Nonequilibrium statistics of barrier crossings with competing pathways

Many biological, chemical, and physical systems are underpinned by stochastic transitions between equilibrium states in a potential energy. Here, we consider such transitions in a minimal model with two possible competing pathways, both starting from a local potential energy minimum and eventually finding the global minimum. There is competition between the distance to travel in state space and the height of the potential energy barriers to be surmounted, for the transition to occur. One pathway has a higher energy barrier to go over, but requires traversing a shorter distance, whereas the other pathway has a lower potential barrier but it is substantially further away in configuration space. The most likely pathway taken depends on the available time for the transition process; when only a relatively short time is available, the most likely path is the one over the higher barrier. We find that upon varying temperature the overall most likely pathway can switch from one to the other. We calculate the statistics of where the barrier crossing occurs and the distribution of times taken to reach the potential minimum. Interestingly, while the configuration space statistics is complex, the time of arrival statistics is rather simple, having an exponential probability density over most of the time range. Taken together, our results show that empirically observed rates in nonequilibrium systems should not be used to infer barrier heights.