Perturbation and Variational Methods in Nonextensive Tsallis Statistics

TL;DR

This paper presents a unified approach to perturbation and variational methods within Tsallis nonextensive statistical mechanics, generalizing key inequalities and illustrating their application to classical mechanics.

Contribution

It introduces a generalized Bogoliubov inequality invariant under the entropic index and applies it to nonextensive systems, advancing theoretical tools in Tsallis statistics.

Findings

Generalized Bogoliubov inequality for Tsallis entropy

Invariant form of the inequality with respect to q

Application example in classical mechanics

Abstract

A unified presentation of the perturbation and variational methods for the generalized statistical mechanics based on Tsallis entropy is given here. In the case of the variational method, the Bogoliubov inequality is generalized in a very natural way following the Feynman proof for the usual statistical mechanics. The inequality turns out to be form-invariant with respect to the entropic index $q$. The method is illustrated with a simple example in classical mechanics. The formalisms developed here are expected to be useful in the discussion of nonextensive systems.