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Front Propagation in Reaction-Superdiffusion Dynamics: Taming Levy Flights with Fluctuations

Summary: Front Propagation in Reaction-Superdiffusion Dynamics:
Taming Le´vy Flights with Fluctuations
D. Brockmann and L. Hufnagel
Max Planck Institute for Dynamics and Self-Organization, Bunsenstrasse 10, 37073 Go¨ttingen, Germany
Kavli Institute for Theoretical Physics, University of California Santa Barbara, California 93106, USA
(Received 21 January 2004; revised manuscript received 25 July 2006; published 27 April 2007)
We investigate front propagation in a reacting particle system in which particles perform scale-free
random walks known as Le´vy flights. The system is described by a fractional generalization of a reaction-
diffusion equation. We focus on the effects of fluctuations caused by a finite number of particles per
volume. We show that, in spite of superdiffusive particle dispersion and contrary to mean-field theoretical
predictions, wave fronts propagate at constant velocities, even for very large particle numbers. We show
that the asymptotic velocity scales with the particle number and obtain the scaling exponent.
DOI: 10.1103/PhysRevLett.98.178301 PACS numbers: 82.40.Ck, 02.50.Ey, 05.40.Fb, 05.70.Ln
One of the fundamental processes involved in nonequi-
librium pattern formation is the spatial propagation of
interfaces or fronts. Front propagation usually emerges
when a local reaction dynamics interplays with diffusion
in space of the reacting agents and has been observed in a
wide range of physical, chemical, and biological systems
[1­6]. One of the most prominent models which displays


Source: Amaral, Luis A.N. - Department of Chemical and Biological Engineering, Northwestern University


Collections: Physics; Biology and Medicine