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 l²/D 1/lnρl in one dimension; a similar scaling behavior occurs in greater than one dimension. In one dimension, we also determine the probability that the kth-closest particle is the first one to hit the absorber. At large k, this probability decays as k^1/3exp(-Ak^2/3), with A = 1.932 99... analytically calculable. As a corollary, the characteristic hitting time T_k for the kth-closest particle scales as k^4/3; this corresponds to superdiffusive but still subballistic motion.