On the theory of ideal Bose-gas at a finite particle number

TL;DR

This paper derives exact and asymptotic expressions for the partition functions and occupation numbers of a finite-particle ideal Bose gas across different ensembles, challenging the common assumption about the chemical potential's range.

Contribution

It provides the first comprehensive analysis of the ideal Bose gas with finite particle number, including exact formulas and a novel insight into the chemical potential's possible values.

Findings

Exact expressions for partition functions and occupation numbers

Asymptotic formulas for large N in canonical and microcanonical ensembles

Chemical potential can be positive, contrary to previous beliefs

Abstract

The ideal Bose-gas with finite number $N$ of particles is investigated. The exact expressions for the partition functions and occupation numbers in the grand canonical, canonical and microcanonical ensembles are found. The asymp\-totic expressions (in the case $N\gg1$) for the partition functions and occupation numbers in the canonical and microcanonical ensembles are evaluated. It is shown that the chemical potential $\mu$ of the ideal Bose-gas can lie in the range $-\infty<\mu<\infty$ oppositely to the widely adopted opinion that the value of this potential is negative.