Extensive statistical mechanics based on nonadditive entropy: Canonical ensemble

TL;DR

This paper revisits the canonical ensemble in nonextensive thermostatistics, demonstrating that with an extensive entropic parameter, Tsallis statistics aligns with equilibrium thermodynamics, especially in the thermodynamic limit.

Contribution

It shows that the entropic parameter must be extensive for proper thermodynamic connection and proves the equivalence of ensembles in the thermodynamic limit within Tsallis statistics.

Findings

Tsallis entropy is extensive in the thermodynamic limit for the perfect gas.

Canonical and microcanonical ensembles are equivalent in the thermodynamic limit.

Finite systems exhibit nonextensivity for both Tsallis and Boltzmann-Gibbs entropies.

Abstract

The original canonical ensemble formalism for the nonextensive entropy thermostatistics is reconsidered. It is shown that the unambiguous connection of the statistical mechanics with the equilibrium thermodynamics is provided if the entropic parameter $1/(q-1)$ is an extensive variable of the state. Based on a particular example of the perfect gas, it is proved that the Tsallis thermostatistics meets all the requirements of equilibrium thermodynamics in the thermodynamic limit. In particular, the entropy of the system is extensive and the temperature is intensive. However, for finite systems both the Tsallis and Boltzmann-Gibbs entropies are nonextensive. The equivalence of the canonical and microcanonical ensembles of Tsallis thermostatistics in the thermodynamic limit is established. The issue associated with physical interpretation of the entropic variable is discussed in detail.