Statistics of Random Placement, Subject to Restrictions, on a Linear Lattice

Abstract

By iterative methods of solving difference equations (or by a Monte Carlo method), statistical parameters are calculated for a number of models concerned with linear lattices. These contain sites which may be in one or two states (empty or occupied), and the progressive occupation of the sites is arranged to occur at random, but subject to specified restrictions. Examples of applications are cited from the field of crystallization or chemical reactions of chain polymers, sorption in graphite, and the close packing of hard spheres in one dimension. Special interest attaches to limiting cases in which the lattice is made to go over into a continuous, infinite straight line. It is shown that when such a line is covered with straight-line segments of constant length by placing these segments on the line at random (but avoiding overlap), the asymptotic packing density approached is approximately 0.748.