Condensation vs Ordering: From the Spherical Models to BEC in the Canonical and Grand Canonical Ensemble

TL;DR

This paper clarifies the nature of Bose-Einstein condensation in different statistical ensembles, showing it as an ordering transition in the canonical ensemble and a fluctuation condensation in the grand canonical ensemble, highlighting ensemble nonequivalence.

Contribution

It reveals the distinct phenomena of BEC in canonical and grand canonical ensembles using analogies with magnetic models, emphasizing ensemble nonequivalence and fluctuation condensation.

Findings

BEC is an ordering transition in the canonical ensemble.

In the grand canonical ensemble, BEC manifests as fluctuation condensation.

The results explain recent photon gas BEC experiments as fluctuation condensation.

Abstract

In this paper we take a fresh look at the long standing issue of the nature of macroscopic density fluctuations in the grand canonical treatment of the Bose-Einstein condensation (BEC). Exploiting the close analogy between the spherical and mean-spherical models of magnetism with the canonical and grand canonical treatment of the ideal Bose gas, we show that BEC stands for different phenomena in the two ensembles: an ordering transition of the type familiar from ferromagnetism in the canonical ensemble and condensation of fluctuations, i.e. growth of macroscopic fluctuations in a single degree of freedom, without ordering, in the grand canonical case. We further clarify that this is a manifestation of nonequivalence of the ensembles, due to the existence of long range correlations in the grand canonical one. Our results shed new light on the recent experimental realization of BEC in a photon gas, suggesting that the observed BEC when prepared under grand canonical conditions is an instance of condensation of fluctuations.