  We show that an analogy between crowding in fluid and jammed phases of hard
spheres captures the density dependence of the kissing number for a family of
numerically generated jammed states. We extend this analogy to jams of mixtures
of hard spheres in $d=3$ dimensions, and thus obtain an estimate of the random
close packing (RCP) volume fraction, $\phi_{\textrm{RCP}}$, as a function of
size polydispersity. We first consider mixtures of particle sizes with discrete
distributions. For binary systems, we show agreement between our predictions
and simulations, using both our own and results reported in previous works, as
well as agreement with recent experiments from the literature. We then apply
our approach to systems with continuous polydispersity, using three different
particle size distributions, namely the log-normal, Gamma, and truncated
power-law distributions. In all cases, we observe agreement between our
theoretical findings and numerical results up to rather large polydispersities
for all particle size distributions, when using as reference our own
simulations and results from the literature. In particular, we find
$\phi_{\textrm{RCP}}$ to increase monotonically with the relative standard
deviation, $s_{\sigma}$, of the distribution, and to saturate at a value that
always remains below 1. A perturbative expansion yields a closed-form
expression for $\phi_{\textrm{RCP}}$ that quantitatively captures a
distribution-independent regime for $s_{\sigma} < 0.5$. Beyond that regime, we
show that the gradual loss in agreement is tied to the growth of the skewness
of size distributions.
