Microcanonical Energy Approach to Generalized Statistical Mechanics

In previous notes, we considered generalized two body scattering: g(f(e1))g(f(e2)) = g(f(e3)) g(f(e4)) ((a)) leading to ln(g(f(ei)) = -(ei-u)/T if one links ((a)) to conservation of energy during scattering. For g=1, one obtains the Maxwell-Boltzmann distribution. Here, f(ei) is the average number of particles with energy ei. Thus, an averaging has already been done and one needs to know the nature of this average. For example, for g=f/(1-f), the Fermi-Dirac case, it seems the averaging allows for different values of N, the total number of particles, whereas (ei-u)/T is associated with allowing for different total energies. Thus, the generalized approach of ((a)) seems to usually be associated with the grand canonical approach in which both total E and N are allowed to vary. In a previous note (1), we tried to develop a microcanonical approach to the Maxwell-Boltzmann distribution which replaced exp(-ei/T) (which is related to the canonical approach) with another energy function more related to fixed energy. This result, however, only held for e of the same order as E (i.e. small N systems). For e<