A geometric probabilistic approach to random packing of hard disks in a plane

Abstract

In this paper, the random packing fraction of hard disks in a plane is analyzed, following a geometric probabilistic approach. First, the random close packing (RCP) of equally sized disks is modelled. Subsequently, following the same methodology, a simple, statistical geometric model is proposed for the random loose packing (RLP) of monodisperse disks. This very basic derivation of RLP leads to a packing value (≈0.66) that is in very good agreement with values that have been obtained previously for 2D disk packings. The present geometrical model also enables a closed-form expression for the contact (coordination) number as a function of the packing density at different states of compaction. These predictions are thoroughly compared with empirical and simulation results, among others the Rényi parking model, yielding good agreement.