Preprint Review Version 1 This version is not peer-reviewed

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

Version 1 : Received: 12 September 2018 / Approved: 13 September 2018 / Online: 13 September 2018 (16:08:39 CEST)

How to cite: Jeldtoft Jensen, H.; Tempesta, P. Group entropies: from phase space geometry to entropy functionals via group theory. Preprints 2018, 2018090247 (doi: 10.20944/preprints201809.0247.v1). Jeldtoft Jensen, H.; Tempesta, P. Group entropies: from phase space geometry to entropy functionals via group theory. Preprints 2018, 2018090247 (doi: 10.20944/preprints201809.0247.v1).

## 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, generalized 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.

## Subject Areas

Phase space volume, formal group theory, entropy