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

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.