Revisiting entropies: formal properties and connections between Boltzmann-Gibbs, Tsallis and R\'enyi

TL;DR

This paper provides a detailed analysis of the formal properties of Boltzmann-Gibbs, Tsallis, and Rényi entropies, highlighting their connections, differences, and foundational theorems in statistical mechanics.

Contribution

It offers a comprehensive and accessible comparison of these entropies, including proofs of their uniqueness and the conditions under which they relate.

Findings

Boltzmann-Gibbs entropy's main properties and uniqueness are summarized.

Tsallis entropy's properties and its contrast with additive entropy are discussed.

Connections between Rényi entropy and other entropic forms are outlined.

Abstract

The aim of the present paper is to present a careful and accessible discussion of the formal aspects of Boltzmann-Gibbs and Tsallis entropies. We begin with a brief overview of Boltzmann-Gibbs entropy, highlighting its main properties and the uniqueness theorems formulated by Shannon and Khinchin. Once these foundational results are established, we introduce the framework of nonadditive statistical mechanics, defining Tsallis entropy, discussing its properties and uniqueness theorem, and contrasting it with the results from additive statistical mechanics. We also show that, in an appropriate limit, the Boltzmann-Gibbs results are recovered. The article concludes with a brief discussion of R\'enyi entropy and its connections to the previously defined entropic forms.