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 + λ h t)2, demonstrating the validity of dynamical scaling in this system. Here \( c_h = {1 \mathord{\left/ {\vphantom {1 8}} \right. \kern-\nulldelimiterspace} 8}\pi \sqrt 3 \) is a universal constant associated with the enclosed area distribution of percolation hulls at the percolation threshold, and λ 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/λ 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) ≃ 2c d [λ d (t+t 0)] τ−2/[A+λ d (t+t 0)] τ , where t 0 is a microscopic timescale and τ = 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 τ → τ c = 379/187 ≃ 2.027. We also find that c d = c h + O(c h 2 ) and λ d = λ h (1 + O(c h )). These predictions are checked by extensive numerical simulations and found to be in good agreement with the data.