Phase-ordering kinetics of one-dimensional nonconserved scalar systems.

We consider the phase-ordering kinetics of one-dimensional scalar systems. For attractive long-range (${\mathit{r}}^{\mathrm{\ensuremath{-}}(1+\mathrm{\ensuremath{\sigma}})}$) interactions with \ensuremath{\sigma}g0, ``energy-scaling'' arguments predict a growth law of the average domain size L\ensuremath{\sim}${\mathit{t}}^{1/(1+\mathrm{\ensuremath{\sigma}})}$ for all \ensuremath{\sigma}g0. Numerical results for \ensuremath{\sigma}=0.5, 1.0, and 1.5 demonstrate both scaling and the predicted growth laws. For purely short-range interactions, an approach of Nagai and Kawasaki [Physica A 134, 483 (1986)] is asymptotically exact. For this case, the equal-time correlations scale, but the time-derivative correlations break scaling. The short-range solution also applies to systems with long-range interactions when \ensuremath{\sigma}\ensuremath{\rightarrow}\ensuremath{\infty}, and in that limit the amplitude of the growth law is exactly calculated.