Entropy Optimization, Generalized Logarithms, and Duality Relations

Abstract

Several generalizations or extensions of the Boltzmann–Gibbs thermostatistics, based on non-standard entropies, have been the focus of considerable research activity in recent years. Among these, the power-law, non-additive entropies (Formula presented.) have harvested the largest number of successful applications. The specific structural features of the (Formula presented.) thermostatistics, therefore, are worthy of close scrutiny. In the present work, we analyze one of these features, according to which the q-logarithm function (Formula presented.) associated with the (Formula presented.) entropy is linked, via a duality relation, to the q-exponential function characterizing the maximum-entropy probability distributions. We enquire into which entropic functionals lead to this or similar structures, and investigate the corresponding duality relations.