Hierarchical Numerical Solution of Smoluchowski Equations with Rough Potentials

We present an efficient and numerically robust algorithm to follow diffusive processes on rough potential energy surfaces. The hierarchical nature of the algorithm (hierarchical discrete approximation or HDA) fully explores the fine- and coarse-grained structure of the underlying interaction potential. The present approach does not impose any restriction on the topology of the potential. The hierarchical grid allows to capture the roughness of the potential and achieve significant reduction of computational time using fewer grid points compared to other DA methods. HDA is shown to be accurate and efficient by comparing with results from the conventional DA and from the "mean first passage time" (MFPT) method. Using potential-optimized grids HDA monotonically converges to results from an analytical treatment for a very rough interaction potential (107 minima). Contrary to MFPT the solution from HDA is numerically stable. Because of the hierarchical structure of the method HDA can be extended to multidimensional problems.