# Entropy

^{1}

^{2}

^{3}

^{4}

## Definition

## 1. Introduction

## 2. Basics

#### 2.1. Definitions and Properties of Entropy

#### 2.2. Additivity versus Extensivity

- Exponential class $W\left(N\right)\sim A{\mu}^{N}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}(A>0;\phantom{\rule{0.166667em}{0ex}}\mu >1)$:This is the typical case within the BG theory. We have ${S}_{BG}\left(N\right)=klnW\left(N\right)\sim Nln\mu +lnA\propto N$, therefore ${S}_{BG}$ is extensive, as thermodynamically required.
- Power-law class $W\left(N\right)\sim B{N}^{\rho}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}(B>0;\phantom{\rule{0.166667em}{0ex}}\rho >0)$:We should not use ${S}_{BG}$ since it implies ${S}_{BG}\left(N\right)=klnW\left(N\right)\sim \rho lnN+lnB\propto lnN$, thus violating thermodynamics. We verify instead that ${S}_{q=1-1/\rho}\left(N\right)=k{ln}_{q=1-1/\rho}W\left(N\right)\propto N$, as thermodynamically required.
- Stretched exponential class $W\left(N\right)\sim C{\nu}^{{N}^{\gamma}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}(C>0;\phantom{\rule{0.166667em}{0ex}}\nu >1;\phantom{\rule{0.166667em}{0ex}}0<\gamma <1)$:In this instance, no value of q exists which would imply an extensive entropy ${S}_{q}$. We can instead used ${S}_{\delta}$ with $\delta =1/\gamma $. Indeed, ${S}_{\delta =1/\gamma}\left(N\right)=k{[lnW\left(N\right)]}^{\delta}\propto N$, as thermodynamically required.
- Logarithmic class $W\left(N\right)\sim DlnN\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}(D>0)$:In this case, no values of $(q,\delta )$ exist which imply an extensive entropy ${S}_{q,\delta}$. Instead, we can use the Curado entropy [33] ${S}_{\lambda}^{C}\left(N\right)=k\left[{e}^{\lambda \phantom{\rule{0.166667em}{0ex}}W\left(N\right)}-{e}^{\lambda}\right]$ with $\lambda =1/D$. Indeed, we can verify that ${S}_{\lambda =1/D}^{C}\left(N\right)\propto N$, as thermodynamically required.

