A Fractional entropy in Fractal phase space: properties and characterization

TL;DR

This paper introduces a two-parameter generalization of entropy based on fractional calculus, explores its properties, and applies it to physical systems, providing new tools for analyzing complex systems.

Contribution

It proposes a novel two-parameter fractional entropy, derives its properties, and applies it to physical models, extending the framework of information measures.

Findings

The generalized entropy is maximized by a distribution involving Lambert's W function.

The entropy satisfies Lesche and thermodynamic stability conditions.

Application to two-level and exponential systems shows differences from Shannon-based measures.

Abstract

A two parameter generalization of Boltzmann-Gibbs-Shannon entropy based on natural logarithm is introduced. The generalization of the Shannon-Kinchinn axioms corresponding to the two parameter entropy is proposed and verified. We present the relative entropy, Jensen-Shannon divergence measure and check their properties. The Fisher information measure, relative Fisher information and the Jensen-Fisher information corresponding to this entropy are also derived. The canonical distribution maximizing this entropy is derived and is found to be in terms of the Lambert's W function. Also the Lesche stability and the thermodynamic stability conditions are verified. Finally we propose a generalization of a complexity measure and apply it to a two level system and a system obeying exponential distribution. The results are compared with the corresponding ones obtained using a similar measure based on the Shannon entropy.