On the Uniqueness Theorem for Pseudo-Additive Entropies

Petr Jizba,Jan Korbel

doi:10.3390/e19110605

Abstract

The aim of this paper is to show that the Tsallis-type (q-additive) entropic chain rule allows for a wider class of entropic functionals than previously thought. In particular, we point out that the ensuing entropy solutions (e.g., Tsallis entropy) can be determined uniquely only when one fixes the prescription for handling conditional entropies. By using the concept of Kolmogorov–Nagumo quasi-linear means, we prove this with the help of Darótzy’s mapping theorem. Our point is further illustrated with a number of explicit examples. Other salient issues, such as connections of conditional entropies with the de Finetti–Kolmogorov theorem for escort distributions and with Landsberg’s classification of non-extensive thermodynamic systems are also briefly discussed.

Highlights

During the last two decades, complex dynamical systems have undergone an important conceptual shift
The statistics associated with power-law tails accounts for a rich class of phenomena observed in many real-world systems ranging from financial markets, physics and biology to geoscience
We have studied the uniqueness of entropic functionals under their composition laws

Summary

Introduction

During the last two decades, complex dynamical systems have undergone an important conceptual shift. If the events are not independent (and do not interact), it may be that some combinations of X and Y are not allowed (i.e., WXY < WX WY ), so that the joint entropy H ( X, Y ) may be less than H ( X ) + H (Y ), a property known as the subadditivity of the entropy This would mean that a system increases its entropy by splitting up into separate systems. In such cases, the above (pseudo-)additivity rules will be replaced with more restrictive entropy chain rules

Degeneracy in Solutions

Part I

Entropic Chain Rule

Theoretical Justification of Degeneracy

Conclusions