Chapter 18 - Diffusion over barriers

Abstract

At this point in the book, we have learned several strategies for computing rate constants. Whether we look back to collision theory, transition state theory, RRKM theory, reactive flux, Kramers theory, Grote-Hynes theory, etc., the velocity at the barrier top, in some guise, was always a part of the final rate expression. The theories in this chapter are completely different. Trajectories from an overdamped (diffusion) process are continuous, but not differentiable, so there are no well-defined velocities. For irreversible phenomena like nucleation, we cannot even use spectral theories. Therefore we must start from entirely different assumptions in deriving the rate. This chapter outlines two general approaches: mean first passage times (MFPTs) and expressions based on committors (splitting probabilities). These closely related approaches yield a flux-over-population rate Farkas (1927) [1] from the steady-state population density with “rescue and replace” boundary conditions. These boundary conditions create a non-equilibrium steady-state current leading from the source (the reactant basin) to the sink (the product state). A similar construct was used in the discussions of classical nucleation theory (Chapter 14) and Kramers theory (Chapter 16). The steady-state rescue and replace construct at first seems to be rather artificial, but when the boundary conditions are imposed appropriately it has a strong theoretical foundation (vide infra).