Random close packing revisited: Ways to pack frictionless disks

We create collectively jammed (CJ) packings of 50-50 bidisperse mixtures of smooth disks in two dimensions (2D) using an algorithm in which we successively compress or expand soft particles and minimize the total energy at each step until the particles are just at contact. We focus on small systems in 2D and thus are able to find nearly all of the collectively jammed states at each system size. We decompose the probability P(phi) for obtaining a collectively jammed state at a particular packing fraction phi into two composite functions: (1) the density of CJ packing fractions rho(phi), which only depends on geometry, and (2) the frequency distribution beta(phi), which depends on the particular algorithm used to create them. We find that the function rho(phi) is sharply peaked and that beta(phi) depends exponentially on phi. We predict that in the infinite-system-size limit the behavior of P(phi) in these systems is controlled by the density of CJ packing fractions-not the frequency distribution. These results suggest that the location of the peak in P(phi) when N ->infinity can be used as a protocol-independent definition of random close packing.