Diffusion, sub-diffusion, and escape: A study of over-damped Brownian motion

Abstract

We study over-damped Brownian motion in various potential energy landscapes. Starting with one-dimensional systems, we derive the diffusion coefficient for motion in a piecewise-defined potential where the barrier height of each section is taken from a probability distribution. When the distribution is exponential the usual transition between diffusion and sub-diffusion is observed. The behaviour of the diffusion coefficient around the transition depends upon the shape of the energy barriers. A coarse-grained simulation scheme, where particles are moved between lattice sites located at the potential minima, is proposed. Good agreement with Brownian dynamics simulations is observed. Motion in one-dimensional rough potentials is studied, and a modified Langevin equation, valid on long time scales, is derived. This result is extended to higher dimensions and used to derive an expression for the effective diffusion coefficient. Comparison to that obtained from a lattice hopping model facilitates the construction of the multi-dimensional coarse-grained simulation scheme. Escape processes in multi-dimensional potentials are then studied. A framework to calculate the mean first-passage time is proposed and found to offer good agreement with simulations, even when the barrier to escape is small compared to the thermal energy. The framework attempts to capture the effect upon the motion of a varying profile in the direction(s) normal to the escape direction. Simulations reveal good agreement when the profile becomes more confining, but only moderate agreement when it becomes less so. Finally, we apply the above framework to a channel described by a confining parabolic potential whose curvature varies periodically along its length. The diffusion coefficient is derived under the assumption of rapid equilibration in the confining direction. Adding a periodic potential along the channel can enhance the rate of diffusion, depending upon its phase relative to the periodic curvature.