#### 2.3. Range of Interactions

#### 2.4. Thermodynamics and Legendre Transformations

#### 2.5. Classification of Entropic Functionals

#### 2.6. Boltzmann–Gibbs and Nonextensive Statistical Mechanics

#### 2.6.1. Boltzmann–Gibbs Statistical Mechanics

#### 2.6.2. q-Generalization of the Boltzmann–Gibbs Theory

## 3. Results and Applications

#### 3.1. In Physics

#### 3.1.1. Nonlinear Dynamical Systems

#### 3.1.2. First-Principle Calculation of q for a Quantum Hamiltonian System

#### 3.1.3. Long-Range Interactions

#### 3.1.4. Overdamped Many-Body Systems

#### 3.1.5. Low Energy Physics

#### 3.1.6. High Energy Physics

#### 3.2. Beyond Physics

#### 3.2.1. Mathematics

#### 3.2.2. Chemistry

#### 3.2.3. Economics

#### 3.2.4. Biology and Life Sciences

#### 3.2.5. Computer Sciences

#### 3.2.6. Random Networks

#### 3.2.7. Image and Signal Processing

#### 3.2.8. Engineering

## 4. Final Remarks

## Funding

## Acknowledgments

## Conflicts of Interest

## Entry Link on the Encyclopedia Platform

## References

- Clausius, R. Uber die Wärmeleitung gasförmiger Körper. Ann. Phys.
**1865**, 125, 353–400. [Google Scholar] [CrossRef] - Clausius, R. The Mechanical Theory of Heat with Its Applications to the Steam Engine and to Physical Properties of Bodies; John van Voorst, 1 Paternoster Row. MDCCCLXVII: London, UK, 1865. [Google Scholar]
- Boltzmann, L. Weitere Studien u̇ber das Wȧrmegleichgewicht unter Gas moleku̇len [Further Studies on Thermal Equilibrium Between Gas Molecules]. Wien. Ber.
**1872**, 66, 275. [Google Scholar] - Boltzmann, L. Uber die Beziehung eines allgemeine mechanischen Satzes zum zweiten Haupsatze der Warmetheorie Sitzungsberichte, K. Akademie der Wissenschaften in Wien, Math. Naturwissenschaften
**1877**, 75, 67–73. [Google Scholar] - Gibbs, J.W. Elementary Principles in Statistical Mechanics—Developed with Especial Reference to the Rational Foundation of Thermodynamics; C. Scribner’s Sons: New York, NY, USA, 1902. [Google Scholar]
- Gibbs, J.W. The collected works. In Thermodynamics; Yale University Press: New Haven, CT, USA, 1948; Volume 1. [Google Scholar]
- Gibbs, J.W. Elementary Principles in Statistical Mechanics; OX Bow Press: Woodbridge, CT, USA, 1981. [Google Scholar]
- von Neumann, J. Thermodynamik quantenmechanischer Gesamtheiten. Nachrichten Ges. Wiss. Gott.
**1927**, 1927, 273–291. [Google Scholar] - Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] - Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 623–656. [Google Scholar] [CrossRef] - Shannon, C.E. The Mathematical Theory of Communication; University of Illinois Press: Urbana, IL, USA, 1949. [Google Scholar]
- Jaynes, E.T. Information theory and statistical mechanics. Phys. Rev.
**1957**, 106, 620–630. [Google Scholar] [CrossRef] - Jaynes, E.T. Information theory and statistical mechanics. II. Phys. Rev.
**1957**, 108, 171–190. [Google Scholar] [CrossRef] - Renyi, A. On measures of information and entropy. In Proceedings of the Fourth Berkeley Symposium; University of California Press: Los Angeles, CA, USA, 1961; Volume 1, pp. 547–561. [Google Scholar]
- Renyi, A. Probability Theory; Dover Publications Inc.: New York, NY, USA, 1970. [Google Scholar]
- Balatoni, J.; Renyi, A. Remarks on entropy. Publ. Math. Inst. Hung. Acad. Sci.
**1956**, 1, 9–40. [Google Scholar] - Renyi, A. On the dimension and entropy of probability distributions. Acta Math. Acad. Sci. Hung.
**1959**, 10, 193–215. [Google Scholar] [CrossRef] - Havrda, J.; Charvat, F. Quantification method of classification processes - Concept of structural α-entropy. Kybernetika
**1967**, 3, 30–35. [Google Scholar] - Lindhard, J.; Nielsen, V. Det Kongelige Danske Videnskabernes Selskab Matematisk-fysiske Meddelelser (Denmark). Stud. Stat. Mech.
**1971**, 38, 1–42. [Google Scholar] - Sharma, B.D.; Mittal, D.P. New non-additive measures of entropy for discrete probability distributions. J. Math. Sci.
**1975**, 10, 28. [Google Scholar] - Sharma, B.D.; Taneja, I.J. Entropy of type (α,β) and other generalized measures in information theory. Metrika
**1975**, 22, 205. [Google Scholar] [CrossRef] - Mittal, D.P. On some functional equations concerning entropy, directed divergence and inaccuracy. Metrika
**1975**, 22, 35. [Google Scholar] [CrossRef] - Jaynes, E.T. Gibbs vs. Boltzmann entropies. Am. J. Phys.
**1965**, 33, 391–398. [Google Scholar] [CrossRef] - Landauer, R. Information is physical. Phys. Today
**1991**, 44, 23. [Google Scholar] [CrossRef] - Tsallis, C. Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys.
**1988**, 52, 479–487, [First appeared as preprint in 1987: CBPF-NF-062/87, ISSN 0029–3865, Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro]. [Google Scholar] [CrossRef] - Tsallis, C. Nonextensive Statistical Mechanics—Approaching a Complex World, 1st ed.; Springer: New York, NY, USA, 2009. [Google Scholar]
- Watanabe, S. Knowing and Guessing; Wiley: New York, NY, USA, 1969. [Google Scholar]
- Barlow, H. Conditions for versatile learning, Helmholtz’s unconscious inference, and the task of perception. Vis. Res.
**1990**, 30, 1561. [Google Scholar] [CrossRef] - Penrose, O. Foundations of Statistical Mechanics: A Deductive Treatment; Pergamon: Oxford, UK, 1970; p. 167. [Google Scholar]
- Tsallis, C.; Cirto, L.J.L. Black hole thermodynamical entropy. Eur. Phys. J. C
**2013**, 73, 2487. [Google Scholar] [CrossRef] - Ubriaco, M.R. Entropies based on fractional calculus. Phys. Lett. A
**2009**, 373, 2516. [Google Scholar] [CrossRef] - Lima, H.S.; Tsallis, C. Exploring the neighborhood of q-exponentials. Entropy
**2020**, 22, 1402. [Google Scholar] [CrossRef] - Tsallis, C. Nonextensive Statistical Mechanics—Approaching a Complex World, 2nd ed.; Springer: New York, NY, USA, 2022; in press. [Google Scholar]
- Holton, G.; Elkana, Y. (Eds.) Albert Einstein: Historical and Cultural Perspectives; Dover Publications: Mineola, NY, USA, 1997; p. 227. [Google Scholar]
- Antoni, M.; Ruffo, S. Clustering and relaxation in Hamiltonian long-range dynamics. Phys. Rev. E
**1995**, 52, 2361–2374. [Google Scholar] [CrossRef] - Anteneodo, C.; Tsallis, C. Breakdown of the exponential sensitivity to the initial conditions: Role of the range of the interaction. Phys. Rev. Lett.
**1998**, 80, 5313. [Google Scholar] [CrossRef] - Nobre, F.D.; Tsallis, C. Classical infinite-range-interaction Heisenberg ferromagnetic model: Metastability and sensitivity to initial conditions. Phys. Rev. E
**2003**, 68, 036115. [Google Scholar] [CrossRef] - Rodriguez, A.; Nobre, F.D.; Tsallis, C. d-Dimensional classical Heisenberg model with arbitrarily-ranged interactions: Lyapunov exponents and distributions of momenta and energies. Entropy
**2019**, 21, 31. [Google Scholar] [CrossRef] - Rodriguez, A.; Nobre, F.D.; Tsallis, C. Quasi-stationary-state duration in d-dimensional long-range model. Phys. Rev. Res.
**2020**, 2, 023153. [Google Scholar] [CrossRef] - Dauxois, T. Fermi, Pasta, Ulam, and a mysterious lady. Phys. Today
**2008**, 6, 55–57. [Google Scholar] [CrossRef] - Bagchi, D.; Tsallis, C. Fermi-Pasta-Ulam-Tsingou problems: Passage from Boltzmann to q-statistics. Phys. A Stat. Mech. Appl.
**2018**, 491, 869–873. [Google Scholar] [CrossRef] - Christodoulidi, H.; Tsallis, C.; Bountis, T. Fermi-Pasta-Ulam model with long-range interactions: Dynamics and thermostatistics. EPL
**2014**, 108, 40006. [Google Scholar] [CrossRef] - Christodoulidi, H.; Bountis, T.; Tsallis, C.; Drossos, L. Dynamics and Statistics of the Fermi–Pasta–Ulam β–model with different ranges of particle interactions. J. Stat. Mech. Theory Exp.
**2016**, 2016, 123206. [Google Scholar] [CrossRef] - Bagchi, D.; Tsallis, C. Sensitivity to initial conditions of d-dimensional long-range-interacting quartic Fermi-Pasta-Ulam model: Universal scaling. Phys. Rev. E
**2016**, 93, 062213. [Google Scholar] [CrossRef] - Landsberg, P.T. Thermodynamics and Statistical Mechanics; Oxford University Press: New York, NY, USA, 1978. [Google Scholar]
- Landsberg, P.T. Thermodynamics and Statistical Mechanics; Dover: New York, NY, USA, 1990. [Google Scholar]
- Tsallis, C.; Cirto, L.J.L. Thermodynamics is more powerful than the role to it reserved by Boltzmann-Gibbs statistical mechanics. Eur. Phys. J. Spec. Top.
