Rate equations and capture numbers with implicit islands correlations

We introduce a numerical method based on the level-set technique to compute capture numbers used in mean-field rate equations that describe epitaxial growth. In our level-set approach, islands grow with a velocity that is computed from solving the diffusion equation for the adatom concentration. The capture number for each island is then calculated by integrating the growth velocity of an island around the island boundary. Thus, our method by construction includes all spatial correlations between islands. The functional form of the capture numbers σ_s is, to first approximation, affinely dependent on the island sizes. Integration of a completely deterministic set of mean-field rate equations for the first time properly reproduces the correct island densities and cluster size distribution.