First Invader Dynamics in Diffusion-Controlled Absorption

We investigate the average time for the earliest particle to hit a spherical absorber when a homogeneous gas of freely diffusing particles with density ρ and diffusivity D is prepared in a deterministic state and is initially separated by a minimum distance l from this absorber. In the high-density limit, this first absorption time scales as l D 1 ln ρl in one dimension; we also obtain the first absorption time in three dimensions. In one dimension, we determine the probability that the k-closest particle is the first one to hit the absorber. At large k, this probability decays as k exp(−Ak), with A = 1.93299 . . . analytically calculable. As a corollary, the characteristic hitting time Tk for the k -closest particle scales as k; this corresponds to superdiffusive but still subballistic motion. First Invader Dynamics in Diffusion-Controlled Absorption 2 1. Statement of the Problem Suppose that a gas of independent random walkers is uniformly and deterministically distributed with density ρ at time t = 0 in the exterior region that is a distance l beyond a spherical absorber of radius a (Fig. 1). For t > 0, the particles diffuse freely and are absorbed when they hit the absorber. This flux, or reaction rate, is fundamental to our understanding of many diffusion-controlled kinetic processes (see e.g., [1–6]). In this work, we are interested in the behavior of this flux at short times. Specifically: (a) What is the average time for the earliest particle in the gas to first hit the absorber? (b) What is the probability that the k-closest particle hits the absorber first?