**2014**, 223, 2161. [Google Scholar] [CrossRef] - Tsallis, C. Approach of complexity in nature: Entropic nonuniqueness. Axioms
**2016**, 5, 20. [Google Scholar] [CrossRef] - Tsallis, C. Beyond Boltzmann-Gibbs-Shannon in physics and elsewhere. Entropy
**2019**, 21, 696. [Google Scholar] [CrossRef] - Tsallis, C. Nonextensive statistics: Theoretical, experimental and computational evidences and connections. Braz. J. Phys.
**1999**, 29, 1–35. [Google Scholar] [CrossRef] - Tsallis, C. Talk at the IMS Winter School on Nonextensive Generalization of Boltzmann-Gibbs Statistical Mechanics and Its Applications; Institute for Molecular Science: Okazaki, Japan, 1999. [Google Scholar]
- Tsallis, C. Nonextensive Statistical Mechanics and Thermodynamics: Historical Background and Present Status. In Nonextensive Statistical Mechanics and Its Applications; Abe, S., Okamoto, Y., Eds.; Series Lecture Notes in Physics; Springer: Berlin/Heidelberg, Germany, 2001. [Google Scholar]
- Hotta, M.; Joichi, I. Composability and generalized entropy. Phys. Lett. A
**1999**, 262, 302. [Google Scholar] [CrossRef] - Enciso, A.; Tempesta, P. Uniqueness and characterization theorems for generalized entropies. J. Stat. Mech.
**2017**, 2017, 123101. [Google Scholar] [CrossRef] - Tsallis, C. Entropy. In Springer Encyclopedia of Complexity and Systems Science; Springer: Berlin/Heidelberg, Germany, 2008; pp. 2859–2883. [Google Scholar]
- Chafai, D. Entropies, convexity, and functional inequalities–On Φ-entropies and Φ-Sobolev inequalities. J. Math. Kyoto Univ.
**2004**, 44, 325–363. [Google Scholar] [CrossRef] - Tempesta, P. Formal groups and Z-entropies. Proc. R. Soc. A
**2016**, 472, 20160143. [Google Scholar] [CrossRef] - Jensen, H.J.; Pazuki, R.H.; Pruessner, G.; Tempesta, P. Statistical mechanics of exploding phase spaces: Ontic open systems. J. Phys. A Math. Theor.
**2018**, 51, 375002. [Google Scholar] [CrossRef] - Landsberg, P.T.; Vedral, V. Distributions and channel capacities in generalized statistical mechanics. Phys. Lett. A
**1998**, 247, 211. [Google Scholar] [CrossRef] - Landsberg, P.T. Entropies galore! In Nonextensive Statistical Mechanics and Thermodynamics. Braz. J. Phys.
**1999**, 29, 46. [Google Scholar] [CrossRef] - Rajagopal, A.K.; Abe, S. Implications of form invariance to the structure of nonextensive entropies. Phys. Rev. Lett.
**1999**, 83, 1711. [Google Scholar] [CrossRef] - Arimoto, S. Information-theoretical considerations on estimation problems. Inf. Control.
**1971**, 19, 181–194. [Google Scholar] [CrossRef] - Curado, E.M.F.; Tempesta, P.; Tsallis, C. A new entropy based on a group-theoretical structure. Ann. Phys.
**2016**, 366, 22–31. [Google Scholar] [CrossRef] - Borges, E.P.; Roditi, I. A family of non-extensive entropies. Phys. Lett. A
**1998**, 246, 399–402. [Google Scholar] [CrossRef] - Abe, S. A note on the q-deformation theoretic aspect of the generalized entropies in nonextensive physics. Phys. Lett. A
**1997**, 224, 326–330. [Google Scholar] [CrossRef] - Kaniadakis, G. Non linear kinetics underlying generalized statistics. Phys. A Stat. Mech. Appl.
**2001**, 296, 405. [Google Scholar] [CrossRef] - Kaniadakis, G. Statistical mechanics in the context of special relativity. Phys. Rev. E
**2002**, 66, 056125. [Google Scholar] [CrossRef] [PubMed] - Kaniadakis, G. Statistical mechanics in the context of special relativity. II. Phys. Rev. E
**2005**, 72, 036108. [Google Scholar] [CrossRef] - Kaniadakis, G.; Lissia, M.; Scarfone, A.M. Deformed logarithms and entropies. Phys. A Stat. Mech. Appl.
**2004**, 340, 41. [Google Scholar] [CrossRef] - Anteneodo, C.; Plastino, A.R. Maximum entropy approach to stretched exponential probability distributions. J. Phys. A
**1999**, 32, 1089. [Google Scholar] [CrossRef] - Hanel, R.; Thurner, S. A comprehensive classification of complex statistical systems and an axiomatic derivation of their entropy and distribution functions. Europhys. Lett.
**2011**, 93, 20006. [Google Scholar] [CrossRef] - Hanel, R.; Thurner, S. When do generalised entropies apply? How phase space volume determines entropy. Europhys. Lett.
**2011**, 96, 50003. [Google Scholar] [CrossRef] - Schwammle, V.; Tsallis, C. Two-parameter generalization of the logarithm and exponential functions and Boltzmann-Gibbs-Shannon entropy. J. Math. Phys.
**2007**, 48, 113301. [Google Scholar] [CrossRef] - Tempesta, P. Beyond the Shannon-Khinchin formulation: The composability axiom and the universal-group entropy. Ann. Phys.
**2016**, 365, 180. [Google Scholar] [CrossRef] - Curado, E.M.F. General aspects of the thermodynamical formalism. Braz. J. Phys.
**1999**, 29, 36. [Google Scholar] [CrossRef] - Curado, E.M.F.; Nobre, F.D. On the stability of analytic entropic forms. Phys. A Stat. Mech. Appl.
**2004**, 335, 94. [Google Scholar] [CrossRef] - Tsekouras, G.A.; Tsallis, C. Generalized entropy arising from a distribution of q-indices. Phys. Rev. E
**2005**, 71, 046144. [Google Scholar] [CrossRef] - Pal, N.R.; Pal, S.K. Object-background segmentation using new definitions of entropy. IEE Proc. E Comput. Digit. Tech.
**1989**, 136, 284–295. [Google Scholar] [CrossRef] - Callen, H.B. Thermodynamics and an Introduction to Thermostatistics, 2nd ed.; Wiley: Hoboken, NJ, USA, 1985. [Google Scholar]
- Balian, R. From Microphysics to Macrophysics; Springer: Berlin, Germany, 1991. [Google Scholar]
- Ferri, G.L.; Martinez, S.; Plastino, A. Equivalence of the four versions of Tsallis’ statistics. J. Stat. Mech. Theory Exp.
**2005**, 2005, P04009. [Google Scholar] [CrossRef] - Ferri, G.L.; Martinez, S.; Plastino, A. The role of constraints in Tsallis’ nonextensive treatment revisited. Phys. A Stat. Mech. Appl.
**2005**, 345, 493. [Google Scholar] [CrossRef] - Tsallis, C.; Mendes, R.S.; Plastino, A.R. The role of constraints within generalized nonextensive statistics. Phys. A Stat. Mech. Appl.
**1998**, 261, 534. [Google Scholar] [CrossRef] - Tsallis, C.; Plastino, A.R.; Alvarez-Estrada, R.F. Escort mean values and the characterization of power-law-decaying probability densities. J. Math. Phys.
**2009**, 50, 043303. [Google Scholar] [CrossRef] - Andrade, J.S., Jr.; da Silva, G.F.T.; Moreira, A.A.; Nobre, F.D.; Curado, E.M.F. Thermostatistics of overdamped motion of interacting particles. Phys. Rev. Lett.
**2010**, 105, 260601. [Google Scholar] [CrossRef] [PubMed] - Ribeiro, M.S.; Nobre, F.D.; Curado, E.M.F. Classes of N-Dimensional nonlinear Fokker-Planck equations associated to Tsallis entropy. Entropy
**2011**, 13, 1928–1944. [Google Scholar] [CrossRef] - Ribeiro, M.S.; Nobre, F.D.; Curado, E.M.F. Time evolution of interacting vortices under overdamped motion. Phys. Rev. E
**2012**, 85, 021146. [Google Scholar] [CrossRef] - Curado, E.M.F.; Souza, A.M.C.; Nobre, F.D.; Andrade, R.F.S. Carnot cycle for interacting particles in the absence of thermal noise. Phys. Rev. E
**2014**, 89, 022117. [Google Scholar] [CrossRef] - Andrade, R.F.S.; Souza, A.M.C.; Curado, E.M.F.; Nobre, F.D. A thermodynamical formalism describing mechanical interactions. EPL
**2014**, 108, 20001. [Google Scholar] [CrossRef] - Nobre, F.D.; Curado, E.M.F.; Souza, A.M.C.; Andrade, R.F.S. Consistent thermodynamic framework for interacting particles by neglecting thermal noise. Phys. Rev. E
**2015**, 91, 022135. [Google Scholar] [CrossRef] - Ribeiro, M.S.; Casas, G.A.; Nobre, F.D. Second law and entropy production in a nonextensive system. Phys. Rev. E
**2015**, 91, 012140. [Google Scholar] [CrossRef] - Vieira, C.M.; Carmona, H.A.; Andrade, J.S., Jr.; Moreira, A.A. General continuum approach for dissipative systems of repulsive particles. Phys. Rev. E
**2016**, 93, 060103(R). [Google Scholar] [CrossRef] - Ribeiro, M.S.; Nobre, F.D. Repulsive particles under a general external potential: Thermodynamics by neglecting thermal noise. Phys. Rev. E
**2016**, 94, 022120. [Google Scholar] [CrossRef] - Tsallis, C.; Haubold, H.J. Boltzmann-Gibbs entropy is sufficient but not necessary for the likelihood factorization required by Einstein. EPL
**2015**, 110, 30005. [Google Scholar] [CrossRef] - Latora, V.; Baranger, M.; Rapisarda, A.; Tsallis, C. The rate of entropy increase at the edge of chaos. Phys. Lett. A
**2000**, 273, 97. [Google Scholar] [CrossRef] - Baldovin, F.; Robledo, A. Nonextensive Pesin identity - Exact renormalization group analytical results for the dynamics at the edge of chaos of the logistic map. Phys. Rev. E
**2004**, 69, 045202(R). [Google Scholar] [CrossRef] - Tirnakli, U.; Tsallis, C.; Beck, C. A closer look at time averages of the logistic map at the edge of chaos. Phys. Rev. E
**2009**, 79, 056209. [Google Scholar] [CrossRef] - Tirnakli, U.; Tsallis, C. Extensive numerical results for integrable case of standard map. Nonlinear Phenom. Complex Syst.
