Math. Model. Nat. Phenom.
Volume 8, Number 2, 2013Anomalous diffusion
|Page(s)||17 - 27|
|Published online||24 April 2013|
Continuous Time Random Walks with Reactions Forcing and Trapping
School of Mathematics and Statistics, University of
New South Wales, Sydney, Australia
Corresponding author. E-mail: email@example.com
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.
Mathematics Subject Classification: 60G22 / 35K57 / 35Q84 / 82C41 / 35R11 / 60J60
Key words: fractional diffusion / reaction-diffusion / random walk / Fokker-Planck equation / stochastic process
© EDP Sciences, 2013
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