Microscopic Dynamics of Nonlinear Fokker-Planck Equations

TL;DR

This paper introduces a nonextensive framework for microscopic dynamics in nonlinear Fokker-Planck equations, providing analytical solutions and insights into anomalous diffusion and thermal noise modeling.

Contribution

It presents a novel nonextensive generalization of the Wiener process to analyze nonlinear Fokker-Planck equations with analytical solutions.

Findings

Analytical solutions for nonextensive Brownian and Ornstein-Uhlenbeck processes.

Explanation of anomalous diffusion via memory effects.

Application to thermal noise in electric circuits.

Abstract

We propose a new approach to describe the effective microscopic dynamics of (power-law) nonlinear Fokker-Planck equations. Our formalism is based on a nonextensive generalization of the Wiener process. This allow us to obtain, in addition to significant physical insights, several analytical results with great simplicity. Indeed, we obtain analytical solutions for a nonextensive version of Brownian free-particle and Ornstein-Uhlenbeck process, and explain anomalous diffusive behaviours in terms of memory effects in a nonextensive generalization of Gaussian white noise. Finally, we apply the develop formalism to model thermal noise in electric circuits.