First Invader Dynamics in Diffusion-Controlled Absorption

Abstract

We investigate the average time for the earliest particle to hit a spherical absorber when a homogeneous gas of freely diffusing particles with density $\rho$ and diffusivity $D$ is prepared in a deterministic state and is initially separated by a minimum distance $\ell$ from this absorber. In the high-density limit, this first absorption time scales as $\frac{{\ell }^2}{D}\frac{1}{\ln \rho \ell }$ in one dimension; we also obtain the first absorption time in three dimensions. In one dimension, we determine the probability that the $k^{\mathrm{th}}$-closest particle is the first one to hit the absorber. At large $k$, this probability decays as $k^{1/3}\exp (-A{k}^{2/3})$, with $A=1.93299\dots$ analytically calculable. As a corollary, the characteristic hitting time $T_k$ for the $k^{\mathrm{th}}$-closest particle scales as $k^{4/3}$; this corresponds to superdiffusive but still subballistic motion.

13 pages, 5 figures, IOP format. Version 2: some minor typos fixed. Version 3: some errors corrected

References in corpus (6)

• Persistence and First-Passage Properties in Non-equilibrium Systems
• Capture of the Lamb: Diffusing Predators Seeking a Diffusing Prey
• Kinetics of a Diffusive Capture Process: Lamb Besieged by a Pride of Lions
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