Microcanonical Ensemble Extensive Thermodynamics of Tsallis Statistics

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.

arXiv comment10 pages, 1 figure, Published version in Physics Letters A. Added references

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• Fluctuations, correlations and the nonextensivity
• Extensive statistical mechanics based on nonadditive entropy: Canonical ensemble
• Renyi statistics in equilibrium statistical mechanics
• Hadron transverse momentum distributions of the Tsallis normalized and unnormalized statistics
• Self-consistent thermodynamics for the Tsallis statistics in the grand canonical ensemble: Nonrelativistic hadron gas
• Ultrarelativistic transverse momentum distribution of the Tsallis statistics
• Comparison of Tsallis statistics with the Tsallis-factorized statistics in the ultrarelativistic $pp$ collisions
• On thermodynamic self-consistency of generic axiomatic-nonextensive statistics
• Hadron transverse momentum distributions in the Tsallis statistics with escort probabilities
• Equilibrium statistical mechanics for incomplete nonextensive statistics
• Critique of multinomial coefficients method for evaluating Tsallis and Renyi entropies
• Microcanonical equations for the Tsallis entropy
• Thermodynamic relations and fluctuations in the Tsallis statistics
• Finite size effects in the thermodynamics of a free neutral scalar field
• Nonextensive statistics based on Landsberg-Vedral entropy
• The statistics of mesoscopic systems and the physical interpretation of extensive and non-extensive entropies