Bose-Einstein Condensation, Fluctuations, and Recurrence Relations in Statistical Mechanics

TL;DR

This paper uses recurrence relations to accurately analyze Bose-Einstein condensation in ideal gases, highlighting the advantages of the canonical ensemble over the grand canonical ensemble for finite systems.

Contribution

It introduces recurrence relations for the partition function to improve calculations of Bose-Einstein condensation properties in finite systems.

Findings

Canonical ensemble provides more accurate condensate properties than grand canonical ensemble.

Recurrence relations simplify calculations for finite Bose systems.

Permutation cycle analysis offers physical insight into recurrence relations.

Abstract

We calculate certain features of Bose-Einstein condensation in the ideal gas by using recurrence relations for the partition function. The grand canonical ensemble gives inaccurate results for certain properties of the condensate that are accurately provided by the canonical ensemble. Calculations in the latter can be made tractable for finite systems by means of the recurrence relations. The ideal one-dimensional harmonic Bose gas provides a particularly simple and pedagogically useful model for which detailed results are easily derived. An analysis of the Bose system via permutation cycles yields insight into the physical meaning of the recurrence relations.