A step beyond Tsallis and Rényi entropies

By M. Masi. 2005

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Abstract Tsallis and Renyi 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 [J. Math. Sci. 10 (1975) 28] and which received attention only recently as an application in statistical mechanics [Physica A 285 (2000) 351, Eur. Phys. J. B 30 (2002) 543] that provides one possible unification. We will show how this generalization that unifies Renyi 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.

Subfields: Economics and Econometrics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b48759f0283ce6a6ffb60b3be9c12b7726ee56b1