Summary: Le´vy Flights in Inhomogeneous Media
D. Brockmann and T. Geisel
Max-Planck-Institut fu¨r Stro¨mungsforschung, Bunsenstrasse 10, Go¨ttingen, Germany
(Received 12 November 2002; published 28 April 2003)
We investigate the impact of external periodic potentials on superdiffusive random walks known as
Le´vy flights and show that even strongly superdiffusive transport is substantially affected by the
external field. Unlike ordinary random walks, Le´vy flights are surprisingly sensitive to the shape of the
potential while their asymptotic behavior ceases to depend on the Le´vy index . Our analysis is based
on a novel generalization of the Fokker-Planck equation suitable for systems in thermal equilibrium.
Thus, the results presented are applicable to the large class of situations in which superdiffusion is
caused by topological complexity, such as diffusion on folded polymers and scale-free networks.
DOI: 10.1103/PhysRevLett.90.170601 PACS numbers: 05.40.a, 02.50.r, 45.10.Hj, 61.41.+e
Diffusion processes are ubiquitous in nature. A freely
diffusive particle is characterized by a mean square dis-
placement which increases linearly in time, hX2
ti / t.
However, a variety of interesting physical systems violate
this temporal behavior. For example, the position Xt of a
superdiffusive particle heuristically evolves as Xt
t1= with 0 < < 2. Superdiffusion has been observed