Stability analysis of the entropies for superstatistics

It seems reasonable to consider concavity (with regard to all probability distributions) and stability (under arbitrarily small deformations of any given probability distribution) as necessary for an entropic form to be a physical one in the thermostatistical sense. Most known entropic forms, e.g. Renyi entropy SqR=(ln∑i piq)/(1−q), violate one and/or the other of these conditions. In contrast, the Boltzmann–Gibbs entropy SBG=−∑i pi ln pi and the nonextensive one Sq=(1−∑i piq)/(q−1) (q>1) satisfy both. SBG and Sq belong in fact to a larger class of entropies S satisfying both, namely those which, through appropriate optimization, yield the Beck–Cohen superstatistics. We briefly review here the proof of stability of S, and illustrate for an important particular case, namely the log-normal superstatistical entropy. The satisfaction of both concavity and stability appears to be very helpful to identify physically admissible entropic forms.