Computing generalized Langevin equations and generalized Fokker–Planck equations
- 7 July 2009
- journal article
- Published by Proceedings of the National Academy of Sciences in Proceedings of the National Academy of Sciences of the United States of America
- Vol. 106 (27), 10884-10889
- https://doi.org/10.1073/pnas.0902633106
Abstract
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.This publication has 12 references indexed in Scilit:
- Random walk in orthogonal space to achieve efficient free-energy simulation of complex systemsProceedings of the National Academy of Sciences of the United States of America, 2008
- Normal mode partitioning of Langevin dynamics for biomoleculesThe Journal of Chemical Physics, 2008
- Collective Langevin dynamics of conformational motions in proteinsThe Journal of Chemical Physics, 2006
- Extracting macroscopic dynamics: model problems and algorithmsNonlinearity, 2004
- Optimal prediction with memoryPhysica D: Nonlinear Phenomena, 2002
- Coarse-grained dynamics of one chain in a polymer meltThe Journal of Chemical Physics, 2000
- A Direct Approach to Conformational Dynamics Based on Hybrid Monte CarloJournal of Computational Physics, 1999
- Generalized Langevin dynamics simulations with arbitrary time-dependent memory kernelsThe Journal of Chemical Physics, 1983
- On the derivation of the generalized Langevin equation for interacting Brownian particlesJournal of Statistical Physics, 1981
- Transport, Collective Motion, and Brownian MotionProgress of Theoretical Physics, 1965