Intermittent random walks for an optimal search strategy: One-dimensional case

Details Intermittent random walks for an optimal search strategy: One-dimensional case Intermittent random walks for an optimal search strategy: One-dimensional case

2006-09-25

arxiv.org/abs/cond-mat/0609641v1

Physics

Condensed Matter

Statistical Mechanics

19 pages, 5 figures; submitted to Journal of Physics: Condensed Matter; special issue on Chemical Kinetics Beyond the Textbook

Scientific paper

10.1088/0953-8984/19/6/065142

We study the search kinetics of an immobile target by a concentration of randomly moving searchers. The object of the study is to optimize the probability of detection within the constraints of our model. The target is hidden on a one-dimensional lattice in the sense that searchers have no a priori information about where it is, and may detect it only upon encounter. The searchers perform random walks in discrete time n=0,1,2, ..., N, where N is the maximal time the search process is allowed to run. With probability \alpha the searchers step on a nearest-neighbour, and with probability (1-\alpha) they leave the lattice and stay off until they land back on the lattice at a fixed distance L away from the departure point. The random walk is thus intermittent. We calculate the probability P_N that the target remains undetected up to the maximal search time N, and seek to minimize this probability. We find that P_N is a non-monotonic function of \alpha, and show that there is an optimal choice \alpha_{opt}(N) of \alpha well within the intermittent regime, 0 < \alpha_{opt}(N) < 1, whereby P_N can be orders of magnitude smaller compared to the "pure" random walk cases \alpha =0 and \alpha = 1.

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