On the classical reaction rate and the first-time problems of Brownian motion

TL;DR

This paper introduces efficient techniques for solving the first-time problems of Brownian motion, clarifies the relationship between different reaction rate coefficients, and proposes new methods for reaction rate determination based on short-time trajectories.

Contribution

It establishes the equivalence of Eyring's transmission coefficient with a boundary-based coefficient and introduces novel methods for reaction rate analysis from short-time trajectory data.

Findings

Eyring's coefficient equals the boundary-based coefficient $$

Proposed methods analyze short-time trajectories for reaction rates

Discussed relation to reactive flux method and reaction coordinates

Abstract

We have developed efficient techniques to solve the first-time problems of Brownian motion. Based on a time-scale separation of recrossings, we show that Eyring's transmission coefficient ($\kappa$) equals to the one ($\kappa_\mathrm{V}$) corresponding to an absorbing boundary consistent with the transition state theory, which is greater than the one ($\kappa_\mathrm{K}$) derived by Kramers. We also propose methods for reaction rate determination by analyzing short-time trajectories from the barrier maximum, and discuss the relation to the reactive flux method and the significance of reaction coordinates.