The Kramers problem in the energy diffusion regime: transient times

Brownian motion driven by thermal fluctuations play an important role in the work of nanomachines and in single-molecule pulling experiments. Some problems in this field are reduced to the so-called Kramers problem concerning the decay rate of a quasistationary state. Provided all Brownian particles start their motion from the potential minimum, the rate increases in time reaching a quasistationary value. We model the process of decay using the Langevin equations. This modeling makes it possible identifying the transient time required for the rate to reach its quasistationary value. We compare the numerical transient times with those known from the literature. It turns out that the difference reaches up to 70% in the case of weak friction. Increasing friction results in decreasing this difference. In order to make our results useful for a larger audience, we present them in the dimensionless form.