Generalized entropies, density of states, and non-extensivity

TL;DR

This paper explores how skewed probability distributions influence the extensivity of generalized entropy in complex systems with strong interactions or history dependence, challenging traditional assumptions.

Contribution

It introduces a macroscopic formalism linking first-order statistics, higher-order statistics, and configuration space growth in complex systems.

Findings

Distribution skewness affects entropy extensivity.

Knowing two factors constrains the third in system behavior.

Unified framework for understanding complex interactions.

Abstract

The concept of entropy connects the number of possible configurations with the number of variables in large stochastic systems. Independent or weakly interacting variables render the number of configurations scale exponentially with the number of variables, making the Boltzmann-Gibbs-Shannon entropy extensive. In systems with strongly interacting variables, or with variables driven by history-dependent dynamics, this is no longer true. Here we show that contrary to the generally held belief, not only strong correlations or history-dependence, but skewed-enough distribution of visiting probabilities, that is, first-order statistics, also play a role in determining the relation between configuration space size and system size, or, equivalently, the extensive form of generalized entropy. We present a macroscopic formalism describing this interplay between first-order statistics, higher-order statistics, and configuration space growth. We demonstrate that knowing any two strongly restricts the possibilities of the third. We believe that this unified macroscopic picture of emergent degrees of freedom constraining mechanisms provides a step towards finding order in the zoo of strongly interacting complex systems.