First-passage-time distribution in a moving parabolic potential with spatial roughness

In this paper, we investigate the first-passage-time distribution (FPTD) within a time-dependent parabolic potential in the presence of roughness with two methods: the Kramers theory and a nonsingular integral equation. By spatially averaging, the rough potential is equivalent to the combination of an effective smooth potential and an effective diffusion coefficient. Based on the Kramers theory, we first obtain Kramers approximations (KAs) of FPTD for both smooth and rough potentials. As expected, KA is valid only for high barriers and small external forces, and generally applicable for high barriers in rough potentials. To overcome the shortcoming of KA, a probability asymptotic approximation (PAA) based on an integral equation is proposed, which uses the transient probability density function (PDF) of the natural boundary conditions instead of the absorbing boundary conditions. We find that PAA fits very well even for large external forces. This enables us to analytically solve the FPTD for large external forces and low barriers as a strong extension to KA. In addition, we show that in the presence of a rough potential, the PAA of FPTD is in good agreement with numerical simulations for low barrier potentials. The PAA makes it possible to investigate the first-passage problem with ultrafast varying potentials and short exiting time. Thus, KA and PAA are complementary in determining the FPTD both for various barriers and external forces. Finally, the mean first-passage time (MFPT) is studied, which illustrates that the PAA of MFPT is effective almost in the whole range of external forces, while the KA of MFPT is valid only for small external forces.