Analysis of Transition Path Ensemble in the Exactly Solvable Models via Overdamped Langevin Equation

TL;DR

This paper develops a theoretical framework using overdamped Langevin dynamics to analyze transition paths in exactly solvable models, providing explicit solutions and confirming the effectiveness of the conditional Langevin equation for sampling transition paths.

Contribution

It introduces a path integral formulation and a conditional Langevin equation for transition path analysis, with explicit solutions in solvable models, advancing understanding of transition mechanisms.

Findings

Conditional Langevin equation effectively samples transition paths.

Minimum action path corresponds to the most probable transition.

Analytic solutions confirm theoretical consistency.

Abstract

Transition of a system between two states is an important but difficult problem in natural science. In this article we study the transition problem in the framework of transition path ensemble. Using the overdamped Langevin method, we introduce the path integral formulation of the transition probability and obtain the equation for the minimum action path in the transition path space. For the effective sampling in the transition path ensemble, we derive a conditional overdamped Langevin equation. In two exactly solvable models, the free particle system and the harmonic system, we present the expression of the conditional probability density and the explicit solutions for the conditional Langevin equation and the minimum action path. The analytic results demonstrate the consistence of the conditional Langevin equation with the desired probability distribution in the transition. It is confirmed that the conditional Langevin equation is an effective tool to sample the transition path ensemble, and the minimum action principle actually leads to the most probable path.