**2020**, 23, 149–152. [Google Scholar] [CrossRef] - Bountis, A.; Veerman, J.J.P.; Vivaldi, F. Cauchy distributions for the integrable standard map. Phys. Lett. A
**2020**, 384, 126659. [Google Scholar] [CrossRef] - Tirnakli, U.; Borges, E.P. The standard map: From Boltzmann-Gibbs statistics to Tsallis statistics. Nat. Sci. Rep.
**2016**, 6, 23644. [Google Scholar] [CrossRef] [PubMed] - Caruso, F.; Tsallis, C. Nonadditive entropy reconciles the area law in quantum systems with classical thermodynamics. Phys. Rev. E
**2008**, 78, 021102. [Google Scholar] [CrossRef] [PubMed] - Cirto, L.J.L.; Assis, V.R.V.; Tsallis, C. Influence of the interaction range on the thermostatistics of a classical many-body system. Physica A
**2014**, 393, 286–296. [Google Scholar] [CrossRef] - Cirto, L.J.L.; Rodriguez, A.; Nobre, F.D.; Tsallis, C. Validity and failure of the Boltzmann weight. EPL
**2018**, 123, 30003. [Google Scholar] [CrossRef] - Moreira, A.A.; Vieira, C.M.; Carmona, H.A.; Andrade, J.S., Jr.; Tsallis, C. Overdamped dynamics of particles with repulsive power-law interactions. Phys. Rev. E
**2018**, 98, 032138. [Google Scholar] [CrossRef] - Lutz, E. Anomalous diffusion and Tsallis statistics in an optical lattice. Phys. Rev. A
**2003**, 67, 051402(R). [Google Scholar] [CrossRef] - Douglas, P.; Bergamini, S.; Renzoni, F. Tunable Tsallis distributions in dissipative optical lattices. Phys. Rev. Lett.
**2006**, 96, 110601. [Google Scholar] [CrossRef] - Lutz, E.; Renzoni, F. Beyond Boltzmann-Gibbs statistical mechanics in optical lattices. Nat. Phys.
**2013**, 9, 615–619. [Google Scholar] [CrossRef] - Tsallis, C.; Bukman, D.J. Anomalous diffusion in the presence of external forces: Exact time-dependent solutions and their thermostatistical basis. Phys. Rev. E
**1996**, 54, R2197. [Google Scholar] [CrossRef] - Plastino, A.R.; Plastino, A. Non-extensive statistical mechanics and generalized Fokker-Planck equation. Physica A
**1995**, 222, 347. [Google Scholar] [CrossRef] - Combe, G.; Richefeu, V.; Stasiak, M.; Atman, A.P.F. Experimental validation of nonextensive scaling law in confined granular media. Phys. Rev. Lett.
**2015**, 115, 238301. [Google Scholar] [CrossRef] - Wong, C.Y.; Wilk, G.; Cirto, L.J.L.; Tsallis, C. From QCD-based hard-scattering to nonextensive statistical mechanical descriptions of transverse momentum spectra in high-energy pp and pp collisions. Phys. Rev. D
**2015**, 91, 114027. [Google Scholar] [CrossRef] - Yalcin, G.C.; Beck, C. Generalized statistical mechanics of cosmic rays: Application to positron-electron spectral indices. Sci. Rep.
**2018**, 8, 1764. [Google Scholar] [CrossRef] - Nivanen, L.; Mehaute, A.L.; Wang, Q.A. Generalized algebra within a nonextensive statistics. Rep. Math. Phys.
**2003**, 52, 437. [Google Scholar] [CrossRef] - Borges, E.P. A possible deformed algebra and calculus inspired in nonextensive thermostatistics. Phys. A Stat. Mech. Appl.
**2004**, 340, 95–101, Corrigenda in Phys. A Stat. Mech. Appl. 2021, 581, 126206. [Google Scholar] [CrossRef] - Umarov, S.; Tsallis, C.; Steinberg, S. On a q-central limit theorem consistent with nonextensive statistical mechanics. Milan J. Math.
**2008**, 76, 307–328. [Google Scholar] [CrossRef] - Umarov, S.; Tsallis, C.; Gell-Mann, M.; Steinberg, S. Generalization of symmetric α-stable Lévy distributions for q > 1. J. Math. Phys.
**2010**, 51, 033502. [Google Scholar] [CrossRef] - Umarov, S.; Tsallis, C. Mathematical Foundations of Nonextensive Statistical Mechanics; World Scientific: Singapore, 2022; in press. [Google Scholar]
- Tsallis, C.; Gell-Mann, M.; Sato, Y. Asymptotically scale-invariant occupancy of phase space makes the entropy S
_{q}extensive. Proc. Natl. Acad. Sci. USA**2005**, 102, 15377. [Google Scholar] [CrossRef] - Gazeau, J.-P.; Tsallis, C. Moebius transforms, cycles and q-triplets in statistical mechanics. Entropy
**2019**, 21, 1155. [Google Scholar] [CrossRef] - Amador, C.H.S.; Zambrano, L.S. Evidence for energy regularity in the Mendeleev periodic table. Phys. A Stat. Mech. Appl.
**2010**, 389, 3866–3869. [Google Scholar] [CrossRef] - Ludescher, J.; Tsallis, C.; Bunde, A. Universal behaviour of interoccurrence times between losses in financial markets: An analytical description. Europhys. Lett.
**2011**, 95, 68002. [Google Scholar] [CrossRef] - Ludescher, J.; Bunde, A. Universal behavior of the interoccurrence times between losses in financial markets: Independence of the time resolution. Phys. Rev.
**2014**, 90, 062809. [Google Scholar] [CrossRef] - Tsallis, C. Economics and finance: q-statistical features galore. Entropy
**2017**, 19, 457. [Google Scholar] [CrossRef] - Ruiz, G.; de Marcos, A.F. Evidence for criticality in financial data. Eur. Phys. J. B
**2018**, 91, 1. [Google Scholar] [CrossRef] - Bogachev, M.I.; Kayumov, A.R.; Bunde, A. Universal internucleotide statistics in full genomes: A footprint of the DNA structure and packaging? PLoS ONE
**2014**, 9, e112534. [Google Scholar] [CrossRef] - Tsallis, C.; Stariolo, D.A. Generalized simulated annealing. Phys. A Stat. Mech. Appl.
**1996**, 233, 395, First appeared in C. Tsallis and D.A. Stariolo, Generalized simulated annealing. Notas de Fisica/CBPF 026 (June 1994). [Google Scholar] [CrossRef] - Price, D.J.D.S. Networks of scientific papers. Science
**1965**, 149, 510–515. [Google Scholar] [CrossRef] - Watts, D.J.; Strogatz, S.H. Collective dynamics of “small-world” networks. Nature
**1998**, 393, 440–442. [Google Scholar] [CrossRef] [PubMed] - Barabasi, A.L.; Albert, R. Emergence of scaling in random networks. Science
**1999**, 286, 509–512. [Google Scholar] [CrossRef] [PubMed] - Newman, M.E.J. The structure of scientific collaboration networks. Proc. Natl. Acad. Sci. USA
**2001**, 98, 404–409. [Google Scholar] [CrossRef] [PubMed] - Soares, D.J.; Tsallis, C.; Mariz, A.M.; da Silva, L.R. Preferential attachment growth model and nonextensive statistical mechanics. EPL
**2005**, 70, 70. [Google Scholar] [CrossRef] - Thurner, S.; Tsallis, C. Nonextensive aspects of self-organized scale-free gas-like networks. EPL
**2005**, 72, 197. [Google Scholar] [CrossRef] - Boccaletti, S.; Latora, V.; Moreno, Y.; Chavez, M.; Hwang, D.U. Complex networks: Structure and dynamics. Phys. Rep.
**2006**, 424, 175–308. [Google Scholar] [CrossRef] - Brito, S.; da Silva, L.R.; Tsallis, C. Role of dimensionality in complex networks. Sci. Rep.
**2016**, 6, 27992. [Google Scholar] [CrossRef] - Cinardi, N.; Rapisarda, A.; Tsallis, C. A generalised model for asymptotically-scale-free geographical networks. J. Stat. Mech. Theory Exp.
**2020**, 2020, 043404. [Google Scholar] [CrossRef] - de Oliveira, R.M.; Brito, S.; da Silva, L.R.; Tsallis, C. Connecting complex networks to nonadditive entropies. Sci. Rep.
**2021**, 11, 1130. [Google Scholar] [CrossRef] - Mohanalin, J.; Beenamol, B.; Kalra, P.K.; Kumar, N. A novel automatic microcalcification detection technique using Tsallis entropy and a type II fuzzy index. Comput. Math. Appl.
**2010**, 60, 2426–2432. [Google Scholar] [CrossRef] - Al-Azawi, R.J.; Al-Saidi, N.M.G.; Jalab, H.A.; Kahtan, H.; Ibrahim, R.W. Efficient classification of COVID-19 CT scans by using q-transform model for feature extraction. PeerJ. Comput. Sci.
**2021**, 7, e553. [Google Scholar] [CrossRef] - Greco, A.; Tsallis, C.; Rapisarda, A.; Pluchino, A.; Fichera, G.; Contrafatto, L. Acoustic emissions in compression of building materials: Q-statistics enables the anticipation of the breakdown point. Eur. Phys. J. Spec. Top.
**2020**, 229, 841–849. [Google Scholar] [CrossRef] - Stavrakas, I.; Triantis, D.; Kourkoulis, S.K.; Pasiou, E.D.; Dakanali, I. Acoustic emission analysis of cement mortar specimens during three point bending tests. Lat. Am. J. Solids Struct.
**2016**, 13, 2283–2297. [Google Scholar] [CrossRef]

