Condensed Matter Physics, 2010, vol. 13, No. 2, p. 23001:18
DOI:10.5488/CMP.13.23001
Title: Stochastic processes crossing from ballistic to
fractional diffusion with memory: exact results
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)

We address the now classical problem of a diffusion
process that crosses over from a ballistic behavior at short times
to a fractional diffusion (sub or superdiffusion) at longer times.
Using the standard nonMarkovian diffusion equation we demonstrate
how to choose the memory kernel to exactly respect the two different
asymptotics of the diffusion process. Having done so we solve for
the probability distribution function as a continuous function which
evolves inside a ballistically expanding domain. This general
solution agrees for long times with the probability distribution
function obtained within the continuous random walk approach but it
is much superior to this solution at shorter times where the effect
of the ballistic regime is crucial.
Key words: fractional diffusion, memory effects,
ballistic processes
PACS: 05.10.Cg, 05.20.Dd, 51.10+y
