Research Article Open Access

Interaction Model in Statistical Mechanics

Farida Kachapova1 and Ilias Kachapov2
  • 1 Auckland University of Technology, New Zealand
  • 2 University of Auckland, New Zealand

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 (tZv)  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.

Journal of Mathematics and Statistics
Volume 13 No. 4, 2017, 339-346

DOI: https://doi.org/10.3844/jmssp.2017.339.346

Submitted On: 28 August 2017 Published On: 4 November 2017

How to Cite: Kachapova, F. & Kachapov, I. (2017). Interaction Model in Statistical Mechanics. Journal of Mathematics and Statistics, 13(4), 339-346. https://doi.org/10.3844/jmssp.2017.339.346

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Keywords

  • Infinite Particle System
  • Gibbs Measure
  • Radius of Interaction
  • Thermodynamic Limit
  • Ising Model
  • Potts Model