Single Particle Degeneracy in the Maxwell-Boltzmann/ Canonical Distribution Case

 The Maxwell-Boltzmann (MB), Fermi-Dirac FD and Bose-Einstein BE distributions may be obtained by maximizing the ln of energy level degeneracies subject to the constraint of Sum over i ei ni = Eave in the MB case (canonical distribution) and this constraint together with a second one Sum over i ni = Nave (grand canonical case) as shown in (1) page 182  . In particular, in the MB case, a single energy level ei with ni particles is said to have the degeneracy (gi) (power ni) / ni ! (which overcounts). This maximization process leads to the weight gi exp(-ei/T). Setting gi=1, however, leads to the same weight exp(-ei/T) so the MB factor is not a result of a single particle level degeneracy i.e. the direction in which the momentum vector points. To shed physical light on this situation, one may consider two particle scattering in the MB, FD and BE cases. In all three cases, e1+e2 = e3+e4 (with e=pp/2m), but only in the MB case does n(e1)n(e2)=n(e3)n(e4). This suggests that the single energy level ei degeneracy levels ei actually play a physical role in two body scattering for the FD and BE cases, but not in the MB case. Thus, calculating probabilities based on counting still involves the physics related to the interactions in the problem. Degeneracies may exist and may or may not need to be part of the counting of arrangements.

    As a specific example, we note that in (1) (gi) (power ni) / ni !  is used to count degenerate arrangements for the MB case (on page 182). This same expression is used in (1) on page 194 to describe a method by Fowler and Darwin to obtain the MB single particle probability exp(-ei/T). In that case, gi is taken to be a parameter which is ultimately set to 1 and not the degeneracy of the single particle level. This begs the question: When should single energy ei degeneracy be included in the counting of states?