Stochastic transitions: Paths over higher energy barriers can dominate in the early stages

The time evolution of many physical, chemical, and biological systems can be modeled by stochastic transitions between the minima of the potential energy surface describing the system of interest. We show that in cases where there are two (or more) possible pathways that the system can take, the time available for the transition to occur is crucially important. The well-known results of the reaction rate theory for determining the rates of transitions apply in the long-time limit. However, at short times, the system can, instead, choose to pass over higher energy barriers with a much higher probability, as long as the distance to travel in phase space is shorter. We construct two simple models to illustrate this general phenomenon. We also apply a version of the geometric minimum action method algorithm of Vanden-Eijnden and Heymann [J. Chem. Phys. 128, 061103 (2008)] to determine the most likely path at both short and long times.