  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.
