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Continuous-time random walks and traveling fronts
Sergei Fedotov1, Vicenç Méndez
1Department of Mathematics, UMIST - University of Manchester Institute Science and Technology, Manchester M60 1QD, United Kingdom.
Summary
This study introduces a geometric method to track reaction fronts in unstable states, applicable to various random walks. It provides a new formula for propagation speed, especially for complex transport phenomena.
Area of Science:
- Mathematical Physics
- Statistical Mechanics
- Nonlinear Dynamics
Background:
- Propagating fronts are crucial in diverse phenomena, from chemical reactions to population dynamics.
- Existing models often rely on differential equations for particle density, limiting applicability.
- Understanding front propagation in unstable states requires robust analytical tools.
Purpose of the Study:
- To develop a novel geometric approach for analyzing reaction-front propagation.
- To derive an integral equation for the action functional governing front dynamics.
- To provide an explicit formula for propagation speed, including anomalous transport.
Main Methods:
- Utilizing a geometric framework for continuous-time random walks.
- Deriving an integral Hamilton-Jacobi type equation for the action functional.
- Avoiding explicit differential equations for particle density.
Main Results:
- A general geometric method for front propagation in unstable states is established.
- An integral Hamilton-Jacobi equation is derived for front position and speed.
- An explicit formula for propagation speed is obtained, applicable to non-Markovian processes.
Conclusions:
- The geometric approach offers a powerful alternative to traditional methods for studying reaction fronts.
- This method is particularly effective for complex systems with anomalous transport.
- The derived Hamilton-Jacobi equation provides new insights into front dynamics and speed determination.
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