Trinity of Varentropy: Finiteness, Fluctuations, and Stability in Power-Law Statistics
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arXiv
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| Formato: | Preprint |
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2026
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| _version_ | 1866911552815235072 |
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| author | Suyari, Hiroki |
| author_facet | Suyari, Hiroki |
| contents | 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 ($C \to \infty$). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_27997 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Trinity of Varentropy: Finiteness, Fluctuations, and Stability in Power-Law Statistics Suyari, Hiroki Statistical Mechanics Information Theory Mathematical Physics 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 ($C \to \infty$). |
| title | Trinity of Varentropy: Finiteness, Fluctuations, and Stability in Power-Law Statistics |
| topic | Statistical Mechanics Information Theory Mathematical Physics |
| url | https://arxiv.org/abs/2603.27997 |