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

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 ($C \to \infty$).

Figures (2)

• Figure 1: Microscopic temperature fluctuations described by the Gamma distribution $f(\beta)$ for varying degrees of freedom $n$ of the heat bath. The mean inverse temperature is fixed at $\langle \beta \rangle = \beta_0$. As $n$ increases ($n \to \infty$), the distribution systematically narrows and converges to a Dirac delta function $\delta(\beta - \beta_0)$, physically corresponding to an exact canonical ensemble coupled to an infinite heat bath.
• Figure 2: Relationship between the non-extensivity parameter $q$ and the dimensionless heat capacity $C$ of the reservoir. As the heat capacity diverges in the standard thermodynamic limit ($C \to \infty$), the fluctuations vanish and the system recovers Boltzmann-Gibbs statistics ($q \to 1$). For finite systems with bounded $C$, the parameter $q > 1$ acts as a thermodynamic stabilizer absorbing the macroscopic fluctuations.