Crystallization of many-particle ground states

5 problems

• Universal optimality of the triangular lattice in the plane

  Open

  In 2022 the $E8$ and Leech lattices were proved to be the best arrangements in 8 and 24 dimensions for every repulsive force from a broad class. The matching claim for the triangular lattice in two dimensions is unproved.

• Body-centered cubic ground state of classical 3D jellium

  Open

  Point charges in a uniform background of opposite charge are expected to form a body-centered cubic crystal, the Wigner crystal. No proof exists that this arrangement has the lowest energy.

• Triangular-lattice crystallization for the 2D Lennard-Jones potential

  Open

  In the plane, particles with Lennard-Jones forces should arrange into the triangular lattice. Existing proofs work only for forces with a much narrower attractive well.

• Periodic ground states for the Lennard-Jones potential in 3D

  Open

  Prove that many particles interacting through the standard Lennard-Jones force (repulsive at short range, attractive at long range) have a lowest-energy arrangement that is a periodic crystal. Computations suggest the hexagonal close-packed crystal.

• Close-packed ground states of sticky hard spheres

  Open

  Model atoms as hard balls that gain energy only when they touch. Prove that the best arrangement of many such balls stacks flat triangular layers, as in fcc or hcp crystals.