The Crystallization Conjecture: A Review

TL;DR

This review discusses the crystallization conjecture, a longstanding open problem in physics and mathematics, concerning whether particles naturally form periodic structures under certain conditions, and highlights related open questions.

Contribution

It provides a comprehensive overview of the current state of research on the crystallization conjecture and identifies key open problems in the field.

Findings

The conjecture remains largely unproven and open.

Mathematically, it involves studying minima of high-dimensional functions.

The review highlights several related open problems.

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 $\mathbb{R}^{3N}$ 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.