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

Abstract

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 αthe searchers step on a nearest-neighbour, and with probability (1-α) 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 α, and show that there is an optimal choice α_{opt}(N) of αwell within the intermittent regime, 0 < α_{opt}(N) < 1, whereby P_N can be orders of magnitude smaller compared to the "pure" random walk cases α=0 and α= 1.

19 pages, 5 figures; submitted to Journal of Physics: Condensed Matter; special issue on Chemical Kinetics Beyond the Textbook: Fluctuations, Many-Particle Effects and Anomalous Dynamics, eds. K.Lindenberg, G.Oshanin and M.Tachiya