On the continuity and Lesche stability of Tsallis and R\'enyi entropies and q-expectation values

TL;DR

This paper investigates the mathematical properties of Tsallis and Rényi entropies, demonstrating their continuity and stability for certain q-values and instability for others, impacting their use in statistical physics.

Contribution

It provides a rigorous analysis of the conditions under which Tsallis and Rényi entropies are continuous and stable, clarifying their mathematical behavior for different q-parameters.

Findings

Entropies are continuous and stable for q>1.

Entropies are not continuous and are unstable for q<1 with uniform distributions.

The stability depends on the value of q, affecting their applicability.

Abstract

It is shown that the R\'enyi and Tsallis entropies and the q-expectation values, are continuous and stable if $q>1$ and are not continuous and instable for uniform finite distributions if $q<1$.