A Proof of Crystallization in Two Dimensions

By Florian Theil. 2006

207 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/e2426bc631817ed5bfd5bff571db6d0e2159757f

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In Computational Theory and Mathematics

In Materials Chemistry

Citations in the corpus

Atomistic to continuum limits for computational materials science (2007, background): “Suppose we now let N go to infinity: why periodic geometries are favored energetically in this limit? In some very specific cases of pair interaction potential, and in some toy models of quantum nature in specific settings, the question can be solved: [21, 23, 25, 74, 110, 111, 118, 135, 137].”
Coulomb Gases and Ginzburg - Landau Vortices (2014, unknown): “n problem is periodic). Some exceptions include the two-dimensional sphere packing problem, for which Radin [Ra] showed that the minimizer is the triangular lattice, and an extension of this by Theil [Th] for a class of very short range Lennard-Jones potentials. The techniques used there do not appl”
Uniform energy distribution for an isoperimetric problem with long-range interactions (2008, background): “In fact, even in a discrete setting with a pair interaction potential, proving periodicity is extremely difficult and was only recently successfully accomplished under certain assumptions in [27].”
The Crystallization Conjecture: A Review (2015, background): “In a famous article [168], Theil deals with smoother, more realistic potentials (which look like VLJ), in dimension two.”
From the Ginzburg-Landau Model to Vortex Lattice Problems (2010, result): “s and other crystallisation problems, it seems natural, although far out of reach, that this could be accomplished by the triangular lattice. Positive answers are found in [Rad] for packing problems, [Th] for some very short-range pairwise interaction potential, and one also ﬁnds the same conjecture”
Higher‐Dimensional Coulomb Gases and Renormalized Energy Functionals (2013, background): “A proof of this is out of our present reach: crystallization problems have up to now been solved only for specific short-range interaction potentials (see [13,34,59,75,77] and references therein) that do not cover Coulomb forces or one-dimensional systems [14,41,68].”