Diffusion-annihilation and the kinetics of the Ising model in one dimension

The relationship between the one-dimensional kinetic Ising model at zero temperature and diffusion-annihilation in one dimension is studied. Explicit asymptotic results for the average domain size, average magnetization squared, and pair-correlation function are derived for the Ising model, for arbitrary initial magnetization. For the case of zero initial magnetization (m₀=0, a number of recent exact results for diffusion-annihilation with random initial conditions are obtained. However, for the casem₀ not equal to zero, the asymptotic behavior turns out to be different from diffusion-annihilation with random initial conditions and at a finite density. In addition, in contrast to the case of diffusion-annihilation, the domain-size distribution scaling functionh(x) is found to depend nontrivially on the initial magnetization. The origin of these differences is clarified and the existence of nontrivial correlations in the initial wall distribution for finite initial magnetization is found to be responsible for these differences. Results of Monte Carlo simulations for the domain size distribution function for different initial magnetizations are also presented.This paper is dedicated to George Weiss on the occasion of his 60th birthday.