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Continuous-time random walks with internal dynamics and subdiffusive reaction-diffusion equations.

S Eule1, R Friedrich, F Jenko

  • 1Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Strasse 9, D-48149 Münster, Germany.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 5, 2009
PubMed
Summary
This summary is machine-generated.

We developed a generalized master equation for continuous-time random walks with deterministic evolution. This enables deriving reaction-diffusion equations for subdiffusive chemical species.

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Area of Science:

  • Physics
  • Chemistry
  • Mathematical Modeling

Background:

  • Continuous-time random walks (CTRWs) are fundamental models in statistical physics.
  • Existing CTRW models often lack the inclusion of deterministic dynamics between transitions.
  • Understanding subdiffusion is crucial in various chemical and physical processes.

Purpose of the Study:

  • To formulate a generalized master equation for CTRWs incorporating deterministic evolution.
  • To apply this formulation to advection-diffusion and jump-diffusion systems.
  • To derive reaction-diffusion equations for subdiffusive chemical species.

Main Methods:

  • Formulation of a generalized master equation.
  • Application to advection-diffusion and jump-diffusion schemes.
  • Derivation of reaction-diffusion equations using a mean-field approximation.

Main Results:

  • A generalized master equation applicable to CTRWs with deterministic intermediate steps was established.
  • The framework was successfully demonstrated using advection-diffusion and jump-diffusion examples.
  • Reaction-diffusion equations for subdiffusive chemical species were derived.

Conclusions:

  • The generalized master equation provides a unified framework for analyzing complex random walk dynamics.
  • This approach facilitates the study of chemical species exhibiting subdiffusive behavior.
  • The mean-field approximation offers a tractable method for deriving macroscopic reaction-diffusion equations.