Computing generalized Langevin equations and generalized Fokker-Planck equations
Computing generalized Langevin equations and generalized Fokker-Planck equations
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DOI：
10.1073/pnas.0902633106
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发表时间：
2009-07-07
影响因子：
11.1
通讯作者：
Kia, Amirali
中科院分区：
文献类型：
--
作者：
Darve, Eric;Solomon, Jose;Kia, Amirali
Mori-Zwanzig 形式是导出描述少量已解析变量演化的微分方程的有效工具。在本文中，我们介绍了其在广义 Langevin 方程和广义非马尔可夫 Fokker-Planck 方程推导中的应用。我们展示了可以从这些方程中提取多长时间的时间尺度速率和亚稳态盆地。提出了数值算法来离散这些方程。一个重要的方面是正交动力学方程的数值解，它是高维空间中的偏微分方程。我们提出了有效的数值方法来求解这个正交动力学方程。此外，我们提出了适用于离散地图的 Mori-Zwanzig 类型的投影形式。数值应用是从哈密顿系统领域提出的。
The Mori-Zwanzig formalism is an effective tool to derive differential equations describing the evolution of a small number of resolved variables. In this paper we present its application to the derivation of generalized Langevin equations and generalized non-Markovian Fokker-Planck equations. We show how long time scales rates and metastable basins can be extracted from these equations. Numerical algorithms are proposed to discretize these equations. An important aspect is the numerical solution of the orthogonal dynamics equation which is a partial differential equation in a high dimensional space. We propose efficient numerical methods to solve this orthogonal dynamics equation. In addition, we present a projection formalism of the Mori-Zwanzig type that is applicable to discrete maps. Numerical applications are presented from the field of Hamiltonian systems.