Trinity of Varentropy: Finiteness, Fluctuations, and Stability in Power-Law Statistics

TL;DR

This paper develops a thermodynamic framework for power-law distributions using renormalized entropy, linking the nonlinearity parameter to heat capacity, and demonstrating stability and finiteness in systems with strong correlations.

Contribution

It introduces a stable thermodynamic limit for power-law statistics based on renormalized entropy and clarifies the physical origin of the nonlinearity parameter q.

Findings

The renormalized entropy remains finite for strongly correlated systems.

The relation |q-1| ≈ 1/C connects the nonlinearity parameter to heat capacity.

Power-law statistics describe finite systems with finite heat capacity.

Abstract

Power-law distributions are widely observed in complex systems, yet establishing their thermodynamic consistency remains a theoretical challenge. In this paper, we present a thermodynamic framework for power-law statistics based on the \textit{renormalized entropy} $s_{2-q}$. Derived from the asymptotic scaling of the combinatorial $q$-factorial, this quantity yields a stable thermodynamic limit, remaining finite ($O(N^0)$) for systems with strong correlations. Furthermore, we clarify the physical origin of the nonlinearity parameter $q$ through the concept of \textit{Varentropy} (Variance of Entropy). By unifying the macroscopic variational principle with the microscopic Superstatistics framework, we derive the relation $|q-1| \simeq 1/C$, where $C$ is the heat capacity of the reservoir. This result suggests that power-law statistics provides a thermodynamic description of finite systems, where the finite heat capacity of the heat bath necessitates a generalization beyond the standard Boltzmann-Gibbs limit ($C \to \infty$).