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Front dynamics in reaction-diffusion systems with Levy flights: a fractional diffusion approach.
D del-Castillo-Negrete1, B A Carreras, V E Lynch
1Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-8071, USA.
Physical Review Letters
|August 9, 2003
Summary
Reaction-diffusion systems with anomalous diffusion, caused by asymmetric Levy flights, show accelerated front dynamics. This study replaces Gaussian diffusion with fractional operators, revealing universal power-law decay in front tails.
Area of Science:
- Physics
- Mathematics
- Chemistry
Background:
- Reaction-diffusion models typically assume Gaussian diffusion.
- Anomalous, non-Gaussian diffusion is increasingly observed in natural systems.
- Understanding reactive systems with anomalous diffusion is crucial.
Purpose of the Study:
- Investigate front dynamics in reaction-diffusion systems with anomalous diffusion.
- Analyze the effects of asymmetric Levy flights on diffusion processes.
- Model non-Gaussian diffusion using fractional calculus.
Main Methods:
- Replaced the Laplacian diffusion operator with a fractional diffusion operator of order alpha.
- Utilized Levy alpha-stable distributions to model anomalous diffusion.
- Performed numerical simulations of the fractional Fisher-Kolmogorov equation.
Main Results:
- Anomalous diffusion leads to exponential acceleration of the reaction front.
- Demonstrated a universal power-law decay (x^-alpha) in the front's tail.
- Fractional diffusion operator accurately models Levy alpha-stable distributions.
Conclusions:
- Asymmetric Levy flights significantly alter reaction-diffusion front dynamics.
- Fractional calculus provides a powerful framework for studying anomalous diffusion in reactive systems.
- Findings challenge the traditional Gaussian diffusion assumption in reaction-diffusion modeling.