Dynamical and spatial disorder in an intermittent search process

We consider a one-dimensional model of an intermittent search process in a medium exhibiting frozen disorder. A tracer, searching for Poisson-distributed targets, alternates diffusive and ballistic motions, but can only find a target when diffusing. Preliminary theoretical results [1] are now confirmed, completed and extended, and their derivations are presented for the first time. We study the mean search time 〈T〉 according to the laws of the searcher waiting times in the diffusive and ballistic regimes. In particular, we obtain a lower bound of 〈T〉, which in certain circumstances is also an approximation and is valid for a very broad class of waiting time distributions. Explicit results and other approximations are presented in the case of exponential waiting times, and we study the optimization of 〈T〉, depending on the mean durations of the diffusive and ballistic phases. Theoretical formulae are supported by numerical simulations. We show that the intermittent behaviour can allow one to minimize the search time in comparison with the purely diffusive behaviour, and that it is possible, by an adequate choice of the parameters, to increase very significantly the efficiency of the search.