Kinetics of diffusion-controlled annihilation with sparse initial conditions

We study diffusion-controlled single-species annihilation with sparse initial conditions. In this random process, particles undergo Brownian motion, and when two particles meet, both disappear. We focus on sparse initial conditions where particles occupy a subspace of dimension δ that is embedded in a larger space of dimension d. We find that the co-dimension Δ = d − δ governs the behavior. All particles disappear when the co-dimension is sufficiently small, Δ ≤ 2; otherwise, a finite fraction of particles indefinitely survive. We establish the asymptotic behavior of the probability S(t) that a test particle survives until time t. When the subspace is a line, δ = 1, we find inverse logarithmic decay, $S\sim {\left(lnt\right)}^{-1}$, in three dimensions, and a modified power-law decay, $S\sim \left(lnt\right)t^{-1/2}$, in two dimensions. In general, the survival probability decays algebraically when Δ < 2, and there is an inverse logarithmic decay at the critical co-dimension Δ = 2. (paper)