Linear Relaxation Processes Governed by Fractional Symmetric Kinetic Equations

TL;DR

This paper derives and analyzes fractional symmetric kinetic equations that describe linear relaxation processes influenced by stable stochastic forces, extending classical models of Brownian motion with fractional derivatives.

Contribution

It introduces fractional symmetric Fokker-Planck and Einstein-Smoluchowski equations for systems driven by Levy stable noise, providing analytical and numerical insights into relaxation dynamics.

Findings

Analytical solutions for relaxation in force-free and linear oscillator cases.

Numerical simulations using Langevin equations with Levy noise show agreement with analytical results.

Extension of classical kinetic equations to include fractional derivatives for Levy-stable stochastic influences.

Abstract

We get fractional symmetric Fokker - Planck and Einstein - Smoluchowski kinetic equations, which describe evolution of the systems influenced by stochastic forces distributed with stable probability laws. These equations generalize known kinetic equations of the Brownian motion theory and contain symmetric fractional derivatives over velocity and space, respectively. With the help of these equations we study analytically the processes of linear relaxation in a force - free case and for linear oscillator. For a weakly damped oscillator we also get kinetic equation for the distribution in slow variables. Linear relaxation processes are also studied numerically by solving corresponding Langevin equations with the source which is a discrete - time approximation to a white Levy noise. Numerical and analytical results agree quantitatively.