Random packing of lines in a lattice cube

Abstract

A study is made of the random sequential packing of complete lines in a cube of integer lattice points, with side N. For N<or=15 exact packing fractions are computed. It is found that if line occupation attempts arrive as a spatial Poisson process the packing has two distinct phases; initially where large numbers of potential adsorption sites are blocked, and subsequently where no further blocking occurs so that filling is exponential in time. It is shown that the ratio of the durations of the blocking to the nonblocking phases falls to zero as N-->infinity . In this limit, the packing fraction at time t is theta(t)=3/4(1-e-t). The rapid switch between phases in large systems creates a dramatic fall in the packing rate at the start of the process. This becomes a discontinuity as N-->infinity and is a consequence of the high aspect ratio of the packing objects. It provides a physical explanation for the diverging coefficients in expansions of theta(t) about t=0 for objects with diverging aspect ratio. After considering the three-dimensional case, the analysis is extended to d -dimensional cubes, for which it is conjectured that theta=d/2d-1 in the limit N-->infinity .