Phase Ordering Kinetics of One-Dimensional Non-Conserved Scalar Systems

TL;DR

This paper investigates the phase-ordering kinetics in one-dimensional scalar systems with both short-range and long-range interactions, confirming theoretical growth laws through numerical simulations and analytical methods.

Contribution

It provides numerical validation of energy-scaling predictions for long-range interactions and extends the exact asymptotic analysis to short-range cases and the infinite-range limit.

Findings

Confirmed growth laws for long-range interactions via simulations

Established asymptotic exactness of Nagai and Kawasaki approach for short-range interactions

Derived exact amplitude of growth law in the infinite-range limit

Abstract

We consider the phase-ordering kinetics of one-dimensional scalar systems. For attractive long-range ($r^{-(1+\sigma)}$) interactions with $\sigma>0$, ``Energy-Scaling'' arguments predict a growth-law of the average domain size $L \sim t^{1/(1+\sigma)}$ for all $\sigma >0$. Numerical results for $\sigma=0.5$, $1.0$, and $1.5$ demonstrate both scaling and the predicted growth laws. For purely short-range interactions, an approach of Nagai and Kawasaki 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 $\sigma \rightarrow \infty$, and in that limit the amplitude of the growth law is exactly calculated.