Equilibrium statistical mechanics for incomplete nonextensive statistics

TL;DR

This paper investigates the properties of incomplete nonextensive statistics in thermodynamics, demonstrating that it aligns with equilibrium thermodynamics when a specific extensive variable is used, and becomes equivalent to Tsallis statistics.

Contribution

It provides exact analytical results showing conditions under which incomplete nonextensive statistics satisfies thermodynamic principles and its equivalence to Tsallis statistics.

Findings

Thermodynamic limit with extensive z satisfies equilibrium thermodynamics.

Incomplete nonextensive statistics becomes equivalent to Tsallis statistics under certain conditions.

When z is intensive, thermodynamic requirements are violated.

Abstract

The incomplete nonextensive statistics in the canonical and microcanonical ensembles is explored in the general case and in a particular case for the ideal gas. By exact analytical results for the ideal gas it is shown that taking the thermodynamic limit, with $z=q/(1-q)$ being an extensive variable of state, the incomplete nonextensive statistics satisfies the requirements of equilibrium thermodynamics. The thermodynamical potential of the statistical ensemble is a homogeneous function of the first degree of the extensive variables of state. In this case, the incomplete nonextensive statistics is equivalent to the usual Tsallis statistics. If $z$ is an intensive variable of state, i.e. the entropic index $q$ is a universal constant, the requirements of the equilibrium thermodynamics are violated.