On the uniqueness theorem for pseudo-additive entropies

TL;DR

This paper examines the conditions under which pseudo-additive entropies like Tsallis entropy are uniquely determined, emphasizing the importance of fixing the rule for conditional entropies, supported by mathematical theorems and examples.

Contribution

It clarifies the uniqueness conditions for Tsallis-type entropies and highlights the role of conditional entropy prescriptions, connecting to thermodynamical classifications.

Findings

Uniqueness of Tsallis entropy depends on conditional entropy handling.

Mathematical theorems support the necessity of fixed prescriptions.

Examples illustrate the impact on non-extensive thermodynamics.

Abstract

We discuss the idea that the Tsallis-type (q-additive) entropic chain rule allows for a wider class of entropic functionals than previously thought. In particular, we point out that the ensuing entropy solutions (e.g., Tsallis entropy) can be determined uniquely only when one fixes the prescription for handling conditional entropies. Our point is substantiated with the Dar\'otzy's mapping theorem and DeFinetti-Kolmogorov theorem for escort distributions and illustrated with a number of examples. Connection with Landsberg's classification of non-extensive thermodynamical systems is also briefly discussed.