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

TL;DR

This paper explores how group theory can generalize entropy functionals beyond the classical Boltzmann-Gibbs form, linking phase space geometry, composability, and extensivity to broader entropy classes.

Contribution

It introduces a group-theoretic framework for entropy, replacing additivity with a composability axiom, and classifies entropy functionals based on phase space volume dependence.

Findings

Classifies entropies based on phase space volume growth

Relates group entropies to Gibbs' paradox

Highlights relevance to information 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.