EMS Surveys in Mathematical Sciences

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Volume 2, Issue 2, 2015, pp. 255–306
DOI: 10.4171/EMSS/13

Published online: 2015-11-22

The crystallization conjecture: a review

Xavier Blanc[1] and Mathieu Lewin[2]

(1) Université Denis Diderot (Paris 7), France
(2) Université de Cergy-Pontoise, France

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.

Keywords: Crystallization conjecture, lattice, thermodynamic limit, Epstein zeta function, Wigner problem

Blanc Xavier, Lewin Mathieu: The crystallization conjecture: a review. EMS Surv. Math. Sci. 2 (2015), 255-306. doi: 10.4171/EMSS/13