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Tsallis, C. Entropy. Encyclopedia. Available online: https://encyclopedia.pub/entry/18759 (accessed on 19 September 2026).
Tsallis C. Entropy. Encyclopedia. Available at: https://encyclopedia.pub/entry/18759. Accessed September 19, 2026.
Tsallis, Constantino. "Entropy" Encyclopedia, https://encyclopedia.pub/entry/18759 (accessed September 19, 2026).
Tsallis, C. (2022, January 25). Entropy. In Encyclopedia. https://encyclopedia.pub/entry/18759
Tsallis, Constantino. "Entropy." Encyclopedia. Web. 25 January, 2022.
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Entropy
This is a peer-reviewed entry published in Encyclopedia Journal, full entry can be found at 10.3390/encyclopedia2010018

The concept of entropy constitutes, together with energy, a cornerstone of contemporary physics and related areas. It was originally introduced by Clausius in 1865 along abstract lines focusing on thermodynamical irreversibility of macroscopic physical processes. In the next decade, Boltzmann made the genius connection—further developed by Gibbs—of the entropy with the microscopic world, which led to the formulation of a new and impressively successful physical theory, thereafter named statistical mechanics. The extension to quantum mechanical systems was formalized by von Neumann in 1927, and the connections with the theory of communications and, more widely, with the theory of information were respectively introduced by Shannon in 1948 and Jaynes in 1957. Since then, over fifty new entropic functionals emerged in the scientific and technological literature. The most popular among them are the additive Renyi one introduced in 1961, and the nonadditive one introduced in 1988 as a basis for the generalization of the Boltzmann–Gibbs and related equilibrium and nonequilibrium theories, focusing on natural, artificial and social complex systems. Along such lines, theoretical, experimental, observational and computational efforts, and their connections to nonlinear dynamical systems and the theory of probabilities, are currently under progress. Illustrative applications, in physics and elsewhere, of these recent developments are briefly described in the present synopsis.

thermodynamics statistical mechanics information theory nonlinear dynamical systems strong and weak chaos nonadditive entropies nonextensive statistical mechanics long-range interactions scale-free networks complex systems

References

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  2. 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.
  3. Boltzmann, L. Weitere Studien u̇ber das Wȧrmegleichgewicht unter Gas moleku̇len [Further Studies on Thermal Equilibrium Between Gas Molecules]. Wien. Ber. 1872, 66, 275.
  4. 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.
  5. 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.
  6. Gibbs, J.W. The collected works. In Thermodynamics; Yale University Press: New Haven, CT, USA, 1948; Volume 1.
  7. Gibbs, J.W. Elementary Principles in Statistical Mechanics; OX Bow Press: Woodbridge, CT, USA, 1981.
  8. von Neumann, J. Thermodynamik quantenmechanischer Gesamtheiten. Nachrichten Ges. Wiss. Gott. 1927, 1927, 273–291.
  9. Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J. 1948, 27, 379–423.
  10. Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J. 1948, 27, 623–656.
  11. Shannon, C.E. The Mathematical Theory of Communication; University of Illinois Press: Urbana, IL, USA, 1949.
  12. Jaynes, E.T. Information theory and statistical mechanics. Phys. Rev. 1957, 106, 620–630.
  13. Jaynes, E.T. Information theory and statistical mechanics. II. Phys. Rev. 1957, 108, 171–190.
  14. 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.
  15. Renyi, A. Probability Theory; Dover Publications Inc.: New York, NY, USA, 1970.
  16. Balatoni, J.; Renyi, A. Remarks on entropy. Publ. Math. Inst. Hung. Acad. Sci. 1956, 1, 9–40.
  17. Renyi, A. On the dimension and entropy of probability distributions. Acta Math. Acad. Sci. Hung. 1959, 10, 193–215.
  18. Havrda, J.; Charvat, F. Quantification method of classification processes - Concept of structural α-entropy. Kybernetika 1967, 3, 30–35.
  19. Lindhard, J.; Nielsen, V. Det Kongelige Danske Videnskabernes Selskab Matematisk-fysiske Meddelelser (Denmark). Stud. Stat. Mech. 1971, 38, 1–42.
  20. Sharma, B.D.; Mittal, D.P. New non-additive measures of entropy for discrete probability distributions. J. Math. Sci. 1975, 10, 28.
  21. Sharma, B.D.; Taneja, I.J. Entropy of type (α,β) and other generalized measures in information theory. Metrika 1975, 22, 205.
  22. Mittal, D.P. On some functional equations concerning entropy, directed divergence and inaccuracy. Metrika 1975, 22, 35.
  23. Jaynes, E.T. Gibbs vs. Boltzmann entropies. Am. J. Phys. 1965, 33, 391–398.
  24. Landauer, R. Information is physical. Phys. Today 1991, 44, 23.
  25. 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].
  26. Tsallis, C. Nonextensive Statistical Mechanics—Approaching a Complex World, 1st ed.; Springer: New York, NY, USA, 2009.
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