Ensemble equivalence for distinguishable particles

TL;DR

This paper addresses the non-physical results in the statistical mechanics of distinguishable particles, proposing a new entropy and partition function definition that restores ensemble equivalence and resolves inconsistencies.

Contribution

It introduces a revised formalism for distinguishable particles that ensures ensemble equivalence and resolves issues of non-extensivity and large fluctuations.

Findings

The standard partition function leads to non-physical fluctuations.

Swendsen's entropy proposal restores ensemble equivalence.

The new formalism is consistent for localized particles.

Abstract

Statistics of distinguishable particles has become relevant in systems of colloidal particles and in the context of applications of statistical mechanics to complex networks. When studying these type of systems with the standard textbook formalism, non-physical results such as non-extensive entropies are obtained. In this paper, we will show that the commonly used expression for the partition function of a system of distinguishable particles leads to huge fluctuations of the number of particles in the grand canonical ensemble and, consequently, to non-equivalence of statistical ensembles. We will see how a new proposed definition for the entropy of distinguishable particles by Swendsen [J. Stat. Phys. 107, 1143 (2002)] solves the problem and restores ensemble equivalence. We also show that the new proposal for the partition function does not produce any inconsistency for a system of distinguishable localized particles, where the monoparticular partition function is not extensive.