Generating Transition Paths by Langevin Bridges

TL;DR

This paper introduces a new stochastic method for generating conditioned transition paths using Langevin dynamics, applicable to systems with explicit solvents, and demonstrates its effectiveness on a quartic oscillator.

Contribution

A novel stochastic approach to generate transition paths conditioned on start and end states, with exact and approximate solutions for different time regimes.

Findings

Paths can be exactly generated by a non-local stochastic differential equation.

Approximate equations can be used for longer times with reweighting to ensure correct statistics.

Method successfully applied to a one-dimensional quartic oscillator.

Abstract

We propose a novel stochastic method to generate paths conditioned to start in an initial state and end in a given final state during a certain time $t_{f}$. These paths are weighted with a probability given by the overdamped Langevin dynamics. We show that these paths can be exactly generated by a non-local stochastic differential equation. In the limit of short times, we show that this complicated non-solvable equation can be simplified into an approximate stochastic differential equation. For longer times, the paths generated by this approximate equation can be reweighted to generate the correct statistics. In all cases, the paths generated by this equation are statistically independent and provide a representative sample of transition paths. In case the reaction takes place in a solvent (e.g. protein folding in water), the explicit solvent can be treated. The method is illustrated on the one-dimensional quartic oscillator.