Non-Detection Probability of a Diffusing Target by a Stationary Searcher in a Large Region

Abstract

We revisit one of the classical search problems in which a diffusing target encounters a stationary searcher. Under the condition that the searcher's detection region is much smaller than the search region in which the target roams diffusively, we carry out an asymptotic analysis to derive the decay rate of the non-detection probability. We consider two different geometries of the search region: a disk and a square, respectively. We construct a unified asymptotic expression valid for both of these two cases. The unified asymptotic expression shows that the decay rate of the nondetection probability, to the leading order, is proportional to the diffusion constant, is inversely proportional to the search region, and is inversely proportional to the logarithm of the ratio of the search region to the searcher's detection region. Furthermore, the second term in the unified asymptotic expansion indicates that the decay rate of the non-detection probability for a square region is slightly smaller than that for a disk region of the same area. We also demonstrate that the asymptotic results are in good agreement with numerical solutions.

Absorption Rate into a Small Sphere for a Diffusing Particle Confined in a Large Sphere

“…In this paper, we would like to extend our earlier work [6] to high dimensions. More specifically, we investigate the absorption rate into a small sphere such as a cookie-cutter sensor for a difusing particle (i.e.…”

“…This problem has been addressed by Eagle [5] where the search region is a two-dimensional disk. Recently we have revisited this problem and have derived a unified asymptotic expression for the decay-rate of the non-detection problability which is valid for the cases where the search region is either a disk or a square [6].…”