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Quasicontinuum Fokker-Planck equation
Francis J Alexander1, Philip Rosenau
1Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
This study introduces a regularized Fokker-Planck equation for discrete systems, offering improved short-time accuracy over the standard Kramers-Moyal expansion. This new method better preserves state-space discreteness and enhances analytical tractability for complex stochastic systems.
Area of Science:
- Statistical Physics
- Computational Chemistry
- Stochastic Processes
Background:
- The standard Kramers-Moyal expansion, while useful, often loses critical information about discrete state-space properties.
- Existing methods struggle with accurate short-time behavior in discrete stochastic systems.
Purpose of the Study:
- To develop a regularized Fokker-Planck equation for discrete-state systems.
- To improve the accuracy of short-time behavior compared to the Kramers-Moyal counterpart.
- To create a well-posed equation amenable to existing analytical and numerical tools.
Main Methods:
- Regularization of the Fokker-Planck equation for discrete systems.
- Focusing on chemical reaction kinetics and a 2D random walk as model problems.
- Developing a quasicontinuum Fokker-Planck equation.
Main Results:
- The regularized equation preserves crucial aspects of state-space discreteness.
- It exhibits more accurate short-time behavior than the standard Kramers-Moyal expansion.
- The approach is shown to be well-posed and more amenable to analysis.
Conclusions:
- The proposed regularized Fokker-Planck equation offers a more accurate and robust framework for discrete-state stochastic systems.
- This method facilitates the application of continuum-based analytical and numerical techniques.
- The approach is extendable to more complex discrete stochastic systems.
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