A coarse-grained approach to modeling Brownian motion in multi-dimensional rough potentials

Diffusion in one-dimensional, spatially rough energy landscapes is treated using Zwanzig's formalism, which is based upon the Smoluchowski Equation. With relatively long periods of time spent around the minima interrupted by quick transitions to neighbouring wells, the motion is reminiscent of a particle hopping between sites on a lattice. A Coarse-Grained simulation scheme based upon mean first-passage times and hopping probabilities is introduced and good agreement with Brownian dynamics simulations and theoretical predictions is observed. Extending Zwanzig's formalism to higher dimensions, we derive an expression for the effective diffusion coefficient. From this we obtain expressions for the mean first-passage time and hopping probabilities, and thereby extend our Coarse-Grained scheme to higher dimensions. Again, our scheme is found to replicate the results of Brownian dynamics simulations, which coincide with our theoretical predictions. We conclude by simulating motion in two-dimensional disordered potential energy landscapes, where analytic results are difficult to obtain.