Kinetics of Diffusion-Controlled Annihilation with Sparse Initial Conditions
E. Ben-Naim and P.L. Krapivsky
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 $\delta$ that is embedded in a larger space of dimension
$d$. We find that the co-dimension $\Delta=d-\delta$ governs the
behavior. All particles disappear when the co-dimension is
sufficiently small, $\Delta\leq 2$; 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, $\delta=1$, 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 $\Delta <2$, and there is an inverse logarithmic decay at the
critical co-dimension $\Delta=2$.
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