2024 - The finite crystallization problem

Abstract

In this talk, I present the finite crystallization problem. I review some of the results in the literature and present a new technique to prove two-dimensional crystallization results in the square lattice for finite particle systems. We consider configurational energies featuring twobody short-ranged particle interactions and three-body angular potentials favoring bond-angles of the square lattice. To each configuration, we associate its bond graph which is then suitably modified by identifying chains of successive atoms. This method, called stratification, reduces the crystallization problem to a simple minimization that corresponds to a proof via slicing of the isoperimetric inequality.