Kinetics of Diffusion-Controlled Annihilation with Sparse Initial Conditions

TL;DR

This paper investigates how particles undergoing diffusion annihilate over time when initially arranged sparsely in a subspace, revealing how the co-dimension influences survival probabilities and asymptotic behaviors.

Contribution

It introduces the role of co-dimension in diffusion-controlled annihilation with sparse initial conditions and characterizes the asymptotic survival probabilities across different dimensions.

Findings

Particles vanish when co-dimension ≤ 2; otherwise, some survive indefinitely.

Survival probability decays algebraically for co-dimension < 2.

Inverse logarithmic decay occurs at the critical co-dimension 2.

Abstract

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 $\Delta=2$.