Two-parameter generalization of the logarithm and exponential functions and Boltzmann-Gibbs-Shannon entropy

Abstract

The $q$-sum $x{\oplus }_{q}y\equiv x+y+(1-q)xy$ ($x{\oplus }_{1}y=x+y$) and the $q$-product $x{\otimes }_{q}y\equiv [x^{1-q}+y^{1-q}-1{]}^{\frac{1}{1-q}}$ ($x{\otimes }_{1}y=xy$) emerge naturally within nonextensive statistical mechanics. We show here how they lead to two-parameter (namely, $q$ and $q^{\prime }$) generalizations of the logarithmic and exponential functions (noted respectively ${\ln }_{q,q^{\prime }}x$ and $e_{q,q^{\prime }}^x$), as well as of the Boltzmann-Gibbs-Shannon entropy $S_{BGS}\equiv -k{\sum }_{i=1}^W{p}_i\ln p_i$ (noted $S_{q,q^{\prime }}$). The remarkable properties of the $(q,q^{\prime })$-generalized logarithmic function make the entropic form $S_{q,q^{\prime }}\equiv k{\sum }_{i=1}^W{p}_i{\ln }_{q,q^{\prime }}(1/p_i)$ to satisfy, for large regions of $(q,q^{\prime })$, important properties such as {\it expansibility}, {\it concavity} and {\it Lesche-stability}, but not necessarily {\it composability}.

9 pages, 4 figures

References in corpus (1)

• A possible deformed algebra and calculus inspired in nonextensive thermostatistics

Cited by in corpus (4)

• Generalized exponential function and discrete growth models
• Rigid rotators and diatomic molecules via Tsallis statistics
• Probability distribution of (Schwämmle and Tsallis) two-parameter entropies and the Lambert W-function
• Composability in a certain family of entropies