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

摘要

The unified description of diffusion processes that cross over from aballistic behavior at short times to normal or anomalous diffusion (sub- orsuperdiffusion) at longer times is constructed on the basis of a non-Markoviangeneralization of the Fokker-Planck equation. The necessary non-Markoviankinetic coefficients are determined by the observable quantities (mean- andmean square displacements). Solutions of the non-Markovian equation describingdiffusive processes in the physical space are obtained. For long times, thesesolutions agree with the predictions of the continuous random walk theory; theyare, however, much superior at shorter times when the effect of the ballisticbehavior is crucial.