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 k/sub 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/sub 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/sub s/ as a function of coverage, thetaequivalentf/sub 1/, we find that f/sub s//f/sub s+1/ = D(theta)s+C(theta,s), where C(theta,s)/s..-->..0 as s..-->..infinity. This corresponds to faster than exponential decay. Also, if rhoequivalentk/sub 1//k/sub 0/, then one has D(theta)approx.(2rhotheta)/sup -1/ as theta..-->..0. The filled-cluster-size distribution n/sub s/ has the same characteristics. Motivated by the behavior of these families of f/sub s//f/sub s+1/-vs-s curves, we develop the natural extension of f/sub s/ to s< or =0. Explicit values for f/sub s/ and related quantities for ''almost random'' filling, k/sub 0/ = k/sub 1/, are obtained from a direct statistical analysis.