  Finding the optimal random packing of non-spherical particles is an open
problem with great significance in a broad range of scientific and engineering
fields. So far, this search has been performed only empirically on a
case-by-case basis, in particular, for shapes like dimers, spherocylinders and
ellipsoids of revolution. Here, we present a mean-field formalism to estimate
the packing density of axisymmetric non-spherical particles. We derive an
analytic continuation from the sphere that provides a phase diagram predicting
that, for the same coordination number, the density of monodisperse random
packings follows the sequence of increasing packing fractions: spheres < oblate
ellipsoids < prolate ellipsoids < dimers < spherocylinders. We find the maximal
packing densities of 73.1% for spherocylinders and 70.7% for dimers, in good
agreement with the largest densities found in simulations. Moreover, we find a
packing density of 73.6% for lens-shaped particles, representing the densest
random packing of the axisymmetric objects studied so far.
