Exact results for curvature-driven coarsening in two dimensions

TL;DR

This paper derives exact analytical results for the distribution of domain areas during curvature-driven coarsening in two-dimensional systems, confirming dynamical scaling and universality in the process.

Contribution

It provides the first exact formulas for the hull-enclosed area distribution in 2D coarsening, demonstrating universality and scaling behavior.

Findings

Distribution of hull areas follows a universal scaling form.

Dynamical scaling is validated for large times.

Similar distributions are found for critical initial states.

Abstract

We consider the statistics of the areas enclosed by domain boundaries (`hulls') during the curvature-driven coarsening dynamics of a two-dimensional nonconserved scalar field from a disordered initial state. We show that the number of hulls per unit area that enclose an area greater than $A$ has, for large time $t$, the scaling form $N_h(A,t) = 2c/(A+\lambda t)$, demonstrating the validity of dynamical scaling in this system, where $c=1/8\pi\sqrt{3}$ is a universal constant. Domain areas (regions of aligned spins) have a similar distribution up to very large values of $A/\lambda t$. Identical forms are obtained for coarsening from a critical initial state, but with $c$ replaced by $c/2$.