Diffusion annihilation in one dimension and kinetics of the Ising model at zero temperature

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. These results are compared with known results for diffusion annihilation, and it is shown that there is only partial equivalence between the Ising model and diffusion annihilation. The results of Monte Carlo simulations for the domain-size distribution function for different initial magnetizations are also presented. In contrast to the case of diffusion annihilation, the domain-size distribution scaling function h (x) is found to depend nontrivially on the initial magnetization. The exponent characterizing the small-x behavior of h (x) is determined exactly and is shown rigorously to be the same for both the Ising model and diffusion annihilation.