The end-on mechanism for lattice filling: Comparison with the conventional mechanism and application to the car-parking problem

Abstract

We present the exact solution for the sequential, random, irreversible filling of one-dimensional lattices by linear n-mers using the end-on filling mechanism. The results are extrapolated to then → ∞ limit (a variation on the car-parking problem) to yield a saturation coverage (packing density) of 0.7350. The end-on filling mechanism involves two steps for a single filling event. First, the landing site for one endpoint of the filling species is chosen and then the second endpoint is subsequently chosen (fromunfilled sites an appropriate distance from the first endpoint). We compare this mechanism to the conventional, one-step filling mechanism, where both endpoints of the filling species are chosen simultaneously. We present results detailing how the lattice saturation coverage varies for the two mechanisms. In addition, we extend our analysis to consider filling in the presence of a time-dependent, random distribution of inactive sites.