# Two-Scale Annihilation

Source: [Two-Scale Annihilation](<https://www.sota2.com/research/paper/68fa24412c2a3760193a194f>)

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

Data updated: 2025-10-24T09:00:16.554Z

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

Published: 1996-04-16T23:14:01.000Z

## Abstract

The kinetics of single-species annihilation, $A+A\\to 0$, is investigated in which each particle has a fixed velocity which may be either $\\pm v$ with equal probability, and a finite diffusivity. In one dimension, the interplay between convection and diffusion leads to a decay of the density which is proportional to $t^{-3/4}$. At long times, the reactants organize into domains of right- and left-moving particles, with the typical distance between particles in a single domain growing as $t^{3/4}$, and the distance between domains growing as $t$. The probability that an arbitrary particle reacts with its $n^{\\rm th}$ neighbor is found to decay as $n^{-5/2}$ for same-velocity pairs and as $n^{-7/4}$ for $+-$ pairs. These kinetic and spatial exponents and their interrelations are obtained by scaling arguments. Our predictions are in excellent agreement with numerical simulations.

## Recorded benchmark results

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/cond-mat/9604106>)

[PDF](<https://arxiv.org/pdf/cond-mat/9604106.pdf>)