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 (S. Biswas and P. Sen, Phys. Rev. E {\bf 80}, 027101 (2009)) 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 $p$ \to 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 $z \simeq 1$. For intermediate values of $p$, the domain growth, magnetisation and persistence show model I like behaviour up to a macroscopic crossover time $ t_1 \sim 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 $t = apL + b(1-p)^3L^2$, 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.