Condensate Fluctuations in Trapped Bose Gases: Canonical vs. Microcanonical Ensemble

TL;DR

This paper analyzes particle number fluctuations in ideal Bose-Einstein condensates within canonical and microcanonical ensembles, deriving formulas that connect fluctuations to the properties of the trapping potential and exploring specific cases.

Contribution

It introduces a novel analytical approach using Mellin-Barnes transformation to relate condensate fluctuations to the poles of a Zeta function, providing new insights into ensemble differences.

Findings

Derived simple expressions linking fluctuations to Zeta function poles.

Explored microcanonical fluctuations in 1D and 3D harmonic traps.

Established a connection between partition theory and Bose-Einstein statistics.

Abstract

We study the fluctuation of the number of particles in ideal Bose-Einstein condensates, both within the canonical and the microcanonical ensemble. Employing the Mellin-Barnes transformation, we derive simple expressions that link the canonical number of condensate particles, its fluctuation, and the difference between canonical and microcanonical fluctuations to the poles of a Zeta function that is determined by the excited single-particle levels of the trapping potential. For the particular examples of one- and three-dimensional harmonic traps we explore the microcanonical statistics in detail, with the help of the saddle-point method. Emphasizing the close connection between the partition theory of integer numbers and the statistical mechanics of ideal Bosons in one-dimensional harmonic traps, and utilizing thermodynamical arguments, we also derive an accurate formula for the fluctuation of the number of summands that occur when a large integer is partitioned.