Microcanonical Ensemble Extensive Thermodynamics of Tsallis Statistics

TL;DR

This paper establishes the microscopic foundations of Tsallis statistics within the microcanonical ensemble, demonstrating that under certain conditions, Tsallis entropy becomes extensive and consistent with classical thermodynamics.

Contribution

It provides a thermodynamic derivation of Tsallis statistics showing its consistency with fundamental thermodynamic principles in the microcanonical ensemble.

Findings

Tsallis entropy is extensive in the thermodynamic limit.

The principle of additivity and zero law are satisfied in this framework.

Thermodynamic identities are derived using the Euler theorem.

Abstract

The microscopic foundation of the generalized equilibrium statistical mechanics based on the Tsallis entropy is given by using the Gibbs idea of statistical ensembles of the classical and quantum mechanics. The equilibrium distribution functions are derived by the thermodynamic method based upon the use of the fundamental equation of thermodynamics and the statistical definition of the functions of the state of the system. It is shown that if the entropic index $\xi=1/(q-1)$ in the microcanonical ensemble is an extensive variable of the state of the system, then in the thermodynamic limit $\tilde{z}=1/(q-1)N=const$ the principle of additivity and the zero law of thermodynamics are satisfied. In particular, the Tsallis entropy of the system is extensive and the temperature is intensive. Thus, the Tsallis statistics completely satisfies all the postulates of the equilibrium thermodynamics. Moreover, evaluation of the thermodynamic identities in the microcanonical ensemble is provided by the Euler theorem. The principle of additivity and the Euler theorem are explicitly proved by using the illustration of the classical microcanonical ideal gas in the thermodynamic limit.