Equilibrium and Recrossings of the Transition State: What Can Be Learned from Diffusion?

Abstract

We examine the assumptions and conclusion of (generalized) transition state theory (GTST) by considering the activation process in the diffusion (Langevin) limit. We find the asymptotic structure of the leading eigenfunctions and eigenvalues of the Fokker-Planck operator with a bistable potential and hence the long-time quasi-equilibrium behavior of the phase space probability density function (pdf). Defining reactant and product as small neighborhoods of the stable states, we examine all possible recrossings of the transition state region (TSR) and find their contribution to the mean first passage time (MFPT) from one state to the other. We show that the mean number of recrossings of the TSR is 1, so the MFPT from one state to the other is twice that to the stochastic separatrix, which we use as a generalized transition state (GTS). The activation rate, that is, the rate at which trajectories arrive from one state to the other, is then shown to be one-half of the arrival rate at the GTS and in the limit of a high harrier is independent of the choice of the size of the domains that define the states. We conclude that to obtain the correct rate in (G)TST (i) the quasi-equilibrium density (qepdf) rather than the equilibrium density (epdf) has to be used, (ii) the qepdf contains a boundary layer near the stochastic separatrix, but otherwise the reactant qepdf is nearly equal the epdf, and (iii) all recrossings of the (G)TS are accounted for if the (G)TS is the stochastic separatrix, but not otherwise. We also consider the case of a single metastable state.