechanics studies models of physical systems with many particles, which interact with an external force and with one another. 2) Xt t ∈ Zν is an independent random field on this probability space
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