Fractional Kinetic Equations for Levy Motion.
Authors:
A. V. Chechkin and V. Yu. Gonchar
By using classical procedure for derivation of kinetic equations for the Brownian
motion, we get fractional kinetic equations for the Levy motion. Namely, we derive
fractional Einstein - Smoluchowsky equation for the distribution function in coordinate
space as well as fractional Fokker - Plank equation for the distribution fucntion in the
phase space. The former equation contains space - fractional Riesz derivative, whereas the
latter one contains velocity - fractional Riesz derivative. We get and study analytic
solutions for the two problems: (i) linear relaxation of a monochromatic beam of Levy
particles, and (ii) harmonic Levy oscillator. We also simulate random Langevin equations
with a white Levy noise source and compare analytic results with the results of numerical
simulation. The finite sample size effects, which are very important for the Levy motion,
are also investigated. For the harmonic Levy oscillator, besides strict solutions, two
limit cases are studied both analytically and numerically, namely, overdamped oscillator
and weakly damped oscillator. Both cases are of special interest in view of further
studies of an anharmonic Levy oscillator.
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