From the Boltzmann equation with non-local correlations to a standard non-linear Fokker-Planck equation

TL;DR

This paper explores the theoretical link between a non-linear Fokker-Planck equation derived from non-additive entropy and the Boltzmann equation with non-additive correlations, ensuring consistency with classical cases.

Contribution

It establishes a formal connection between non-local Boltzmann equations and non-linear Fokker-Planck equations using $q$-algebra, with specific constraints on the entropic index.

Findings

Derived a non-additive Fokker-Planck equation consistent with the Boltzmann equation.

Ensured the model reduces to the standard Fokker-Planck equation as $q$ approaches 1.

Identified the entropic index $q$ as a constant characteristic parameter.

Abstract

In this work, we study the formal connections between the non-linear Fokker-Planck Equation associated with the non-additive entropy and the Boltzmann Equation with the non-additive correlation functional. The collisional term following the $q$-algebra is adopted. In the derivation of the non-additive Fokker-Planck Equation, two constraints are imposed on the final result: i) that the entropic index $q$ is a characteristic parameter of the non-additive systems with a value that does not change with time, and ii) that for $q \rightarrow 1$ a smooth transition for the standard Fokker-Planck Equation is obtained.