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Anomalous diffusion with linear reaction dynamics: from continuous time random walks to fractional reaction-diffusion
B I Henry1, T A M Langlands, S L Wearne
1Department of Applied Mathematics, School of Mathematics, University of New South Wales, Sydney NSW 2052, Australia. B.Henry@unsw.edu.au
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 10, 2006
Summary
This study introduces fractional reaction-diffusion equations for anomalously diffusing species, incorporating linear reaction dynamics. Different models show how reaction kinetics and diffusion terms are affected by continuous time random walks.
Area of Science:
- Physics
- Chemistry
- Mathematical Modeling
Background:
- Anomalous diffusion is crucial for understanding transport in complex systems.
- Continuous time random walks (CTRW) model anomalous diffusion at the mesoscopic level.
- Integrating reaction dynamics into CTRW models is essential for realistic simulations.
Purpose of the Study:
- To investigate the impact of linear reaction dynamics on anomalously diffusing species modeled by CTRW.
- To derive and compare fractional reaction-diffusion equations under different reaction scenarios.
- To extend CTRW models for more general reaction dynamics.
Main Methods:
- Utilized continuous time random walks (CTRW) to model anomalous diffusion.
- Incorporated linear reaction dynamics (instantaneous and per capita rate) into the CTRW framework.
- Derived long-time asymptotic limits to obtain fractional reaction-diffusion equations.
Main Results:
- Instantaneous addition/removal of walkers leads to fractional derivatives acting on both diffusion and reaction terms.
- Per capita rate addition/removal results in fractional derivatives acting on diffusion but standard kinetics.
- Comparison with a phenomenological model highlights differences in fractional derivative application.
Conclusions:
- The manner of incorporating reaction dynamics significantly alters the resulting fractional reaction-diffusion equation.
- CTRW models provide a flexible framework for studying anomalous diffusion with complex reaction kinetics.
- Further extensions of CTRW models can accommodate more general reaction dynamics in anomalous diffusion systems.
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