**Figure 1.**Typical behaviors of $W\left(N\right)$ (number of nonzero-probability states of a system with N random variables) in the $N\to \infty $ limit and entropic functionals which, under the assumption of equal probabilities for all states with nonzero probability, yield extensive entropies for specific values of the corresponding (nonadditive) entropic indices. In what concerns the exponential class $W\left(N\right)\sim A{\mu}^{N}$, ${S}_{BG}$ is not the unique entropy that yields entropic extensivity; the (additive) Renyi entropic functional ${S}_{q}^{R}$ also is extensive for all values of q. Analogously, in what concerns the stretched-exponential class $W\left(N\right)\sim C{\nu}^{{N}^{\gamma}}$, the (nonadditive) entropic functional ${S}_{\delta}$ is not unique. All the entropic families illustrated in this table contain ${S}_{BG}$ as a particular case, except ${S}_{\lambda}^{C}$, which is appropriate for the logarithmic class $W\left(N\right)\sim DlnN$. In the limit $N\to \infty $, the inequalities ${\mu}^{N}\gg {\nu}^{{N}^{\gamma}}\gg {N}^{\rho}\gg lnN\gg 1$ are satisfied, hence ${lim}_{N\to \infty}{\nu}^{{N}^{\gamma}}/{\mu}^{N}={lim}_{N\to \infty}{N}^{\rho}/{\mu}^{N}={lim}_{N\to \infty}lnN/{\mu}^{N}=0$. This exhibits that, in all these nonadditive cases, the occupancy of the full phase space corresponds essentially to zero Lebesgue measure, similarly to a whole class of (multi) fractals. If the equal probabilities hypothesis is not satisfied, specific analysis becomes necessary and the results might be different.

