A Fractional Entropy in Fractal Phase Space: Properties and Characterization

Chandrashekar Radhakrishnan,Segar Jambulingam,Ravikumar Chinnarasu

doi:10.1155/2014/460364

Abstract

A two-parameter generalization of Boltzmann-Gibbs-Shannon entropy based on natural logarithm is introduced. The generalization of the Shannon-Khinchin 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, the relative Fisher information, and the Jensen-Fisher information corresponding to this entropy are also derived. Also the Lesche stability and the thermodynamic stability conditions are verified. We propose a generalization of a complexity measure and apply it to a two-level system and a system obeying exponential distribution. Using different distance measures we define the statistical complexity and analyze it for two-level and five-level system.

Highlights

Entropy is a very important quantity and plays a key role in many aspects of statistical mechanics and information theory
The most widely used form of entropy was given by Boltzmann and Gibbs from the statistical mechanics point of view and by Shannon from information theory point of view
In [4] a new expression for the entropy was proposed as a generalization of the BoltzmannGibbs entropy and the necessary properties like concavity, Lesche stability, and thermodynamic stability were verified

Summary

A Fractional Entropy in Fractal Phase Space

A two-parameter generalization of Boltzmann-Gibbs-Shannon entropy based on natural logarithm is introduced. The generalization of the Shannon-Khinchin 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, the relative Fisher information, and the Jensen-Fisher information corresponding to this entropy are derived. The Lesche stability and the thermodynamic stability conditions are verified. We propose a generalization of a complexity measure and apply it to a two-level system and a system obeying exponential distribution. Using different distance measures we define the statistical complexity and analyze it for two-level and five-level system

Introduction

Generalized Entropy and Its Axiomatic Characterization

Generalized Divergence Measures

Thermodynamic Properties

Complexity Measures

Conclusions