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

Abstract

We consider processes where the sites of an infinite, uniform, one-dimensional lattice are filled irreversibly and cooperatively, with the rates $k_i$, 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 $k_2$=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 $f_s$ as a function of coverage, θ≡$f_1$, we find that $f_s$/$f_{s+1\mathrm{}}$=D(θ)s+C(θ,s), where C(θ,s)/s→0 as s→∞. This corresponds to faster than exponential decay. Also, if ρ≡$k_1$/$k_0$, then one has D(θ)∼(2ρθ${)}^{\mathrm{-}1}$ as θ→0. The filled-cluster-size distribution $n_s$ has the same characteristics. Motivated by the behavior of these families of $f_s$/$f_{s+1\mathrm{}}$-vs-s curves, we develop the natural extension of $f_s$ to s≤0. Explicit values for $f_s$ and related quantities for ‘‘almost random’’ filling, $k_0$=$k_1$, are obtained from a direct statistical analysis.