Fokker-Planck equation with memory: The crossover from ballistic to diffusive processes in many-particle systems and incompressible media

Abstract

The unied 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 coeffcients are determined by the observable quantities (mean- andmean 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 the continuous random walk theory; they are, however, much superior at shorter times when the effect of the ballistic behavior is crucial. © V. Ilyin, I. Procaccia, A. Zagorodny, 2013.