Is nonextensive statistics applicable to continuous Hamiltonian systems?

TL;DR

This paper investigates the applicability of nonextensive statistics to continuous Hamiltonian systems, analyzing entropy formulations and their implications for physical properties, and finds limitations for certain parameter ranges.

Contribution

It provides an explicit analysis of nonextensive entropy in continuous systems and discusses the restrictions on its applicability based on the parameter q.

Findings

Nonextensive formalism is valid mainly for q<1 in continuous Hamiltonian systems.

Explicit distribution functions are derived for the ideal gas case.

Application to phenomena with power law decay is problematic for q<1.

Abstract

The homogeneous entropy for continuous systems in nonextensive statistics reads $S^{H}_{q}=k_B\,{(1 - (K \int d\Gamma \rho^{1/q}(\Gamma))^{q})}/({1-q})$, where $\Gamma$ is the phase space variable. Optimization of $S^{H}_{q}$ combined with normalization and energy constraints gives an implicit expression of the distribution function $\rho (\Gamma)$ which can be computed explicitly for the ideal gas. From this result, we compute properties such as the energy fluctuations and the specific heat. Similar results are also presented using the formulation based on the Tsallis entropy. From the analysis, we discuss the validity of the application of the nonextensive formalism to continuous Hamiltonian systems which is found to be restricted to the range $q<1$, which renders problematic its applicability to the class of phenomena exhibiting power law decay.