Crystallization in classical and quantum particle systems

The emergence of crystalline order is ubiquitous in physics, biology, and mathematics: atoms self-assemble into three-dimensional crystals; messy-looking proteins assemble into perfectly regular two-dimensional virus shells; densest sphere packings appear to be crystalline. Related effects are experimentally or numerically observed in quantum mechanics and continuum mechanics (vortex lattices in Bose-Einstein condensates; Wigner crystallization in the homogeneous electron gas at low density; Voronoi tesselations in optimal transport). Mathematically, these phenomena are subtle. An important first step is to prove that crystallized states asymptotically achieve the lowest energy, packing density, transportation cost divided by the number N of assembled objects or the available volume V. But this does not yet prove long-range crystalline order: disordered states might achieve the same energy/density/cost at leading order, and may even be favoured by lower-order terms.

In this mini-course I will focus on ``strong crystallization theorems’’ which say that optimizers necessarily must exhibit crystalline order. These are available in rare examples, whose previous understanding by pioneers like Heitmann, Radin, Theil, Mainini, Stefanelli (2D classical systems) and Colombo, Di Marino, De Pascale (1D Wigner crystallization) relies on clever but highly technical estimates. I will present much simpler poofs, based respectively on methods of discrete differential geometry (2D Heitmann-Radin model; joint work with Lucia De Luca) respectively convex duality (1D Wigner crystallization). These methods appear to be very promising for future research.

I will begin this mini-course with an elementary introduction to the underlying physical models and relevant scaling limits.