A Proof of Crystallization in Two Dimensions

@article{Theil2006APO,
  title={A Proof of Crystallization in Two Dimensions},
  author={Florian Theil},
  journal={Communications in Mathematical Physics},
  year={2006},
  volume={262},
  pages={209-236}
}
  • F. Theil
  • Published 1 February 2006
  • Computer Science
  • Communications in Mathematical Physics
Many materials have a crystalline phase at low temperatures. The simplest example where this fundamental phenomenon can be studied are pair interaction energies of the type where y(x) ∈ℝ2 is the position of particle x and V(r) ∈ ℝ is the pair-interaction energy of two particles which are placed at distance r. Due to the Mermin-Wagner theorem it can't be expected that at finite temperature this system exhibits long-range ordering. We focus on the zero temperature case and show rigorously that… 
On the Crystallization of 2D Hexagonal Lattices
It is a fundamental problem to understand why solids form crystals at zero temperature and how atomic interaction determines the particular crystal structure that a material selects. In this paper we
Distribution of Cracks in a Chain of Atoms at Low Temperature
We consider a one-dimensional classical many-body system with interaction potential of Lennard–Jones type in the thermodynamic limit at low temperature $$1/\beta \in (0,\infty )$$ 1 / β ∈ ( 0
Long range order in atomistic models for solids
The emergence of long-range order at low temperatures in atomistic systems with continuous symmetry is a fundamental, yet poorly understood phenomenon in Physics. To address this challenge we study a
Crystallization in the hexagonal lattice for ionic dimers
We consider finite discrete systems consisting of two different atomic types and investigate ground-state configurations for configurational energies featuring two-body short-ranged particle
Local variational study of 2d lattice energies and application to Lennard–Jones type interactions
In this paper, we focus on finite Bravais lattice energies per point in two dimensions. We compute the first and second derivatives of these energies. We prove that the Hessian at the square and the
Optimality of the triangular lattice for Lennard-Jones type lattice energies: a computer-assisted method
It is well-known that any Lennard-Jones type potential energy must a have periodic ground state given by a triangular lattice in dimension 2. In this paper, we describe a computer-assisted method
Crystallization in block copolymer melts with a dominant phase
In this paper we derive a new model for diblock copolymer melts with a dominant phase that is simple enough to be amenable not only to numerics but also to analysis, yet sophisticated enough to
Finite Crystallization and Wulff shape emergence for ionic compounds in the square lattice
We present two-dimensional crystallization results in the square lattice for finite particle systems consisting of two different atomic types. We identify energy minimizers of configurational
Surface energies in a two-dimensional mass-spring model for crystals
We study an atomistic pair potential-energy E((n))(y) that describes the elastic behavior of two-dimensional crystals with n atoms where y is an element of R(2xn) characterizes the particle
From the Ginzburg-Landau Model to Vortex Lattice Problems
TLDR
It is shown that the vortices of minimizer of Ginzburg-Landau, blown-up at a suitable scale, converge to minimizers of W, thus providing a first rigorous hint at the Abrikosov lattices, which is a next order effect compared to the mean-field type results.
...
...

References

SHOWING 1-7 OF 7 REFERENCES
New integral estimates for deformations in terms of their nonlinear strains
AbstractIf u is a bi-Lipschitzian deformation of a bounded Lipschitz domain Ω in ℓn (n≧2), we show that the LP norm (p≧1, p≠n) of a certain “nonlinear strain function” e(u) associated with u
A Theorem on Geometric Rigidity and the Derivation of Nonlinear Plate Theory from Three-Dimensional Elasticity
The energy functional of nonlinear plate theory is a curvature functional for surfaces first proposed on physical grounds by G. Kirchhoff in 1850. We show that it arises as a Γ‐limit of
Singular perturbations as a selection criterion for periodic minimizing sequences
SummaryMinimizers of functionals like $$\int_0^1 { \in ^2 u^2 _{xx} } + (u_x^2 - 1)^2 + u^2 dx$$ subject to periodic (or Dirichlet) boundary conditions are investigated. While for ε=0 the infimum is
The infinite-volume ground state of the Lennard-Jones potential
We consider a finite chain of particles in one dimension, interacting through the Lennard-Jones potential. We prove the ground state is unique, and approaches uniform spacing in the infinite-particle
The ground state for soft disks
We consider some two-dimensional models of point particles interacting through short-range two-body potentials and prove that their zero temperature, zero pressure states are crystalline.
Rotation and strain