The concept of "random close packing" was shown by Torquato et al. (2000) to be mathematically ill-defined idea that is better replaced by the notion of "maximally random jammed."
Random close packing of circles in two dimensions has a theoretical packing density of 0.886441 (Zaccone 2022).
Random close packing of spheres in three dimensions gives a packing density of only 
(Bernal and Mason 1960, Jaeger and Nagel 1992, Zaccone 2022), significantly smaller
than the optimal packing density for cubic or
hexagonal close packing of 0.74048. Zaccone (2022) give an exact packing density
of
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(1)
|
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(2)
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using Percus-Yevick theory, or
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(3)
|
using a "very accurate" Carnahan-Starling expression.
Donev et al. (2004) showed that a maximally random jammed state of M&Ms chocolate candies has a packing density of about 68%, or 4% greater than spheres. Furthermore, Donev et al. (2004) also showed by computer simulations other ellipsoid packings resulted in random packing densities approaching that of the densest sphere packings, i.e., filling nearly 74% of space.
and 
." Phys. Rev. Lett. 128, 028002, pp. 1-5,
2022.