Regarding the transition from Maxwell-Boltzmann to Bose-Einstein statistics: There is no transition. If you're dealing with Bosons, it's Bose-Einstein statistics at any energy. It's just that at higher energies (or, lower densities in phase-space), Bose-Einstein and Fermi-Dirac statistics become indistinguishable, with Maxwell-Boltzmann statistics being their common "high energy" limit.