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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, $Δ\leq 2$; 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, $S\sim (\ln t)^{-1}$, in three dimensions, and a modified power-law decay, $S\sim (\ln t)\,t^{-1/2}$, 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 $Δ=2$.