# Anomalous Brownian motion via linear Fokker-Planck equations

## TL;DR

This paper demonstrates that anomalous Brownian motion can be modeled using non-Markovian linear Fokker-Planck equations, challenging the notion that only nonlinear equations with non-Boltzmann entropies can describe such phenomena.

## Contribution

It introduces a non-Markovian linear Fokker-Planck framework to analyze anomalous diffusion, expanding the theoretical understanding beyond nonlinear models.

## Key findings

- Anomalous diffusion characterized by mean square displacement in non-Markovian models.
- Analysis of mean square momentum of a free particle with inertial effects.
- Demonstration of anomalous diffusion in harmonic oscillator systems.

## Abstract

According to a traditional point of view Boltzmann entropy is intimately related to linear Fokker-Planck equations (Smoluchowski, Klein-Kramers, and Rayleigh equations) that describe a well-known nonequilibrium phenomenon: (normal) Brownian motion of a particle immersed in a thermal bath. Nevertheless, current researches have claimed that non-Boltzmann entropies (Tsallis and Renyi entropies, for instance) may give rise to anomalous Brownian motion through nonlinear Fokker-Planck equations. The novelty of the present article is to show that anomalous diffusion could be investigated within the framework of non-Markovian linear Fokker-Planck equations. So on the ground of this non-Markovian approach to Brownian motion, we find out anomalous diffusion characterized by the mean square displacement of a free particle and a harmonic oscillator in absence of inertial force as well as the mean square momentum of a free particle in presence of inertial force.

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Source: https://tomesphere.com/paper/1701.02670