Thermodynamic identities and particle number fluctuations in weakly interacting Bose--Einstein condensates

TL;DR

This paper derives exact thermodynamic identities for weakly interacting Bose gases, enabling analytical predictions of particle number fluctuations across ensembles, revealing normal fluctuation behavior contrary to the ideal Bose gas.

Contribution

It introduces auxiliary partition functions to relate particle fluctuations in different ensembles and provides the first analytical prediction of microcanonical fluctuations in weakly interacting Bose gases.

Findings

Particle number fluctuations are normal, scaling linearly with volume.

Analytical predictions match numerical simulations for weakly interacting gases.

Fluctuation behavior differs fundamentally from the ideal Bose gas case.

Abstract

We derive exact thermodynamic identities relating the average number of condensed atoms and the root-mean-square fluctuations determined in different statistical ensembles for the weakly interacting Bose gas confined in a box. This is achieved by introducing the concept of {\it auxiliary partition functions} for model Hamiltonians that do conserve the total number of particles. Exploiting such thermodynamic identities, we provide the first, completely analytical prediction of the microcanonical particle number fluctuations in the weakly interacting Bose gas. Such fluctuations, as a function of the volume V of the box are found to behave normally, at variance with the anomalous scaling behavior V^{4/3} of the fluctuations in the ideal Bose gas.