**Figure 2.**The classical systems for which $\alpha /d>1$ correspond to an extensive total energy and typically involve absolutely convergent series, whereas the so-called nonextensive systems ($0\le \alpha /d<1$ for the classical ones) correspond to a superextensive total energy and typically involve divergent series. The marginal systems ($\alpha /d=1$ here) typically involve conditionally convergent series, which therefore depend on the boundary conditions, i.e., typically on the external shape of the system. Capacitors constitute a notorious example of the $\alpha /d=1$ case. The model usually referred to in the literature as the Hamiltonian-Mean-Field (HMF) one [35] lies on the $\alpha =0$ axis (for all values of $d>0$). The models usually referred to as the d-dimensional $\alpha $-XY [36], $\alpha $-Heisenberg [37,38,39] and $\alpha $-Fermi–Pasta–Ulam (or $\alpha $-Fermi–Pasta–Ulam–Tsingou problem [40,41]) [42,43,44] models lie parallel to the vertical axis at abscissa d (for all values of $\alpha \ge 0$). The standard Lennard–Jones gas is located at $(d,\alpha )=(3,6)$. From [33].

**Figure 3.**Schematic representation of the different scaling regimes for classical d-dimensional systems. In the case of attractive long-ranged interactions (i.e., $0\le \alpha /d\le 1$, $\alpha $ characterizing the interaction range in a potential with the form $1/{r}^{\alpha}$; for example, Newtonian gravitation corresponds to $(d,\alpha )=(3,1)$), we may distinguish three classes of thermodynamic variables, namely, those scaling with ${L}^{\theta}$, named pseudo-intensive (L is a characteristic linear length, $\theta $ is a system-dependent parameter), those scaling with ${L}^{d+\theta}$ with $\theta =d-\alpha $, the pseudo-extensive ones (the energies), and those scaling with ${L}^{d}$ (which are always extensive). In the case of short-ranged interactions (i.e., $\alpha >d$), we have $\theta =0$ and the energies recover their standard ${L}^{d}$ extensive scaling, falling within the same class of S, N, V, M, etc., whereas the previous pseudo-intensive variables become truly intensive ones (independent of L); this is the region, with only two classes of variables that is covered by the traditional textbooks of thermodynamics. From [26,30,33,47,48,49].

**Figure 4.**It has been proved [54] 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 ${S}_{q}^{R}$ [14], the Tempesta $(a,b,\alpha )$-entropy ${S}_{a,b,\alpha}^{T}$ (Equation (9.1) in [57]), the Jensen–Pazuki–Pruessner–Tempesta entropy ${S}_{\gamma ,\alpha}^{JPPT}$ [58] 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 ${S}_{q,r}^{SM}$ [20,21,22], Landsberg–Vedral–Rajagopal–Abe entropy ${S}_{q}^{LVRA}$ [59,60,61], Tsallis–Mendes–Plastino entropy ${S}_{q}^{TMP}$, Arimoto entropy ${S}_{q}^{Ar}$ [62], Curado–Tempesta–Tsallis entropy ${S}_{a,b,r}^{CTT}$ [63], Borges–Roditi entropy ${S}_{q,{q}^{\prime}}^{BR}$ [64], Abe entropy ${S}_{q}^{Ab}$ [65], Kaniadakis entropy ${S}_{\kappa}^{K}$ [66,67,68], Kaniadakis–Lissia–Scarfone entropy ${S}_{\kappa ,r}^{KLS}$ [69], Anteneodo–Plastino entropy ${S}_{\eta}^{AP}$ [70], Hanel–Thurner entropy ${S}_{c,d}^{HT}$ [71,72], ${S}_{q,\delta}$ [30], Schwammle–Tsallis entropy ${S}_{q,{q}^{\prime}}^{ST}$ [73], the Tempesta $(\alpha ,\beta ,q)$-entropy ${S}_{\alpha ,\beta ,q}^{T}$ [74], the Curado b-entropy ${S}_{b}^{C}$ [75,76], the Curado $\lambda $-entropy ${S}_{\lambda}^{C}$ [33], and the exponential c-entropy ${S}_{c}^{E}$ [77]; one more exponential form is, in fact, available in the literature, namely the Pal and Pal non-composable trace-form entropy ${S}^{PP}$ [78]. The entropic form ${S}_{\lambda}^{C}$ is a rare case which does not include ${S}_{BG}$ and is neither trace-form nor composable. From [33].

**Figure 5.**Time evolution of ${S}_{q}$ for $a=2$. The interval $[-1,1]$ is partitioned into W equal cells. The initial distribution consists of $M={10}^{6}$ points placed at random inside a randomly picked cell. We indicate three typical values of q and the two cases $W={10}^{4}$ and $W={10}^{5}$. Results are averages over 100 runs. From [95].

**Figure 6.**Numerical confirmation (full circles) of the q-generalized Pesin-like identity ${K}_{q}^{\left(k\right)}={\lambda}_{q}^{\left(k\right)}$ at the logistic-map edge of chaos. On the ordinate, we plot the q-logarithm of ${\xi}_{{t}_{k}}$ (equal to ${\lambda}_{q}^{\left(k\right)}\phantom{\rule{0.166667em}{0ex}}t$), and, on the abscissa, ${S}_{q}$ (equal to ${K}_{q}^{\left(q\right)}\phantom{\rule{0.166667em}{0ex}}t$), both for $q=0.2445...$ The dashed line is a linear fit. Inset: The full lines are from the analytic result. From [96].

