Phase ordering in one-dimensional systems with long-range interactions

We study the dynamics of phase ordering of a nonconserved, scalar order parameter in one dimension, with long-range interactions characterized by a power law ${\mathit{r}}^{\mathrm{-}\mathit{d}\mathrm{-}\mathrm{\sigma }}$. In contrast to higher-dimensional systems, the point nature of the defects allows simpler analytic and numerical methods. We find that, at least for σ>1, the model exhibits evolution to a self-similar state characterized by a length scale which grows with time as ${\mathit{t}}^{1/(1+\mathrm{\sigma })}$, and that the late-time dynamics is independent of the initial length scale. The insensitivity of the dynamics to the initial conditions is consistent with the scenario of an attractive, nontrivial renormalization-group fixed point which governs the late-time behavior. For σ≤1 we find indications in both the simulations and an analytic method that this behavior may be dependent on system size.