Group entropies: from phase space geometry to entropy functionals via group theory

Abstract

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 number 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, relate to the Gibbs' paradox discussion and explain why group entropies may be particularly relevant from an information theoretic perspective.

12 pages - invited contribution to the journal Entropy's special issue "Nonadditive Entropies and Complex Systems"

References in corpus (4)

• The large deviation approach to statistical mechanics
• Classification of complex systems by their sample-space scaling exponents
• The Lazard formal group, universal congruences and special values of zeta functions
• A new class of entropic information measures, formal group theory and information geometry

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• Multivariate Group Entropies, Super-exponentially Growing Complex Systems and Functional Equations
• A new class of entropic information measures, formal group theory and information geometry
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• Permutation group entropy: a new route to complexity for real-valued processes
• Complexity-based permutation entropies: from deterministic time series to white noise