Cluster-size distributions for irreversible cooperative filling of lattices. I. Exact one-dimensional results for coalescing clusters

Abstract

We consider processes where the sites of an infinite, uniform lattice are filled irreversibly and cooperatively, with the rate of adsorption at a site depending on the state of its nearest neighbors (only). The asymmetry between empty and filled sites, associated with irreversibility, leads one to consider the closed infinite coupled hierarchies of rate equations for probabilities of connected and singly, doubly, etc., disconnected empty subconfigurations and results in an empty-site-shielding property. The latter allows exact solutions, via truncation, of these equations in one dimension and is used here to determine probabilities of filled s-tuples, $f_s$ ($f_1$≡θ is the coverage), and thus of clusters of exactly s filled sites, $n_s$≡$f_s$-2$f_{s+1\mathrm{}}$+$f_{s+2\mathrm{}}$ for s≤13 and 11, respectively. When all rates are nonzero so that clusters can coalesce, the $f_s$ and $n_s$ distributions decay exponentially as s→∞, and we can accurately estimate the asymptotic decay rate λ(θ)≡ ${\lim }_{\mathrm{s}\to \mathrm{\infty }}$ ${\mathrm{f}}_{\mathrm{s}+1\mathrm{}}$/${\mathrm{f}}_{\mathrm{s}}$≡ ${\lim }_{\mathrm{s}\to \mathrm{\infty }}$ ${\mathrm{n}}_{\mathrm{s}+1\mathrm{}}$/${\mathrm{n}}_{\mathrm{s}}$, where 0=λ(0)≤λ(θ)≤λ(1)=1. Divergent behavior of the average cluster size, as θ→1, is also considered. In addition, we develop a novel technique to determine directly the asymptotic decay rate λ(θ) and indicate its extension to higher-dimensional irreversible cooperative filling (and to other dynamic processes on lattices).