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Group Entropies: From Phase Space Geometry to Entropy Functionals via Group Theory

Centre for Complexity Science and Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
Institute of Innovative Research, Tokyo Institute of Technology, 4259, Nagatsuta-cho, Yokohama 226-8502, Japan
Departamento de Física Teórica, Universidad Complutense de Madrid, 28040 Madrid, Spain
Instituto de Ciencias Matemáticas (ICMAT), 28049 Madrid, Spain
Author to whom correspondence should be addressed.
Entropy 2018, 20(10), 804;
Received: 12 September 2018 / Revised: 9 October 2018 / Accepted: 10 October 2018 / Published: 19 October 2018
(This article belongs to the Special Issue Nonadditive Entropies and Complex Systems)
PDF [263 KB, uploaded 19 October 2018]


The entropy of Boltzmann-Gibbs, as proved by Shannon and Khinchin, is based on four axioms, where the fourth one concerns additivity. The group theoretic entropies make use of formal group theory to replace this axiom with a more general composability axiom. As has been pointed out before, generalised entropies crucially depend on the number of allowed degrees of freedom N. The functional form of group entropies is restricted (though not uniquely determined) by assuming extensivity on the equal probability ensemble, which leads to classes of functionals corresponding to sub-exponential, exponential or super-exponential dependence of the phase space volume W on N. We review the ensuing entropies, discuss the composability axiom and explain why group entropies may be particularly relevant from an information-theoretical perspective. View Full-Text
Keywords: generalised entropies; formal groups; phase space growth rate generalised entropies; formal groups; phase space growth rate
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Jeldtoft Jensen, H.; Tempesta, P. Group Entropies: From Phase Space Geometry to Entropy Functionals via Group Theory. Entropy 2018, 20, 804.

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