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, $n_h(A,t)dA$, with enclosed area in the range $(A,A+dA)$ is described, for large time $t$, by the scaling form $n_h(A,t) = 2c_h/(A+łambda_h t)^2$, demonstrating the validity of dynamical scaling in this system. Here $c_h=1/8\pi3$ is a universal constant associated with the enclosed-area distribution of percolation hulls at the percolation threshold, and $łambda_h$ is a material parameter. The distribution, $n_d(A,t)$, of domain areas is apparently very similar to that of hull areas up to very large values of $A/łambda_h t$. Identical forms are obtained for coarsening from a critical initial state, but with $c_h$ replaced by $c_h/2$. The similarity of the two distributions (of areas enclosed by hulls, and of domain areas) is accounted for by the smallness of the parameter $c_h$. By applying a `mean-field' type of approximation, we obtain the form $n_d(A,t) c_dłambda_d(t+t_0)^\tau-2/A+łambda_d(t+t_0)^\tau$, where $t_0$ is a
microscopic time scale and $= 187/91 2.055$, for a disordered initial state, and a similar result for a critical initial state but with $c_d c_d/2$ and $= 379/187 2.027$. We also find that $c_d=c_h + O(c_h^2)$ and $łambda_d = łambda_h(1 + O(c_h))$. These predictions are checked by extensive numerical simulations and found to be in good agreement with the data.
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