Revisiting entropies: formal properties and connections between Boltzmann-Gibbs, Tsallis and Rényi

TL;DR

The paper analyzes the formal properties and interconnections of Boltzmann-Gibbs, Tsallis, and Rényi entropies, grounded in the Shannon–Khinchin axioms and their implications for statistical mechanics. It contrasts additive BG entropy with the nonadditive Tsallis entropy $S_q$, detailing their respective concavity/convexity, positivity, and extensivity, and derives their canonical and microcanonical distributions via variational methods, including $q$-exponentials. Rényi entropy is presented as an additive, log-based alternative that recovers BG in the limit $q\to1$, but with limitations for thermodynamic consistency. The discussion highlights a unified, axiom-driven view of nonextensive statistics, outlines key applications across physics and interdisciplinary domains, and emphasizes both the mathematical coherence and practical constraints of these generalized entropies. Overall, the work provides an accessible synthesis of how BG, Tsallis, and Rényi entropies relate, offering a formal foundation for studying complex systems with long-range correlations and nonlocal effects.

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 Rényi entropy and its connections to the previously defined entropic forms.

Figures (7)

• Figure 1: Em azul a reta secante $x_{0}x_{1}$ que liga os pontos $(x_{0},f(x_{0}))$ e $(x_{1},f(x_{1}))$. Em preta a linha que determina o gráfico $f(x)$.
• Figure 2: Diagrama ilustrando o experimento com as três possíveis resultados e as respectivas probabilidades associadas: $p_{1}$, $p_{2}$ e $p_{3}$.
• Figure 3: Diagrama ilustrando o experimento. Inicialmente, há duas possibilidades de escolhas. A escolha com probabilidade $p'_{1}=1/2$ leva ao resultado 1. Já a escolha $p'_{2}=1/2$ se subdivide em duas outras escolhas possíveis, com probabilidades $p_{21}=1/3$ e $p_{22}=2/3$, cada uma levando aos resultados 2 e 3, respectivamente.
• Figure 4: Diagrama ilustrando o experimento com as quatro possibilidades de resultados, cada um associado às probabilidades $p_{1}$, $p_{2}$, $p_{3}$ e $p_{4}$.
• Figure 5: Diagrama ilustrando o experimento. Inicialmente têm-se duas possibilidades de resultados. A escolha com probabilidade $p_{L}=7/12$ se subdivide em outras duas escolhas com probabilidades $p_{1}/p_{L}=4/7$ e $p_{2}/p_{L}=3/7$ que levam aos resultados 1 e 2. Já a escolha $p_{M}=5/12$ se subdivide em duas outras escolhas possíveis com probabilidades $p_{3}/p_{M}=2/5$ e $p_{4}/p_{M}=3/5$, cada uma leva aos resultados 3 e 4, respectivamente.

Theorems & Definitions (6)

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