**Figure 7.**Data collapse of probability density functions (in log-linear plots) for $T={2}^{2n}$, where $2n$ is an odd number (

**top**) or an even number (

**bottom**). As n increases, good fits with ${q}_{attractor}$-Gaussian $\propto {e}_{{q}_{attractor}}^{-\beta {y}^{2}}$ (with $({q}_{attractor},\beta )=(1.63,6.2)$ (

**top**), and $({q}_{stat},\beta )=(1.70,6.2)$ (

**bottom**)) are obtained for increasingly large regions. Insets: Linear-linear plots of the data for a better visualization of the central part. From [97].

**Figure 8.**Phase portrait of the standard map for representative values of K. In each case, black dots represent the region of chaotic sea in the available phase space and all other colors represent different stability islands. From [100].

**Figure 9.**Lyapunov exponent results of the phase portrait of the standard map. The same representative K values are used. For each case, Lyapunov exponents are calculated for 200,000 initial conditions. In the calculation, each initial condition is iterated 107 times. From [100].

**Figure 12.**The index ${q}_{entropy}$ as a function of the inverse central charge $1/c$. The universality classes of some specific models are indicated, see [94]. The BG value ${q}_{entropy}=1$ is recovered in the $c\to \infty $ limit.

**Figure 13.**A typical single-initial-condition one-momentum distribution $P\left(p\right)$ for $N={10}^{6}$, $u=0.69$, $\tau =1$ (corresponding to 5 molecular-dynamics steps), calculated in the region $[{t}_{min};{t}_{max}]=[200,000;500,000]$ for $\alpha =0.9$ (top plot), and $\alpha =2.0$ (bottom plot). The upper temperature indicated in the $\alpha =0.9$ inset coincides with that analytically calculated within BG statistical mechanics, namely ${T}_{kin}\equiv 2K\left(t\right)/N\simeq 0.475$. The horizontal line of the $\alpha =2.0$ inset corresponds to the numerically calculated time average; indeed, analytical solutions are only available for $\alpha <1$ and in the $\alpha \to \infty $ limit. The continuous curves correspond to $P\left(\tilde{p}\right)/{P}_{0}={e}_{{q}_{n}}^{-{\beta}_{{q}_{n}}^{\left({P}_{0}\right)}\left[\tilde{p}{P}_{0}\right){]}^{2}/2}$ with $({q}_{n},{\beta}_{{q}_{n}}^{\left({P}_{0}\right)})=(1.58,11.2)$ for $\alpha =0.9$ and (1.0,6.4) for $\alpha =2.0$. Notice that, for $\alpha =0.9$, $1/{\beta}_{{q}_{n}}^{\left({P}_{0}\right)}\ne T$. Each distribution has been rescaled with its own ${P}_{0}$. From [102].

**Figure 14.**Distributions of the time-averages of the momenta ${\overline{p}}_{i}$ and of the energies ${\overline{E}}_{i}$ (with $\tau =1$) for $\alpha /d=0.9$, in $d=1$, 2 and 3 dimensions. The simulations were done for the energy per particle $u=0.69$ and total number of rotators $N={10}^{6}$.

**Top**: Distribution $P\left({\overline{p}}_{i}\right)$ is shown (${P}_{0}\equiv P({\overline{p}}_{i}=0)$), the full line being a q-Gaussian with ${q}_{p}=1.59$ and ${\beta}_{p}=5.6$, and the dashed line being a Gaussian $(q=1)$. The left inset shows the same data in a q-logarithm versus squared-momentum representation; a straight line is obtained as expected (since ${ln}_{q}\left({e}_{q}^{x}\right)=x$).

**Bottom**: The full line represents the q-exponential $P\left({\overline{E}}_{i}\right)=P\left(\mu \right){exp}_{{q}_{E}}[-{\beta}_{E}({\overline{E}}_{i}-\mu )]$, with ${q}_{E}=1.31$ (${\beta}_{E}=48.0$, $\mu =0.69$, and $P\left(\mu \right)=12$); the corresponding exponential (dashed line) is also shown for comparison. Since the density of states is necessary to reproduce the entire range of data, a chemical potential $\mu $ was introduced in the fitting. The bottom inset shows a straight line by using the q-logarithm in the ordinate. The kinetic temperature $T\left(t\right)\equiv 2K\left(t\right)/N$ and time window $\Delta t$, along which the time averages were calculated, coincide in both cases (shown as insets). One notices that, in all plots, the collapse of all dimensions occurs with nearly the same value of q. From [103].

**Figure 15.**Quantum Monte Carlo verification (

**left panels**) [(

**a**) Analytic and numerical distributions; (

**b**) Analytical and numerical functions $q({U}_{0}/{E}_{r})$], and laboratory verification with ${C}_{s}$ atoms (

**right panels**) [(

**Top a**) Analytical and experimental distributions; (

**Bottom b**) Experimental frequency dependence of q; (

**Left a**) Linear-linear representation of the experimental distribution of momenta, the black curve corresponding to the present q-Gaussian, the red curve corresponding to a Maxwellian distribution; (

**Right b**) The same in log-log representation.] of the 2003 Lutz prediction. From [106].

**Figure 16.**Probability density functions of the horizontal components of the randomly fluctuating displacements tracked during two typical increments of shear strain ($\Delta \gamma =7.3\times {10}^{-4}$ and $\Delta \gamma ={10}^{-1}$). The scatters correspond to experimental data, and the solid lines correspond to q-Gaussian fittings. From [110].

**Figure 17.**Evolution of the measured value q as a function of the squared inverse of the strain increment for both the experiments and simulations. The dashed line corresponds to a regression using the function $q(1/\sqrt{\Delta \gamma})=1+atanh(b/\sqrt{\Delta \gamma})$, with $(a,b)=(0.521,0.096)$. Inset: The same plot for data from a simulation that highlights the limit $q=1$ when $\Delta \gamma \to \infty $. The fitted parameters for simulations were $(a,b)=(0.387,0.057)$. From [110].

**Figure 18.**Verification of the scaling law $\alpha =2/(3-q)$ [108] for several regimes of diffusion.

**Top**: Evolution of the measured diffusion exponent $\alpha $ as a function of $1/\sqrt{\Delta \gamma}$; the dashed line is a direct application of the scaling law from the fit of the values shown in Figure 17, $\alpha (1/\sqrt{\Delta \gamma})=2/[3-q(1/\sqrt{\Delta \gamma})]$.

