On one-parameter-dependent generalizations of Boltzmann-Gibbs statistical mechanics

Abstract

Based on a previously postulated entropy, that now becomes a particular case, we show that there exists an infinite set of entropies, with similar properties, that reduce in a common limit to the Boltzmann-Shannon form. The probabilities for the microcanonical ensemble and for the canonical ensemble are obtained. The method used to construct the set is quite simple and quite general and can be applied to generalizations of physical quantities and to other generalized entropies. The existence of an infinite set of `entropies' with, in principle, similar properties, could be a serious drawback for the actual utility of any of them and points to their utter uselessness unless some reason can be given for a special choice of one of them.