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Continuous Time Random Walks with Reactions Forcing and Trapping

Published online by Cambridge University Press:  24 April 2013

C. N. Angstmann ,
I. C. Donnelly  and
B. I. Henry
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Abstract

One of the central results in Einstein’s theory of Brownian motion is that the mean square displacement of a randomly moving Brownian particle scales linearly with time. Over the past few decades sophisticated experiments and data collection in numerous biological, physical and financial systems have revealed anomalous sub-diffusion in which the mean square displacement grows slower than linearly with time. A major theoretical challenge has been to derive the appropriate evolution equation for the probability density function of sub-diffusion taking into account further complications from force fields and reactions. Here we present a derivation of the generalised master equation for an ensemble of particles undergoing reactions whilst being subject to an external force field. From this general equation we show reductions to a range of well known special cases, including the fractional reaction diffusion equation and the fractional Fokker-Planck equation.

Keywords

fractional diffusionreaction-diffusionrandom walkFokker-Planck equationstochastic process
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 2: Anomalous diffusion , 2013 , pp. 17 - 27
Copyright
© EDP Sciences, 2013

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