Pseudo generators for under-resolved molecular dynamics

TL;DR

This paper investigates the projection of stochastic Langevin dynamics onto configuration space, deriving Smoluchowski equations in various regimes, and introduces methods to approximate metastable dynamics using local information.

Contribution

It provides new derivations of Smoluchowski equations in non-Cartesian geometries and develops approaches to approximate metastable dynamics from local data.

Findings

Derivation of Smoluchowski equations from Langevin dynamics in non-Cartesian geometries.

Explicit small-time asymptotics for metastable behavior.

Three approaches to approximate metastable dynamics using time-local information.

Abstract

Many features of a molecule which are of physical interest (e.g. molecular conformations, reaction rates) are described in terms of its dynamics in configuration space. This article deals with the projection of molecular dynamics in phase space onto configuration space. Specifically, we study the situation that the phase space dynamics is governed by a stochastic Langevin equation and study its relation with the configurational Smoluchowski equation in the three different scaling regimes: Firstly, the Smoluchowski equations in non-Cartesian geometries are derived from the overdamped limit of the Langevin equation. Secondly, transfer operator methods are used to describe the metastable behaviour of the system at hand, and an explicit small-time asymptotics is derived on which the Smoluchowski equation turns out to govern the dynamics of the position coordinate (without any assumptions on the damping). By using an adequate reduction technique, these considerations are then extended to one-dimensional reaction coordinates. Thirdly, we sketch three different approaches to approximate the metastable dynamics based on time-local information only.