Estimating diffusivity along a reaction coordinate in the high friction limit: Insights on pulse times in laser-induced nucleation

In the high friction limit of Kramers' theory, the diffusion coefficient for motion along the reaction coordinate is a crucial parameter in determining reaction rates from mean first passage times. The Einstein relation between mean squared displacement, time, and diffusivity is inaccurate at short times because of ballistic motion and inaccurate at long times because trajectories drift away from maxima in the potential of mean force. Starting from the Smoluchowski equation for a downward parabolic barrier, we show how drift induced by the potential of mean force can be included in estimating the diffusivity. A modified relation between mean squared displacement, time, and diffusivity now also includes a dependence on the barrier curvature. The new relation provides the diffusivity at the top of the barrier from a linear regression that is analogous to the procedure commonly used with Einstein's relation. The new approach has particular advantages over previous approaches when evaluations of the reaction coordinate are costly or when the reaction coordinate cannot be differentiated to compute restraining forces or velocities. We use the new method to study the dynamics of barrier crossing in a Potts lattice gas model of nucleation from solution. Our analysis shows that some current hypotheses about laser-induced nucleation mechanisms lead to a nonzero threshold laser pulse duration below which a laser pulse will not affect nucleation. We therefore propose experiments that might be used to test these hypotheses.