Generalized Statistics Variational Perturbation Approximation using q-Deformed Calculus

TL;DR

This paper introduces a generalized variational perturbation approximation framework based on q-deformed calculus within Tsallis nonadditive statistics, providing a new approach for approximating generalized free energy with numerical demonstrations.

Contribution

It develops a novel q-deformed variational perturbation approximation method for Tsallis statistics, extending prior models with a new mathematical framework and numerical validation.

Findings

Derived a q-deformed generalized VPA using Hellmann-Feynman theorem

Established generalized Bogoliubov inequality for Tsallis entropy

Numerical examples illustrate the model's qualitative differences

Abstract

A principled framework to generalize variational perturbation approximations (VPA's) formulated within the ambit of the nonadditive statistics of Tsallis statistics, is introduced. This is accomplished by operating on the terms constituting the perturbation expansion of the generalized free energy (GFE) with a variational procedure formulated using \emph{q-deformed calculus}. A candidate \textit{q-deformed} generalized VPA (GVPA) is derived with the aid of the Hellmann-Feynman theorem. The generalized Bogoliubov inequality for the approximate GFE are derived for the case of canonical probability densities that maximize the Tsallis entropy. Numerical examples demonstrating the application of the \textit{q-deformed} GVPA are presented. The qualitative distinctions between the \textit{q-deformed} GVPA model \textit{vis-\'{a}-vis} prior GVPA models are highlighted.