Reactive trajectories and the transition path process

TL;DR

This paper characterizes the probability law and statistical properties of transition paths in stochastic differential equations, providing new formulas and insights relevant to chemical reactions and thermally activated processes.

Contribution

It introduces a transition path process with a singular drift to describe reactive trajectories and offers methods to recover its statistics from empirical data.

Findings

Derived the probability law of transition paths via a new SDE

Provided formulas for transition path density and current

Showed how to empirically sample and analyze transition paths

Abstract

We study the trajectories of a solution $X_t$ to an It\^o stochastic differential equation in $\Rm^d$, as the process passes between two disjoint open sets, $A$ and $B$. These segments of the trajectory are called transition paths or reactive trajectories, and they are of interest in the study of chemical reactions and thermally activated processes. In that context, the sets $A$ and $B$ represent reactant and product states. Our main results describe the probability law of these transition paths in terms of a transition path process $Y_t$, which is a strong solution to an auxiliary SDE having a singular drift term. We also show that statistics of the transition path process may be recovered by empirical sampling of the original process $X_t$. As an application of these ideas, we prove various representation formulas for statistics of the transition paths. We also identify the density and current of transition paths. Our results fit into the framework of the transition path theory by E and Vanden-Eijnden.