Group entropies, correlation laws and zeta functions

TL;DR

This paper introduces the concept of group entropy to unify various entropy definitions, explores their mathematical structure via formal group theory, and links statistical mechanics to zeta functions, including a relation between Tsallis entropy and the Riemann zeta function.

Contribution

It proposes a new framework of group entropy that generalizes existing entropies and connects statistical mechanics with zeta functions, expanding theoretical understanding.

Findings

Unified various entropy definitions under group entropy framework.

Established a link between Tsallis entropy and the Riemann zeta function.

Introduced new entropic functionals related to correlation laws in out-of-equilibrium systems.

Abstract

The notion of group entropy is proposed. It enables to unify and generalize many different definitions of entropy known in the literature, as those of Boltzmann-Gibbs, Tsallis, Abe and Kaniadakis. Other new entropic functionals are presented, related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium, when the dynamics is weakly chaotic. The associated thermostatistics are discussed. The mathematical structure underlying our construction is that of formal group theory, which provides the general structure of the correlations among particles and dictates the associated entropic functionals. As an example of application, the role of group entropies in information theory is illustrated and generalizations of the Kullback-Leibler divergence are proposed. A new connection between statistical mechanics and zeta functions is established. In particular, Tsallis entropy is related to the classical Riemann zeta function.