Non-Extensive Entropic Distance Based on Diffusion: Restrictions on Parameters in Entropy Formulae

Tamás Biró,Zsolt Schram

doi:10.3390/e18020042

Abstract

Based on a diffusion-like master equation we propose a formula using the Bregman divergence for measuring entropic distance in terms of different non-extensive entropy expressions. We obtain the non-extensivity parameter range for a universal approach to the stationary distribution by simple diffusive dynamics for the Tsallis and the Kaniadakis entropies, for the Hanel-Thurner generalization, and finally for a recently suggested log-log type entropy formula which belongs to diverging variance in the inverse temperature superstatistics.

Highlights

Over the last decades, there have been several suggestions for generalizations of the Boltzmann–Gibbs–Shannon (BGS) entropy formula [1,2,3,4,5,6]
In the present paper we investigate whether such entropy formulas define an entropic distance between two probability distributions, which has the following useful properties: 1. it is positive for any two different distributions; 2. it is zero for comparing any distribution with itself; 3. it is symmetric
In the following we show this behavior for the traditional logarithmic entropy formula and a generally state-dependent nearest neighbour master equation, and propose a generalization of the symmetrized entropic distance measure based on the deformed logarithm function

Summary

Introduction

There have been several suggestions for generalizations of the Boltzmann–Gibbs–Shannon (BGS) entropy formula [1,2,3,4,5,6]. Most formulas can be grouped into categories either by their mathematical form (trace form or a function of the trace form) [7], or by the scaling properties for large systems, usually providing large entropies, S, even if not necessarily proportional to the logarithm of the number of states, ln W [8,9,10,11]. In the following we show this behavior for the traditional logarithmic entropy formula and a generally state-dependent nearest neighbour master equation (defined below), and propose a generalization of the symmetrized entropic distance measure based on the deformed logarithm function. We conclude that some proposals encertain a shrinking of the entropic distance during an approach to the stationary distribution only for a restricted range of the non-extensivity parameter(s) used in the entropy formula

Probability Distributions

Master Equation

Entropic Distance

Conclusions