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 ϕ_RCP=0.658 96 in d =3 and ϕ_RCP=0.886 48 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 ϕ_RCP via the coordination number z , (iii) disordered packings with ϕ >ϕ_RCP are possible at the expense of creating some order, and z =12 at the fcc limit acts as a boundary condition.