Exact results for two-dimensional coarsening

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 + lambda (h)t)(2), demonstrating the validity of dynamical scaling in this system. Here c(h) = 1/8 pi root 3 is a universal constant associated with the enclosed area distribution of percolation hulls at the percolation threshold, and lambda(h) is a material parameter. The distribution of domain areas, n(d) (A, t), is apparently very similar to that of hull areas up to very large values of A/lambda (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 c(h) . By applying a 'mean-field' type of approximation we obtain the form n(d) (A, t) similar or equal to 2c (d) [lambda (d) (t+t(0))](tau-2)/[A+lambda (d) (t+t(0))](tau), where t(0) is a microscopic timescale and tau = 187/91 similar or equal to 2.055, for a disordered initial state, and a similar result for a critical initial state but with c(d) -> c(d) /2 and tau -> tau(c) = 379/187 similar or equal to 2.027. We also find that c(d) = c(h) + O(c(h)(2)) and lambda(d) = lambda(h) (1 + O(c(h) )). These predictions are checked by extensive numerical simulations and found to be in good agreement with the data.