# Reaction Kinetics of Clustered Impurities

Source: [Reaction Kinetics of Clustered Impurities](<https://www.sota2.com/research/paper/68fa24132c2a37601939efdf>)

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

Data updated: 2025-10-27T02:56:38.015Z

Authors: E. Ben-Naim

Published: 1994-12-09T20:27:06.000Z

## Abstract

We study the density of clustered immobile reactants in the diffusion-controlled single species annihilation. An initial state in which these impurities occupy a subspace of codimension d' leads to a substantial enhancement of their survival probability. The Smoluchowski rate theory suggests that the codimensionality plays a crucial role in determining the long time behavior. The system undergoes a transition at d'=2. For d'\<2, a finite fraction of the impurities survive: ni(t) ~ ni(infinity)+const x log(t)/t^{1/2} for d=2 and ni(t) ~ ni(infinity)+const/t^{1/2} for d\>2. Above this critical codimension, d'\>=2, the subspace decays indefinitely. At the critical codimension, inverse logarithmic decay occurs, ni(t) ~ log(t)^{-a(d,d')}. Above the critical codimension, the decay is algebraic ni(t) ~ t^{-a(d,d')}. In general, the exponents governing the long time behavior depend on the dimension as well as the codimension.

## Recorded benchmark results

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/cond-mat/9412048>)

[PDF](<https://arxiv.org/pdf/cond-mat/9412048.pdf>)