Transition Path Dynamics of Non-Markovian Systems across a Rough Potential Barrier

Transition paths refer to rare events in physics, chemistry, and biology where the molecules cross barriers separating stable molecular conformations. The conventional analysis of the transition path times employs a diffusive and memoryless transition over a smooth potential barrier. However, it is widely acknowledged that the free energy profile between two minima in biomolecular processes is inherently not smooth. In this article, we discuss a theoretical model with a parabolic rough potential barrier and obtain analytical results of the transition path distribution and mean transition path times by incorporating absorbing boundary conditions across the boundaries under the driving of Gaussian white noise. Further, the influence of anomalous dynamics in rough potential driven by a power-law memory kernel is analyzed by deriving a time-dependent scaled diffusion coefficient that coarse-grains the effects of roughness, and the system's dynamics is reduced to a scaled diffusion on a smooth potential. Our theoretical results are tested and validated against numerical simulations. The findings of our study show the influence of the boundary conditions, barrier height, barrier roughness, and memory effect on the transition path time distributions in a rough potential, and the validity of the scaling diffusion coefficient has been discussed.