#
Senses along Which the Entropy S_{q} Is Unique

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## Abstract

**:**

## 1. Introduction

## 2. On Uniqueness

#### 2.1. Santos 1997 Theorem

#### 2.2. The 1997 Connection to Weak Chaos in the Logistic Map

#### 2.3. Connection with Jackson Derivative

#### 2.4. Abe 2000 Theorem

#### 2.5. Beck-Cohen 2003 Superstatistics

#### 2.6. Lattice-Boltzmann Models for Fluids

#### 2.7. Topsoe 2005 Factorizability in Game Theory

#### 2.8. Amari-Ohara-Matsuzoe 2012 Conformally Invariant Geometry

#### 2.9. Enciso–Tempesta 2017 Theorem

#### 2.10. The Shore–Johnson–Axioms Controversy (2005–2019)

#### 2.11. Plastino-Tsallis-Wedemann-Haubold 2022

#### 2.12. Plastino-Plastino 2023 Connection with the Micro-Canonical Ensemble

## 3. Closely Related Issues

#### 3.1. The Values of the Entropic Indices Might Depend on the Class of States of the System

#### 3.2. Entropic Functional vs. Entropy of a System

## 4. Summary

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**It has been proven [23] that ${S}_{q}$ is the unique entropic form which simultaneously is trace-form, composable, and recovers ${S}_{BG}$ as a particular instance. ${S}_{q}$ (hence ${S}_{BG}$), the Renyi entropy ${S}_{q}^{R}$ [24], the Tempesta $(a,b,\alpha )$-entropy ${S}_{a,b,\alpha}^{T}$ (Equation (9.1) in [25]), the Jensen–Pazuki–Pruessner–Tempesta entropy ${S}_{\gamma ,\alpha}^{JPPT}$ [26] and many others belong to the class of group entropies and are therefore composable. To facilitate the identification, we are here using the following notations: Sharma–Mittal entropy ${S}_{q,r}^{SM}$ [27], Landsberg-Vedral-Rajagopal-Abe entropy ${S}_{q}^{LVRA}$ [28,29,30], Tsallis–Mendes–Plastino entropy ${S}_{q}^{TMP}$, Arimoto entropy ${S}_{q}^{Ar}$ [31], Curado–Tempesta–Tsallis entropy ${S}_{a,b,r}^{CTT}$ [32], Borges–Roditi entropy ${S}_{q,{q}^{\prime}}^{BR}$ [33], Abe entropy ${S}_{q}^{Ab}$ [34], Kaniadakis entropy ${S}_{\kappa}^{K}$ [35], Kaniadakis–Lissia–Scarfone entropy ${S}_{\kappa ,r}^{KLS}$ [36], Anteneodo–Plastino entropy ${S}_{\eta}^{AP}$ [37], Hanel–Thurner entropy ${S}_{c,d}^{HT}$ [38,39], ${S}_{q,\delta}$ [40], Schwammle–Tsallis entropy ${S}_{q,{q}^{\prime}}^{ST}$ [41], the Tempesta $(\alpha ,\beta ,q)$-entropy ${S}_{\alpha ,\beta ,q}^{T}$ [42], the Curado b-entropy ${S}_{b}^{C}$ [43,44], the Curado $\lambda $-entropy ${S}_{\lambda}^{C}$ [45] (see [12]), and the exponential c-entropy ${S}_{c}^{E}$ (see [10,46]). The entropic form ${S}_{\lambda}^{C}$ is one among the rare cases which do not include ${S}_{BG}$ and is neither trace-form nor composable. From [1,12].

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Tsallis, C.
Senses along Which the Entropy *S _{q}* Is Unique.

*Entropy*

**2023**,

*25*, 743. https://doi.org/10.3390/e25050743

**AMA Style**

Tsallis C.
Senses along Which the Entropy *S _{q}* Is Unique.

*Entropy*. 2023; 25(5):743. https://doi.org/10.3390/e25050743

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2023. "Senses along Which the Entropy *S _{q}* Is Unique"

*Entropy*25, no. 5: 743. https://doi.org/10.3390/e25050743