Directing a random walker with optimal potentials

In a two-dimensional discrete environment we obtain numerically the potential surface that minimizes the diffusion time for a particle that is guided toward a goal point, for a given temperature. The optimal potential shape is a branched one from the confluence of three factors that helps direct diffusion: the reduction of the dimensionality of the walk, the optimization of the potential shape in one dimension, and the minimization of the paths. We discuss the possible applications of the result to robotic navigation.