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Review

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

1
Centre for Complexity Science and Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
2
Institute of Innovative Research, Tokyo Institute of Technology, 4259, Nagatsuta-cho, Yokohama 226-8502, Japan
3
Departamento de Física Teórica, Universidad Complutense de Madrid, 28040 Madrid, Spain
4
Instituto de Ciencias Matemáticas (ICMAT), 28049 Madrid, Spain
*
Author to whom correspondence should be addressed.
Entropy 2018, 20(10), 804; https://doi.org/10.3390/e20100804
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)
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
MDPI and ACS Style

Jeldtoft Jensen, H.; Tempesta, P. Group Entropies: From Phase Space Geometry to Entropy Functionals via Group Theory. Entropy 2018, 20, 804. https://doi.org/10.3390/e20100804

AMA Style

Jeldtoft Jensen H, Tempesta P. Group Entropies: From Phase Space Geometry to Entropy Functionals via Group Theory. Entropy. 2018; 20(10):804. https://doi.org/10.3390/e20100804

Chicago/Turabian Style

Jeldtoft Jensen, Henrik; Tempesta, Piergiulio. 2018. "Group Entropies: From Phase Space Geometry to Entropy Functionals via Group Theory" Entropy 20, no. 10: 804. https://doi.org/10.3390/e20100804

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