# Kinetics of Diffusion-Controlled Annihilation with Sparse Initial Conditions

Source: [Kinetics of Diffusion-Controlled Annihilation with Sparse Initial Conditions](<https://www.sota2.com/research/paper/68fa34b41e209131ce89b756>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-24T20:50:53.202Z

Authors: E. Ben-Naim, P.L. Krapivsky

Published: 2016-07-27T21:25:47.000Z

## 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 $\\Delta=2$.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1607.08268>)

[PDF](<https://arxiv.org/pdf/1607.08268.pdf>)