A stochastic step flow model with growth in 1+1 dimensions

Mathematical implications of adding Gaussian white noise to the Burton– Cabrera–Frank model for N terraces (‘gaps’) on a crystal surface are studied under external material deposition for large N. The terraces separate straight, non-interacting line defects (steps) with uniform spacing initially (t = 0). As the growth tends to vanish, the gaps become uncorrelated. First, simple closed-form expressions for the gap variance are obtained directly for small fluctuations. The leading-order, linear stochastic differential equations are prototypical for discrete asymmetric processes. Second, the Bogoliubov– Born–Green–Kirkwood–Yvon (BBGKY) hierarchy for joint gap densities is formulated. Third, a self-consistent ‘mean field’ is defined via the BBGKY hierarchy. This field is then determined approximately through a terrace decorrelation hypothesis. Fourth, comparisons are made of directly obtained and mean-field results. Limitations and issues in the modeling of noise are outlined. PACS numbers: 81.15.Aa, 05.10.Gg, 68.35.−p, 05.40.−a, 02.10.Yn