Reactions, Distributions and Entropy in Generalized Statistical Mechanics

For a number of years, articles have appeared in the literature describing statistical mechanics which goes beyond that of the Boltzmann-Gibbs-Shannon picture (1),(2),(3). In addition, it seems two particular entropy density Sd functional forms in terms of the distribution f((e-u)/T), where u is the chemical potential, appear frequently. The first is   Sd=- f ln(g(f)) and the second, Kaniadakis (1) Sd = - Integral df ln(g(f)) where ln(g(f)) = -(e-u)/T. In this note, we argue first that one is not abandoning Shannon’s entropy in general statistical mechanics, but rather is applying it to the number of reactions of a particle with energy ei, rather than to particle number. Thus, Shannon’s entropy is - g ln(g). This entropy, however, applies to reactions, not particles. Secondly, we try to show that it is Kaniadakis form of the particle entropy which is consistent with both thermodynamics and the Jaynes idea of maximizing particle entropy to find the distribution function.