Statistical mechanics of an ideal Bose gas in a confined geometry

TL;DR

This paper investigates the thermodynamic behavior of an ideal Bose gas confined in a finite geometry, analyzing Bose-Einstein condensation and specific heat, and how these properties approach the infinite volume limit.

Contribution

It provides a detailed analysis of Bose gas behavior in confined geometries, including analytical and numerical results on specific heat and chemical potential.

Findings

Bose-Einstein condensation can occur without a phase transition in finite systems.

The specific heat approaches the known infinite-volume result as the system size increases.

The chemical potential approaches zero in the large-volume limit.

Abstract

We study the behaviour of an ideal non-relativistic Bose gas in a three-dimensional space where one of the dimensions is compactified to form a circle. In this case there is no phase transition like that for the case of an infinite volume, nevertheless Bose-Einstein condensation signified by a sudden buildup of particles in the ground state can occur. We use the grand canonical ensemble to study this problem. In particular, the specific heat is evaluated numerically, as well as analytically in certain limits. We show analytically how the familiar result for the specific heat is recovered as we let the size of the circle become large so that the infinite volume limit is approached. We also examine in detail the behaviour of the chemical potential and establish the precise manner in which it approaches zero as the volume becomes large.