**Inset**: A typical diffusion curve showing the mean square displacement fluctuations $\langle {x}^{2}\rangle $ as a function of the shear strain $\gamma $; this allows the assessment of the diffusion exponent $\alpha \equiv \mu $ for each strain window tested. In the case shown, it corresponds to the smallest strain window, the rightmost point in the curve in the main panel. Note that for a constant strain rate, $\gamma $ is proportional to time.

**Bottom**: Measure of the deviation of the data relative to the scaling law prediction ${\alpha}_{P}=2/(3-q)$, as a function of $1/\sqrt{\Delta \gamma}$, showing a remarkable agreement of the order of $\pm 2\%$. From [110].

**Figure 19.**Experimental transverse momentum distribution of hadrons in $pp$ collisions at central rapidity y compared with theoretical q-exponentials with $q\simeq 1.13\pm 0.02$ and $T\simeq (0.14\pm 0.01)\phantom{\rule{0.166667em}{0ex}}GeV$. The corresponding Boltzmann–Gibbs (purely exponential) fit is illustrated as the dashed curve. The data and the analytical curves have been divided by a constant factor as indicated, for a better visualization. The ratios data/fit are shown at the bottom, where a nearly log-periodic behavior is observed on top of the q-exponential one. See [111] for details.

**Figure 20.**The measured AMS-02 data are very well fitted by linear combination of escort and non-escort distributions (solid lines); ${q}_{1}=13/11$ and ${q}_{2}=1/(2-{q}_{1})=11/9$. From [112].

**Figure 22.**

**Left panel**: Cumulative distributions of absolute normalized returns corresponding to different time scales $\Delta t$ for the 100 American companies with the highest market capitalization (points), and the fitted cumulative q-Gaussian distributions (lines). In order to better visualize the associated results, each q-Gaussian CDF and the respective experimental data have been multiplied by a positive factor $c\ne 1$.

**Right panel**: Dependence of the index q on the time scale $\Delta t$, for the estimated q-Gaussian pdfs of normalized absolute returns in the left panel. Inset: log-log representation exhibiting a power-law dependence of the type $q-1\propto {\left(\Delta t\right)}^{-\tau}$ , with $\tau =0.081\pm 0.004$. From [124].

**Figure 23.**PDFs of the inter-nucleotide intervals A-A, T-T (open symbols); G-G, C-C (full symbols) in the DNA sequences from Homo Sapiens and Bacteria full genomes (in scaled form). Dashed lines show the best fits by a q-exponential distribution $A=1/{[1+(q-1)\beta (l/L)]}^{\frac{1}{q-1}}$. While in Bacteria the approximation by a single q-exponential with $q\sim 1.1$ and $\beta \sim 1.5$ is possible, in H. Sapiens, a sum of two q-exponentials with $q\sim 1.11$ and $\beta \sim 1.5$ and 0.1 makes the best fit. To avoid overlapping, the PDFs for Bacteria are shifted downwards by two decades. For comparison, dotted lines show corresponding exponential PDFs. From [125].

**Figure 24.**Sample of a network with 100 sites for the following model parameters $(d,{\alpha}_{A},{\alpha}_{G},\eta ,{w}_{0})=(2,1,5,1,1)$, where ${\alpha}_{A}$ and ${\alpha}_{G}$ respectively characterize the attachment and geographical ranges; $\eta $ and ${w}_{0}$ are parameters of the distribution of widths. As illustrated for this choice of parameters, hubs (highly connected nodes) naturally emerge in the network. Each link has a specific width ${w}_{ij}$ and the total energy ${\epsilon}_{i}$ associated with the site i will be given by half of the sum over all link widths connected to the site i (see zoom of site i). From [136].

**Figure 25.**In all cases, the energy distribution is well fitted with the form ${p}_{q}\propto {e}_{q}^{-{\beta}_{q}\epsilon}$. (

**a**) q as a function of ${\alpha}_{A}/d$ (black solid line); $q=4/3$ for $0\le {\alpha}_{A}/d\le 1$ and decreases exponentially with ${\alpha}_{A}/d$ for ${\alpha}_{A}/d>1$, down to ${q}_{\infty}=1$. (

**b**) ${\beta}_{q}$ as a function of ${\alpha}_{A}/d$ for $\eta =1,2,3$ and ${w}_{0}=1,5,10$, for typical values of ${\alpha}_{A}/d$; ${\beta}_{q}$ increases with $\eta $ and decreases with ${w}_{0}$ and ${\alpha}_{A}/d$. From [136].

**Figure 26.**Without q-entropy enhancement with $q\ne 1$, detection of microcalcifications is meager: 80.21% Tps (true positives) with 8.1 Fps (false positives), whereas upon introduction of the q-entropy, the results surge to 96.55% Tps with 0.4 Fps. Detection results from the experiment: (

**a**) mdb236, (

**b**) output with the Mcs enhanced, (

**c**) output with the Mcs extracted, (

**d**) mdb216, (

**e**) output with the Mcs enhanced, (

**f**) output with the Mcs extracted. From [137].

**Figure 27.**Sample scans from the dataset before and after enhancement showing infected lungs. (

**a**) Original Computer Tomography scans, with red circles highlighting some regions where fibrosis can be seen; (

**b**) enhanced Computer Tomography scans using $q=0.5$. From [138].

**Figure 28.**Cumulative pdfs of the inter-event times of acoustic emission for the last three loadings of the specimens.

**Left**: C1, C2 (concrete) and

**Right**: B1, B2 (basalt). The q-exponential fittings are also shown. From [139].

**Figure 29.**The values of the entropic index of the q-exponential fits, reproducing the complementary cumulative pdfs obtained from experimental data about the AE inter-event time series for both the concrete (C1, C2—full circles) and basalt (B1, B2—full squares) specimens, are reported as function of $1/{\beta}_{q}$. Linear fits of the reported values are also shown. The macroscopic failure occurs when $1/{\beta}_{q}$ vanishes. The values for q and $1/{\beta}_{q}$ extracted from analogous failure tests with cement mortar specimens in [140] are also reported (triangles). From [139].

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tsallis, C. Entropy. *Encyclopedia* **2022**, *2*, 264-300.
https://doi.org/10.3390/encyclopedia2010018

**AMA Style**

Tsallis C. Entropy. *Encyclopedia*. 2022; 2(1):264-300.
https://doi.org/10.3390/encyclopedia2010018

**Chicago/Turabian Style**

Tsallis, Constantino. 2022. "Entropy" *Encyclopedia* 2, no. 1: 264-300.
https://doi.org/10.3390/encyclopedia2010018