1207.6190 (Valery Ilyin et al.)

Fokker-Planck equation with memory: the cross over from ballistic to
diffusive processes in many particle systems and incompressible media    [PDF]

Valery Ilyin, Itamar Procaccia, Anatoly Zagorodny

The unified description of diffusion processes that cross over from a ballistic behavior at short times to normal or anomalous diffusion (sub- or superdiffusion) at longer times is constructed on the basis of a non-Markovian generalization of the Fokker-Planck equation. The necessary non- Markovian kinetic coefficients are determined by the observable quantities (mean- and mean square displacements). Solutions of the non-Markovian equation describing diffusive processes in the physical space are obtained. For long times these solutions agree with the predictions of continuous random walk theory; they are however much superior at shorter times when the effect of the ballistic behavior is crucial.

View original: http://arxiv.org/abs/1207.6190