The ground state for soft disks

The crystallization conjecture: a review

“…Subsequent works aimed at generalizing this result to potentials which are similar to (20), but are closer to physically realistic interactions. For instance, in [194], Radin considered a potential satisfying (20) for r ∈ [0, 1], which is non-decreasing for r ≥ 1, and tends to 0 fast enough as r → +∞. In a famous article [229], Theil dealt with smoother, more realistic potentials (which look like V LJ ), in dimension two.…”

“…It has been proposed by Wulff [249], and proved rigorously for a hard sphere model 4 in dimension two by Au Yeung, Friesecke and Schmidt in [17,220]. This work is based on results by Radin et al [194,129]. We refer for instance to [29,28,52] for similar results on the Ising model.…”

Atomistic to Continuum limits for computational materials science

“…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]. But in almost all cases relevant in practice, it is an unsolved theoretical question.…”

From atoms to crystals: a mathematical journey

“…Already for simpler models than the ab initio models treated here, i.e. models based upon two-body interactions, the results are rare: [203,174,175], [118] in one dimension, [184] in two dimensions. For the type of models we are interested in for this survey, the only result we are aware of is an oversimplified one-dimensional TF-type model, settled in [25].…”