Stochastic model for barrier crossings and fluctuations in local timescale

TL;DR

This paper develops a stochastic differential equation framework with local time considerations to analyze barrier crossing rates in metastable systems, connecting excursion theory with transition path analysis.

Contribution

It introduces a novel SDE with a singular drift term involving local time, linking excursion theory to barrier crossing rate calculations without assuming a transition state.

Findings

Rate factorizes into local time and excursion measure terms

Excursion theory provides a general framework for barrier crossing analysis

Mathematical structure aligns with transition state theory expressions

Abstract

The problem of computing the rate of diffusion-aided activated barrier crossings between metastable states is one of broad relevance in physical sciences. The transition path formalism aims to compute the rate of these events by analysing the statistical properties of the transition path between the two metastable regions concerned. In this paper, we show that the transition path process is a unique solution to an associated stochastic differential equation (SDE), with a discontinuous and singular drift term. The singularity arises from a local time contribution, which accounts for the fluctuations at the boundaries of the metastable regions. The presence of fluctuations at the local time scale calls for an excursion theoretic consideration of barrier crossing events. We show that the rate of such events, as computed from excursion theory, factorizes into a local time term and an excursion measure term, which bears empirical similarity to the transition state theory rate expression. Since excursion theory makes no assumption about the presence of a transition state in the potential energy landscape, the mathematical structure underlying this factorization ought to be general. We hence expect excursion theory (and local times) to provide some physical and mathematical insights in generic barrier crossing problems.