Effective diffusion in rough potential energy landscapes

Diffusion in spatially rough, confining, one-dimensional energy landscapes is treated using Zwanzig's proposed formalism, which is based upon the Smoluchowski Equation. Disagreement between its predictions and the results of numerical simulations is observed. We use the configurational partition function to amend Zwanzig's formalism, and resolve the disagreement. The analogous over-damped Langevin Equation is proposed, and a numerical simulation scheme offering potentially significant reductions in computational time is derived. The case of random roughness is treated. We then extend the above into higher dimensions and calculate effective diffusion coefficients for motion in non-confining, rough, multi-dimensional potential energy landscapes. This leads us to propose an expression for the mean first-passage time from one potential minimum to any one of the immediately adjacent potential minima. As before, simulation schemes offering significant improvements upon those derived from the unmodified Langevin Equation are obtained. Good agreement between our theory's predictions and both numerical simulation schemes - unmodified and modified - is observed.