Monotonically equivalent entropies and solution of additivity equation

TL;DR

This paper characterizes a family of generalized entropies that serve as Lyapunov functions for Markov chains, focusing on properties like universality, trace-form, and additivity, and explores their equivalences and classifications.

Contribution

It identifies all generalized entropies with key properties and classifies pairs of equivalent entropies, including Renyi-Tsallis, based on additivity and trace-form.

Findings

All such entropies form a one-parametric family.

Classes of equivalent entropies include Renyi-Tsallis.

Identified two main families: additive trace-form and Renyi-Tsallis.

Abstract

Generalized entropies are studied as Lyapunov functions for the Master equation (Markov chains). Three basic properties of these Lyapunov functions are taken into consideration: universality (independence of the kinetic coefficients), trace-form (the form of sum over the states), and additivity (for composition of independent subsystems). All the entropies, which have all three properties simultaneously and are defined for positive probabilities, are found. They form a one-parametric family. We consider also pairs of entropies $S_{1}$, $S_{2}$, which are connected by the monotonous transformation $S_{2}=F(S_{1})$ (equivalent entropies). All classes of pairs of universal equivalent entropies, one of which has a trace-form, and another is additive (these entropies can be different one from another), were found. These classes consist of two one-parametric families: the family of entropies, which are equivalent to the additive trace-form entropies, and the family of Renyi-Tsallis entropies.