Inequivalence of the momentum distribution functions in the canonical and microcanonical ensembles for two-dimensional fermionic and bosonic systems
We considered the problem of determining the microcanonical momentum distribution function (MMDF). The MMDF is the momentum distribution function (MDF) in the microcanonical ensemble. We solved for the MDF for isolated systems with fixed energy value, where it is deemed that the MMDF is more appropriate to use than the canonical momentum distribution function (CMDF) which is the familiar Fermi-Dirac and Bose-Einstein distributions. For the system of two-dimensional fermions, we have demonstrated inequivalence between the MMDF and the CMDF when the energy of the system is low. For the system of two-dimensional bosons, no inequivalence can be seen.