  We prove a lower bound on the entropy of sphere packings of $\mathbb R^d$ of
density $\Theta(d \cdot 2^{-d})$. The entropy measures how plentiful such
packings are, and our result is significantly stronger than the trivial lower
bound that can be obtained from the mere existence of a dense packing. Our
method also provides a new, statistical-physics-based proof of the $\Omega(d
\cdot 2^{-d})$ lower bound on the maximum sphere packing density by showing
that the expected packing density of a random configuration from the hard
sphere model is at least $(1+o_d(1)) \log(2/\sqrt{3}) d \cdot 2^{-d}$ when the
ratio of the fugacity parameter to the volume covered by a single sphere is at
least $3^{-d/2}$. Such a bound on the sphere packing density was first achieved
by Rogers, with subsequent improvements to the leading constant by Davenport
and Rogers, Ball, Vance, and Venkatesh.
