Generalized Time Reversal Elastic Scattering Balance and Statistical Mechanics

Addendum: March 9, 2020 In equation 10, we discuss a new probability g, the scattering probability to the power of f the probability for a particle with energy ei. In a note from March 9, 2020, we change (10) to be g to the power of g. This then leads to Shannon's entropy -g ln(g) where g(f), but describes a scattering entropy. We argue in the note from March 9, 2020 that Kaniadakis entropy density - Integral df g(f) yields a particle number entropy consistent with thermodyanmics. In previous notes, we argued the Maxwell-Boltzmann (MB), Fermi-Dirac (FD) and Bose-Einstein (BE) distributions may be obtained by a time reversal elastic two body scattering balance. This approach does not require counting, any notion of entropy or partition functions. The basic ideas for the MB case is to use f(e1)f(e2)=f(e3)f(e4) ((1a)) and e1+e2=e3+e4 ((1b)) where f(e) is the distribution function. Taking ln of ((1a)) and equating to ((1b)) yields the MB distribution. For the FD and BE cases, ((1a)) is modified to account for scattering restrictions or enhancements. In this note, we wish to consider a more generalized case. We first consider that ((1a)) does not hold for MB type particles, but rather that g(f(e1)) g(f(e2)) = g(f(e3)) g(f(e4)) ((2)) is in place. Conservation of energy is retained. This leads to issues with the usual ensemble approach which leads to the canonical (and grand canonical) partition functions via saddle point integration and Shannon’s entropy. As a result, a new type of entropy may be needed. In addition, we consider ((2)) modified for the FD and BE cases