ABSTRACT
Consider N independently diffusing particles that reversibly bind to a target. We study a problem recently introduced by Grebenkov of finding the first passage time (FPT) for K of the N particles to be simultaneously bound to the target. Since binding is reversible, bound particles may unbind before the requisite K particles bind to the target. This so-called "impatience" leads to a delicate temporal coupling between particles. Recent work found the mean of this FPT in the case that N = K = 2 in a one-dimensional spatial domain. In this paper, we approximate the full distribution of the FPT for any N ≥ K ≥ 1 in a broad class of domains in any space dimension. We prove that our approximation (i) is exact in the limit that the target and/or binding rate is small and (ii) is an upper bound in any parameter regime. Our approximation is analytically tractable and we give explicit formulas for its mean and distribution. These results reveal that the FPT can depend sensitively and nonlinearly on both K and N. The analysis is accompanied by detailed numerical simulations.
