The unique, universal entropy for complex systems
For researchers in complex systems and statistical mechanics, this work provides a foundational resolution to the long-standing problem of defining a unique entropy for complex systems, though it is primarily theoretical and incremental in nature.
The paper establishes an axiomatic foundation for entropy in complex systems, proving that the coupled entropy maximized by coupled stretched exponential distributions is the unique, universal entropy that satisfies requirements of measuring uncertainty at the informational scale with log-log slope -1 and extensivity across Hanel-Thurner scaling classes. It shows that Tsallis q-statistics misaligns physical modeling and reviews information thermodynamic applications.
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.