On the entropy minimization problem in Statistical Mechanics

TL;DR

This paper rigorously analyzes the entropy minimization problem in Statistical Mechanics for different particle distributions, providing a complete study for Maxwell--Boltzmann entropy using recent convex function series results.

Contribution

It offers a rigorous mathematical treatment of entropy minimization in Statistical Mechanics, especially for Maxwell--Boltzmann entropy, extending prior formal methods.

Findings

Rigorous formulation of entropy minimization for Bose--Einstein, Fermi--Dirac, and Maxwell--Boltzmann entropies.

Complete analysis of the Maxwell--Boltzmann entropy case.

Application of recent convex function series results to statistical mechanics problems.

Abstract

In the works on Statistical Mechanics and Statistical Physics, when deriving the distribution of particles of ideal gases, one uses the method of Lagrange multipliers in a formal way. In this paper we treat rigorously this problem for Bose--Einstein, Fermi--Dirac and Maxwell--Boltzmann entropies and present a complete study in the case of the Maxwell--Boltzmann entropy. Our approach is based on recent results on series of convex functions.