  We create collectively jammed (CJ) packings of 50-50 bidisperse mixtures of
smooth disks in 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 $N \to \infty$ can be used as a
protocol-independent definition of random close packing.
