Explicit Analytical Solution for Random Close Packing in $d=2$ and $d=3$

We present an analytical derivation of the volume fractions for random close packing (RCP) in both $d=3$ and $d=2$, based on the same methodology. Using suitably modified nearest neighbor statistics for hard spheres, we obtain ${\varphi }_{\mathrm{RCP}}=0.65896$ in $d=3$ and ${\varphi }_{\mathrm{RCP}}=0.88648$ in $d=2$. These values are well within the interval of values reported in the literature using different methods (experiments and numerical simulations) and protocols. This statistical derivation suggests some considerations related to the nature of RCP: (i) RCP corresponds to the onset of mechanical rigidity where the finite shear modulus emerges, (ii) the onset of mechanical rigidity marks the maximally random jammed state and dictates ${\varphi }_{\mathrm{RCP}}$ via the coordination number $z$, (iii) disordered packings with $\varphi >{\varphi }_{\mathrm{RCP}}$ are possible at the expense of creating some order, and $z=12$ at the fcc limit acts as a boundary condition.