First-passage behavior of the random-barrier model

The previously proposed transport equation for the random-barrier model, which is the diffusion equation with resetting to positions visited in the past, is used here to calculate the first-passage times. The results obtained are compared with those obtained using the normal diffusion equation with an effective diffusion coefficient. It is shown that, under certain conditions, the equation with the effective diffusion coefficient can greatly overestimate the time of the first passage. In particular, the rate constant of a bimolecular diffusion-controlled reaction calculated from this equation can be significantly lower than the actual rate. This result can serve as one of the possible explanations for the high rates of diffusion-controlled reactions observed in an experiment.