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How a finite potential barrier decreases the mean first passage time

We consider the mean first passage time of a random walker moving in a potential landscape on a finite interval, starting and end points being at different potentials. From analytical calculations and Monte Carlo simulations we demonstrate that the mean first passage time for a piecewise linear curve between these two points is minimised by introduction of a potential barrier. Due to thermal fluctuations this barrier may be crossed. It turns out that the corresponding expense for this activation is less severe than the gain from an increased slope towards the end point. In particular, the resulting mean first passage time is shorter than for a linear potential drop between the two points.