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Transport of Brownian particles confined to a weakly corrugated channel

We investigate the average velocity of Brownian particles driven by a constant external force when constrained to move in two-dimensional, weakly-corrugated channels. We consider both the geometric confinement of the particles between solid walls as well as the soft confinement induced by a periodic potential. Using perturbation methods we show that the leading order correction to the marginal probability distribution of particles in the case of soft confinement is equal to that obtained in the case of geometric confinement, provided that the (configuration) integral over the cross-section of the confining potential is equal to the width of the solid channel. We then calculate the probability distribution and average velocity in the case of a sinusoidal variation in the width of the channels. The reduction on the average velocity is larger in the case of soft channels at small Péclet numbers and for relatively narrow channels and the opposite is true at large Péclet numbers and for wide channels. In the limit of large Péclet numbers the convergence to bulk velocity is faster in the case of soft channels. The leading order correction to the average velocity and marginal probability distribution agree well with Brownian Dynamics simulations for the two types of confinement and over a wide range of Péclet numbers.