The crystallization conjecture: a review

  • Xavier Blanc

    Université Denis Diderot (Paris 7), France
  • Mathieu Lewin

    Université de Cergy-Pontoise, France

Abstract

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 where 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.

Cite this article

Xavier Blanc, Mathieu Lewin, The crystallization conjecture: a review. EMS Surv. Math. Sci. 2 (2015), no. 2, pp. 255–306

DOI 10.4171/EMSS/13