Morphometric approach to many-body correlations in hard spheres

We model the thermodynamics of local structures within the hard sphere liquid at arbitrary volume fractions through the \textit{morphometric} calculation of $n$-body correlations. We calculate absolute free energies of local geometric motifs in excellent quantitative agreement with molecular dynamics simulations across the liquid and supercooled liquid regimes. We find a bimodality in the density library of states where five-fold symmetric structures appear lower in free energy than four-fold symmetric structures, and from a single reaction path predict a relaxation barrier which scales linearly in the compressibility factor. The method provides a new route to assess changes in the free energy landscape at volume fractions dynamically inaccessible to conventional techniques.