Sphere packings, singularities, and statistical mechanics
Miranda Holmes-Cerfon, CIMS

Abstract: What are all the ways to arrange N hard spheres into a rigid cluster? And what can the solution tell us about how materials crystallize? I will explain a numerical algorithm to tackle the first question, using ideas from semidefinite programming, and show that it produces many clusters with geometrically unusual properties. Among these are an abundance of “singular” clusters, those that are linearly flexible but nonlinearly rigid, so called because they correspond to singular solutions to a set of algebraic equations. These are also the clusters one sees with unusually high probability in experiments, which consider colloidal particles interacting with a short-ranged potential. Approximating this probability using standard methods from statistical mechanics gives a result that diverges, however I will show that by considering a “sticky-sphere” limit we can evaluate the leading-order asymptotic term in the probability, a result that has a geometric interpretation in terms of the volumes of semialgebraic sets. Applying the calculations to rigid clusters of up to N = 21 spheres suggests the free-energy landscape of a finite collection of sticky spheres approaches a universal shape, and it brings insight into the pathways to crystallization; these observations are empirical and could benefit from a more theoretical understanding.