Novel ballistic to diffusive crossover in the dynamics of a one-dimensional Ising model with variable range of interaction

The idea that the dynamics of a spin is determined by the size of its neighbouring domains was recently introduced (Biswas and Sen 2009 Phys. Rev. E 80 027101) in a Ising spin model (henceforth, referred to as model I). A parameter p is now defined to modify the dynamics such that a spin can sense domain sizes up to R = pL/2 in a one-dimensional system of size L. For the cutoff factor p -> 0, the dynamics is Ising like and the domains grow with time t diffusively as t(1/z) with z = 2, while for p = 1, the original model I showed ballistic dynamics with z similar or equal to 1. For intermediate values of p, the domain growth, magnetization and persistence show a model-I-like behaviour up to a macroscopic crossover time t(1) similar to pL/2. Beyond t(1), characteristic power-law variations of the dynamic quantities are no longer observed. The total time to reach equilibrium is found to be t = apL + b(1 - p)L-3(2), from which we conclude that the later time behaviour is diffusive. We also consider the case when a random but quenched value of p is used for each spin for which ballistic behaviour is once again obtained.