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

2026-03-29

AI Summary

This work addresses the challenge of reconciling power-law distributions in strongly correlated complex systems with thermodynamic consistency. By constructing a thermodynamic framework for power-law statistics based on the renormalized entropy $s_{2-q}$, it integrates macroscopic variational principles with microscopic superstatistics. Introducing the concept of varentropy—the variance of entropy—it unifies macroscopic and microscopic perspectives and reveals the physical origin of the nonextensivity parameter $q$, establishing the thermodynamic relation $|q - 1| \simeq 1/C$, where $C$ denotes the heat capacity. Through asymptotic analysis of the $q$-order generalized factorial, the study achieves a finite and stable thermodynamic limit of order $O(N^0)$, thereby providing a self-consistent and thermodynamically sound foundation for power-law behavior in systems with finite heat capacity.

Technology Category

• Machine Learning: Information Theory
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

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 ($C \to \infty$).

Problem

• power-law statistics
• thermodynamic consistency
• Varentropy
• nonextensive entropy
• finite heat capacity

Innovation

• renormalized entropy
• Varentropy
• power-law statistics
• Superstatistics
• nonextensive thermodynamics