Alexander Francis J, Rosenau Philip
Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2010 Apr;81(4 Pt 1):041902. doi: 10.1103/PhysRevE.81.041902. Epub 2010 Apr 2.
Building on the work [C. R. Doering, P. S. Hagan, and P. Rosenau, Phys. Rev. A 36, 985 (1987)] we present a regularized Fokker-Planck equation for discrete-state systems with more accurate short-time behavior than its standard, Kramers-Moyal counterpart. This regularization leads to a quasicontinuum Fokker-Planck equation with several key features: it preserves crucial aspects of state-space discreteness ordinarily lost in the standard Kramers-Moyal expansion; it is well posed, and it is more amenable to analytical and numerical tools currently available for continuum systems. In order to expose the basic idea underlying the regularization, it suffices for us to focus on two simple problems--the chemical reaction kinetics of a one-component system and a two-dimensional symmetric random walk on a square lattice. We then describe the path to applying this approach to more complex, discrete-state stochastic systems.
在[C. R. 多林、P. S. 哈根和P. 罗森纳，《物理评论A》36卷，985页（1987年）]的工作基础上，我们提出了一个用于离散态系统的正则化福克 - 普朗克方程，其短时间行为比标准的克莱默斯 - 莫亚尔对应方程更精确。这种正则化导致了一个具有几个关键特征的准连续福克 - 普朗克方程：它保留了通常在标准克莱默斯 - 莫亚尔展开中丢失的状态空间离散性的关键方面；它是适定的，并且更适合目前可用于连续系统的分析和数值工具。为了揭示正则化背后的基本思想，我们只需关注两个简单问题——单组分系统的化学反应动力学和方形晶格上的二维对称随机游走。然后我们描述将这种方法应用于更复杂的离散态随机系统的途径。
