Distributional fixed-point equations for island nucleation in one dimension: the inverse problem

The self-consistency of the distributional fixed-point equation (DFPE) approach to understanding the statistical properties of island nucleation and growth during submonolayer deposition is explored. We perform kinetic Monte Carlo simulations, in which point islands nucleate on a one-dimensional lattice during submonolyer deposition with critical island size $i$, and examine the evolution of the inter-island gaps as they are fragmented by new island nucleation. The DFPE couples the fragmentation probability distribution within the gaps to the consequent gap size distribution (GSD), and we find a good fit between the DFPE solutions and the observed GSDs for $i = 0, 1, 2, 3$. Furthermore, we develop numerical methods to address the inverse problem, namely the problem of obtaining the gap fragmentation probability from the observed GSD, and again find good self-consistency in the approach. This has consequences for its application to experimental situations where only the GSD is observed, and where the growth rules embodied in the fragmentation process must be deduced.