Title: Fokker-Planck equation with memory: the cross over from ballistic to diffusive processes in many particle systems and incompressible media
Author(s):

	V. Ilyin (Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel) ,
	I. Procaccia (Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel) ,
	A. Zagorodny ( Bogolyubov Institute for Theoretical Physics, 252143 Kiev, Ukraine)

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.

Key words: non-Markovian processes, fractional diffusion, ballistic effects
PACS: 05.40 Fb, 05.40 Jc, 51.10 +y

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