Analytical solution for the polydisperse random close packing problem in 2D

An analytical theory for the random close packing density, ${\varphi }_{\text{RCP}}$, of polydisperse hard disks is provided using an equilibrium model of crowding (Zaccone, 2022) which has been justified on the basis of extensive numerical analysis of the maximally random jammed (MRJ) line in the phase diagram of hard spheres (Anzivino et al., 2023). The solution relies on the equations of state for the hard disk fluid and provides predictions for ${\varphi }_{\text{RCP}}$ as a function of the ratio, $s$, of the standard deviation of the distribution of disk diameters to its mean. For a power-law size distribution with $s=0.246$, the theory yields ${\varphi }_{\text{RCP}}=0.892$, which compares well with the most recent numerical estimate ${\varphi }_{\text{RCP}}=0.905$ based on the Monte-Carlo swap algorithms (Ghimenti et al., 2024).