Transition-event durations in one-dimensional activated processes.

Despite their importance in activated processes, transition-event durations—which are much shorter than first passage times—have not received a complete theoretical treatment. The authors therefore study the distribution ρ_b(t) of durations of transition events over a barrier in a one-dimensional system undergoing overdamped Langevin dynamics. The authors show that ρ_b(t) is determined by a Fokker-Planck equation with absorbing boundary conditions and obtain a number of results, including (i) the analytic form of the asymptotic short-time transient behavior, which is universal and independent of the potential function; (ii) the first nonuniversal correction to the short-time behavior leading to an estimate of a key physical time scale; (iii) following previous work, a recursive formulation for calculating, exactly, all moments of ρ_b based solely on the potential function—along with approximations for the distribution based on a small number of moments; and (iv) a high-barrier approximation to the long-time (t→∞) behavior of ρ_b(t). The authors also find that the mean event duration does not depend simply on the barrier-top frequency (curvature) but is sensitive to details of the potential. All of the analytic results are confirmed by transition-path-sampling simulations implemented in a novel way. Finally, the authors discuss which aspects of the duration distribution are expected to be general for more complex systems. [ABSTRACT FROM AUTHOR]