On the classical reaction rate and the first-time problems of Brownian motion
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 ($κ$) equals to the one ($κ_\mathrm{V}$) corresponding to an absorbing boundary consistent with the transition state theory, which is greater than the one ($κ_\mathrm{K}$) 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.