A step beyond Tsallis and Renyi entropies

TL;DR

This paper explores the Sharma-Mittal entropy measure as a unifying framework for Tsallis and Rényi entropies, using generalized logarithm and exponential functions, and discusses its properties and relations to other measures in statistical mechanics.

Contribution

It demonstrates how Sharma-Mittal entropy naturally unifies Tsallis and Rényi entropies through q-formalism and introduces a new perspective using Kolmogorov-Nagumo averages.

Findings

Sharma-Mittal measure unifies Rényi and Tsallis entropies.

A new extension of entropy is proposed that does not obey pseudo-additivity.

Relations between information measures are clarified via Kolmogorov-Nagumo averages.

Abstract

Tsallis and R\'{e}nyi entropy measures are two possible different generalizations of the Boltzmann-Gibbs entropy (or Shannon's information) but are not generalizations of each others. It is however the Sharma-Mittal measure, which was already defined in 1975 (B.D. Sharma, D.P. Mittal, J.Math.Sci \textbf{10}, 28) and which received attention only recently as an application in statistical mechanics (T.D. Frank & A. Daffertshofer, Physica A \textbf{285}, 351 & T.D. Frank, A.R. Plastino, Eur. Phys. J., B \textbf{30}, 543-549) that provides one possible unification. We will show how this generalization that unifies R\'{e}nyi and Tsallis entropy in a coherent picture naturally comes into being if the q-formalism of generalized logarithm and exponential functions is used, how together with Sharma-Mittal's measure another possible extension emerges which however does not obey a pseudo-additive law and lacks of other properties relevant for a generalized thermostatistics, and how the relation between all these information measures is best understood when described in terms of a particular logarithmic Kolmogorov-Nagumo average.