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

We consider processes where the sites of an infinite, uniform, one-dimensional lattice are filled irreversibly and cooperatively, with the rates kᵢ, depending on the number i=0,1,2 of filled nearest neighbors. Furthermore, we suppose that filling of sites with both neighbors already filled is forbidden, so k₂=0. Thus, clusters can nucleate and grow, but cannot coalesce. Exact truncation solutions of the corresponding infinite hierarchy of rate equations for subconfiguration probabilities are possible. For the probabilities of filled s-tuples fₛ as a function of coverage, {θ}{≡}f₁, we find that fₛ/fₛ₊₁=D({θ})s+C({θ},s), where C({θ},s)/s{→}0 as s{→}{∞}. This corresponds to faster than exponential decay. Also, if {ρ}{≡}k₁/k₀, then one has D({θ}){~}(2{ρ}{θ})^-1 as {θ}{→}0. The filled-cluster-size distribution nₛ has the same characteristics. Motivated by the behavior of these families of fₛ/fₛ₊₁-vs-s curves, we develop the natural extension of fₛ to s{≤}0. Explicit values for fₛ and related quantities for ``almost random'' filling, k₀=k₁, are obtained from a direct statistical analysis.