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

TL;DR

This paper introduces a two-parameter generalization of logarithm, exponential, and entropy functions, extending nonextensive statistical mechanics and analyzing their mathematical properties and stability features.

Contribution

It develops a novel two-parameter (q, q') generalization of key functions and entropy, expanding the framework of nonextensive statistical mechanics.

Findings

The generalized functions satisfy key mathematical properties like expansibility and concavity.

The entropy form maintains Lesche-stability over a broad parameter range.

The generalized entropy may not always be composable.

Abstract

The $q$-sum $x \oplus_q y \equiv x+y+(1-q) xy$ ($x \oplus_1 y=x+y$) and the $q$-product $x\otimes_q y \equiv [x^{1-q} +y^{1-q}-1]^{\frac{1}{1-q}}$ ($x\otimes_1 y=x y$) 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 $S_{BGS}\equiv -k \sum_{i=1}^Wp_i \ln p_i$ (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}.