Constructing Gibbs Measure in a Rigorous Way
- 1 Auckland University of Technology, New Zealand
- 2 University of Auckland, New Zealand
Abstract
Equilibrium statistical mechanics studies mathematical models for physical systems with many particles interacting with an external force and with one another. In this paper we describe an interaction model that generalizes several of these models in one model. An infinite model is constructed as the limiting case of finite interaction models, that is as a thermodynamic limit. The key point in constructing a thermodynamic limit is a proof of existence of the limiting probability measure (Gibbs measure). Traditional proofs use DLR formalism and are quite complicated. Here we explain a more transparent and more constructive proof for the case of high temperatures. The paper provides a detailed, step-by-step rigorous construction of a statistical model and corresponding proofs. The paper also includes a version of the central limit theorem for a random field transformed by a renormalization group, in a special case of the interaction model.
DOI: https://doi.org/10.3844/jmssp.2019.308.322
Copyright: © 2019 Farida Kachapova and Ilias Kachapov. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
- 3,326 Views
- 1,340 Downloads
- 0 Citations
Download
Keywords
- Infinite Particle System
- Gibbs Modification
- Radius of Interaction
- Thermodynamic Limit
- Semi-Invariant