Eur. Phys. J. B 64, 403-407 (2008)
https://doi.org/10.1140/epjb/e2008-00020-6

Exact results for two-dimensional coarsening

¹ Instituto de Física, Universidade Federal do Rio Grande do Sul, CP 15051, 91501-970 Porto Alegre RS, Brazil
² School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, UK
³ Université Pierre et Marie Curie – Paris VI, LPTHE UMR 7589, 4 Place Jussieu, 75252 Paris Cedex 05, France
⁴ LPTENS UMR 8549 24, Rue Lhomond, 75321 Paris Cedex 05, France

Corresponding author: ^a alan.bray@manchester.ac.uk

Received: 17 September 2007
Published online: 18 January 2008

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, , with enclosed area in the range (A, A+dA), is described, for large time t, by the scaling form , demonstrating the validity of dynamical scaling in this system. Here is a universal constant associated with the enclosed area distribution of percolation hulls at the percolation threshold, and is a material parameter. The distribution of domain areas, , is apparently very similar to that of hull areas up to very large values of . Identical forms are obtained for coarsening from a critical initial state, but with c<sub>h</sub> replaced by . The similarity of the two distributions (of areas enclosed by hulls, and of domain areas) is accounted for by the smallness of c<sub>h</sub>. By applying a `mean-field' type of approximation we obtain the form , where t₀ is a microscopic timescale and , for a disordered initial state, and a similar result for a critical initial state but with and . We also find that and . These predictions are checked by extensive numerical simulations and found to be in good agreement with the data.