The unique, universal entropy for complex systems

TL;DR

This paper establishes a new axiomatic foundation for entropy in complex systems, identifying the coupled entropy as the unique universal form satisfying key physical and informational criteria.

Contribution

It introduces the coupled entropy as the only universal entropy measure that meets the axiomatic requirements for complex systems, extending previous frameworks.

Findings

Coupled entropy maximized by coupled stretched exponential distributions.

Coupled entropy is non-additive, reflecting long-range dependence.

Evidence suggests Tsallis q-statistics may misalign with physical modeling.

Abstract

An axiomatic foundation regarding the entropy for complex systems is established. Missing from decades of research was the requirement that entropy must measure the uncertainty at the informational scale of the maximizing distribution, where the log-log slope equals $-1$. Additionally, entropy must be extensive across the full universality scaling classes defined by Hanel-Thurner. The coupled entropy, maximized by the coupled stretched exponential distributions, is proven to be the unique, universal entropy that satisfies these requirements. The non-additivity of the entropy is equal to the long-range dependence or nonlinear statistical coupling. The entropy-matched extensivity is a function of the coupling, stretching parameter, and dimensions. Evidence is provided that the Tsallis $q$-statistics creates misalignment in the physical modeling of complex systems. Information thermodynamic applications are reviewed, including measuring complexity, a zeroth law of temperature, the thermodynamic consistency of the coupled free energy, and a model of intelligence in non-equilibrium.