Momentum distribution functions in ensembles: the inequivalence of microcannonical and canonical ensembles in a finite ultracold system

It is demonstrated that in many thermodynamic textbooks the equivalence of the different ensembles is achieved in the thermodynamic limit. In this present work we remark the inequivalence of microcannonical and canonical ensembles in a finite ultracold system at low energies. We calculate the microcanonical momentum distribution function (MDF) in a system of identical fermions (bosons). We find that, the microcanonical MDF deviates from the canonical one, which is the Fermi-Dirac (Bose-Einstein) function, in a finite system at low energies where the single-particle density of states and its inverse are finite.