Generalized entropies and logarithms and their duality relations

TL;DR

This paper develops a comprehensive theory of generalized logarithms and entropies for complex systems that violate certain axioms, revealing a duality that determines escort probabilities and links to entropy scaling.

Contribution

It introduces a duality relation that uniquely determines escort probabilities and derives the functional forms of generalized logarithms from axiom violations.

Findings

Duality relation fixes a unique escort probability.

Generalized logarithms relate to entropy's asymptotic scaling.

Complete theory for non-Shannon entropies in complex systems.

Abstract

For statistical systems that violate one of the four Shannon-Khinchin axioms, entropy takes a more general form than the Boltzmann-Gibbs entropy. The framework of superstatistics allows one to formulate a maximum entropy principle with these generalized entropies, making them useful for understanding distribution functions of non-Markovian or non-ergodic complex systems. For such systems where the composability axiom is violated there exist only two ways to implement the maximum entropy principle, one using escort probabilities, the other not. The two ways are connected through a duality. Here we show that this duality fixes a unique escort probability, which allows us to derive a complete theory of the generalized logarithms that naturally arise from the violation of this axiom. We then show how the functional forms of these generalized logarithms are related to the asymptotic scaling behavior of the entropy.