Is nonextensive statistics applicable to continuous Hamiltonian systems?

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.