A new entropy based on a group-theoretical structure

TL;DR

This paper introduces a multi-parametric entropy based on group theory, generalizing existing entropies and applicable to systems with diverse phase space growth behaviors, including stabilization and freezing phenomena.

Contribution

It presents a novel entropy $S_{a,b,r}$ derived from a group-theoretical framework, extending nonadditive entropies and enabling modeling of complex physical systems.

Findings

Reduces to $S_q$ and $S_{BG}$ in specific parameter limits

Can describe systems with exponential or stabilized phase space growth

Provides a mathematical foundation for systems with frozen degrees of freedom

Abstract

A multi-parametric version of the nonadditive entropy $S_{q}$ is introduced. This new entropic form, denoted by $S_{a,b,r}$, possesses many interesting statistical properties, and it reduces to the entropy $S_q$ for $b=0$, $a=r:=1-q$ (hence Boltzmann-Gibbs entropy $S_{BG}$ for $b=0$, $a=r \to 0$). The construction of the entropy $S_{a,b,r}$ is based on a general group-theoretical approach recently proposed by one of us \cite{Tempesta2}. Indeed, essentially all the properties of this new entropy are obtained as a consequence of the existence of a rational group law, which expresses the structure of $S_{a,b,r}$ with respect to the composition of statistically independent subsystems. Depending on the choice of the parameters, the entropy $S_{a,b,r}$ can be used to cover a wide range of physical situations, in which the measure of the accessible phase space increases say exponentially with the number of particles $N$ of the system, or even stabilizes, by increasing $N$, to a limiting value. This paves the way to the use of this entropy in contexts where a system "freezes" some or many of its degrees of freedom by increasing the number of its constituting particles or subsystems.