Large-Scale Simulations of Diffusion-Limited n-Species Annihilation

TL;DR

This paper reports large-scale computer simulations of diffusion-limited n-species annihilation on a line, revealing a new concentration decay exponent and proposing a scaling relation for correction-to-scaling effects.

Contribution

It introduces the renormalized reaction-cell method for large simulations and provides new insights into the decay exponent for n-species annihilation.

Findings

The concentration decay exponent is (n)=(n-1)/2n, differing from previous beliefs.

Simulation results agree with recent theoretical predictions.

A scaling relation for the correction-to-scaling exponent elta is proposed.

Abstract

We present results from computer simulations for diffusion-limited $n$-species annihilation, $A_i+A_j\to0$ $(i,j=1,2,...,n;i\neq j)$, on the line, for lattices of up to $2^{28}$ sites, and where the process proceeds to completion (no further reactions possible), involving up to $10^{15}$ time steps. These enormous simulations are made possible by the renormalized reaction-cell method (RRC). Our results suggest that the concentration decay exponent for $n$ species is $\a(n)=(n-1)/2n$ instead of $(2n-3)/(4n-4)$, as previously believed, and are in agreement with recent theoretical arguments \cite{tauber}. We also propose a scaling relation for $\Delta$, the correction-to-scaling exponent for the concentration decay; $c(t)\sim t^{-\a}(A+Bt^{-\Delta})$.