Transition path properties for one-dimensional non-Markovian models

Transitions between long-lived states are rare but important. The statistic of successful transitions is considered in transition path theory. We here consider the transition path properties of a generalized Langevin equation with built-in memory. The general form of the approximate theoretical solutions to the transition path time distribution, mean transition path time, and coefficient of variation are obtained from the generalized Smoluchowski equation. Then, the accuracy of our theoretical results is verified by the Forward Fluxing Sampling scheme. Finally, two examples are worked out in detail. We quantify how the potential function and the memory parameters affect the transition path properties. The short time limit of transition path time distribution always has an exponential decay. For the parabolic potential case, the memory strongly affects the long-time behavior of the transition path time distribution. Our results show that the behavior of the mean transition path time is dominated by the smaller of the two memory times when both memory times exceed the intrinsic diffusion time. Interestingly, the results also show that the memory can effect a coefficient of variation of transition path times exceeding unity, in contrast to Markovian case.