Linear Relaxation Processes Governed by Fractional Symmetric Kinetic Equations

Details Linear Relaxation Processes Governed by Fractional Symmetric Kinetic Equations Linear Relaxation Processes Governed by Fractional Symmetric Kinetic Equations

1999-10-07

arxiv.org/abs/cond-mat/9910091v1

Physics

Condensed Matter

Statistical Mechanics

30 pages, LaTeX, 13 figures PostScript

Scientific paper

10.1134/1.1320100

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.

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