On dynamical behaviors in the fractional generalized Langevin equation

TL;DR

This paper investigates the active dynamics described by a fractional generalized Langevin equation with memory effects, deriving the Fokker-Planck equation to analyze statistical properties like mean squared displacement and velocity.

Contribution

It introduces a derivation of the Fokker-Planck equation from a second-order fractional Langevin equation with a memory kernel, enabling detailed statistical analysis.

Findings

Derived the Fokker-Planck equation for the fractional Langevin system.

Calculated mean squared displacement and velocity in various three-time domains.

Provided insights into the statistical behavior of systems with memory effects.

Abstract

The focus of our study in this paper is on the active dynamics and a fractional generalized Langevin equation with a memory kernel K(t). The Fokker-Planck equation is obtained by deriving it from a second-order differential equation. The joint probability density we obtained enables us to calculate various statistical quantities such as mean squared displacement and mean squared velocity in different three-time domains.