  We analyze the general relation between canonical and grand canonical
ensembles in the thermodynamic limit. We begin our discussion by deriving, with
an alternative approach, some standard results first obtained by Kac and
coworkers in the late 1970s. Then, motivated by the Bose-Einstein condensation
(BEC) of trapped gases with a fixed number of atoms, which is well described by
the canonical ensemble and by the recent groundbreaking experimental
realization of BEC with photons in a dye-filled optical micro-cavity under
genuine grand canonical conditions, we apply our formalism to a system of
non-interacting Bose particles confined in a two-dimensional harmonic trap. We
discuss in detail the mathematical origin of the inequivalence of ensembles
observed in the condensed phase, giving place to the so-called grand canonical
catastrophe of density fluctuations. We also provide explicit analytical
expressions for the internal energy and specific heat and compare them with
available experimental data. For these quantities, we show the equivalence of
ensembles in the thermodynamic limit.
