Condensation versus ordering: from the spherical models to Bose-Einstein condensation in the canonical and grand canonical ensemble

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