Generalized Statistical Mechanics and the Partition Function

In previous notes, we argued one may find the number of particles in a state ei by using time reversal elastic two body scattering balance. This approach yields the Maxwell-Boltzmann (MB), Fermi-Dirac (FD) and Bose-Einstein (BE) distributions and has also been applied to more generalized cases. The approach makes no use of entropy or partition functions and seems to be very close to that used by Kaniadakis (1), although he develops a Fokker-Planck equation and defines an entropy. His entropy, however, seems to be a mathematical construct, created by integrating a function of f so that one may later take the derivative with respect to f and obtain the original function. This original function is then equated to -(e-u)/T (u=chemical potential) from a Lagranian multiplier. It seems, however, that if an entropy is defined, one may calculate a Free Energy and try to link this to a canonical (or grand canonical partition function). In (2), a different generalized theory of statistical mechanics is proposed. It too calculates entropy density, but obtains a different functional form. Imposing the “scattering constraint” which both (1) and (2) ultimately seem to use, leads to similar results. In (2), however, a canonical and then grand canonical partition function is developed from the entropy. This seems to lead to different distributions than those predicted by the scattering approach. The objective is this note is to try to analyze these different theories to see why they appear to be different