Calculus of Variations and Geometric Measure Theory

M. Friedrich - L. Kreutz

A proof of finite crystallization via stratification

created by kreutz on 29 Sep 2022
modified on 30 Sep 2022



Inserted: 29 sep 2022
Last Updated: 30 sep 2022

Number: 17
Year: 2022


We devise a new technique to prove two-dimensional crystallization results in the square lattice for finite particle systems. We apply this strategy to energy minimizers of configurational energies featuring two-body 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 in $\ell^1$. As a byproduct, we also prove a fluctuation estimate for minimizers of the configurational energy, known as the $n^{3/4}$-law.