Irreversible random and cooperative processes on lattices: Exact and approximate hierarchy truncation and solution

Hierarchial rate equations are presented for processes where events occur irreversibly and, in general, cooperatively ‘‘filling’’ the sites of a lattice (the hierarchy is infinite for an infinite lattice). We comment on the hierarchial connectivity structure and a shielding property of empty sites. Hierachy truncation techniques are developed based on these. We consider, in detail, two irreversible processes on infinite, uniform lattices with nearest neighbor cooperative effects, modeling: (i) reaction at the sites of a 1D polymer chain; and (ii) chemisorption onto the sites of a 2D square lattice. Our truncation procedure recovers previously obtained exact results for the 1D case and provides approximate results for the 2D case. These are compared in various cooperativity regimes including highly autoinhibitory rates (filling in stages) and autocatalytic rates (island formation).