Ideal relativistic quantum gas and invariant phase space

Abstract

The density of states of an ideal relativistic gas with Bose-Einstein or Fermi-Dirac statistics is written in closed form as a cluster decomposition over the usual phase-space integrals with Boltzmann statistics. We stress the difference between invariant phase space and invariant momentum space; only the former gives the correct description of ideal gases. Our results are formulated for the microcanonical, grand microcanonical, canonical and grand canonical ensembles in invariant phase space and invariant momentum space. A whole section is devoted to a numerical study, which reveals that while under normal conditions thermodynamical quantities (derived from the logarithm of the density of states) acquire only very small corrections, the densities (phase-space integrals) themselves are significantly changed. This holds in particular for elementary particle applications (statistical models, phase-space considerations) where the effects on phase space of Bose-Einstein and/or Fermi-Dirac statistics can almost never be neglected; however, in these applications, the number of equal particles is so low that the cluster decompositions can be explicitly evaluated.