An overview of generalized entropic forms

TL;DR

This paper provides a comprehensive classification of various generalized entropic forms developed over fifty years, analyzing their axiomatic foundations, stability, and composability within statistical physics and information theory.

Contribution

It systematically categorizes main entropic forms based on deformation parameters and discusses their foundational properties and stability criteria.

Findings

Classified entropic forms into three main families.

Analyzed axiomatic foundations and composability rules.

Discussed stability and consistency with Shore-Johnson axioms.

Abstract

The aim of this focus letter is to present a comprehensive classification of the main entropic forms introduced in the last fifty years in the framework of statistical physics and information theory. Most of them can be grouped into three families, characterized by two-deformation parameters, introduced respectively by Sharma, Taneja, and Mittal (entropies of degree $(\alpha,\,\beta$)), by Sharma and Mittal (entropies of order $(\alpha,\,\beta)$), and by Hanel and Thurner (entropies of class $(c,\,d)$). Many entropic forms examined will be characterized systematically by means of important concepts such as their axiomatic foundations {\em \`{a} la} Shannon-Khinchin and the consequent composability rule for statistically independent systems. Other critical aspects related to the Lesche stability of information measures and their consistency with the Shore-Johnson axioms will be briefly discussed on a general ground.