The Crystallization Conjecture: A Review

Xavier Blanc (1) , Mathieu Lewin (2)

1 LJLL - Laboratoire Jacques-Louis Lions
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

In this article we describe the crystallization conjecture. It states that, in appropriate physical conditions, interacting particles always place themselves into periodic configurations, breaking thereby the natural translation-invariance of the system. This famous problem is still largely open. Mathematically, it amounts to studying the minima of a real-valued function defined on $\mathbb{R}^{3N}$ where $N$ is the number of particles, which tends to infinity. We review the existing literature and mention several related open problems, of which many have not been thoroughly studied.

Domaines: Mécanique statistique [cond-mat.stat-mech]; Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Xavier Blanc, Mathieu Lewin. The Crystallization Conjecture: A Review. EMS Surveys in Mathematical Sciences, 2015, 2 (2), pp.255-306. ⟨hal-01139322v2⟩