Two-Scale Annihilation

TL;DR

This paper studies the one-dimensional annihilation process with particles moving at fixed velocities and diffusing, revealing decay dynamics, domain formation, and reaction probabilities through scaling arguments and simulations.

Contribution

It introduces a detailed analysis of the interplay between convection and diffusion in single-species annihilation, deriving new kinetic and spatial exponents and their relations.

Findings

Density decays as t^{-3/4} in one dimension.

Particles form domains with size growing as t^{3/4}.

Reaction probabilities decay as power laws with respect to neighbor distance.

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.