On the configuration of a one-dimensional system of interacting particles with minimum potential energy per particle

From the empirical fact that at sufficiently low temperatures most substances crystallize one may be led to the assumption that for (infinitely) large systems of interacting particles the configuration with minimum potential energy per particle will be a periodic structure. Plausible as this assumption may seem, as far as we are aware rigorous proof has not yet been given. In this paper we restrict ourselves to finite and infinite one-dimensional systems of equal particles interacting with two-body potentials of simple type, among others to convexrepulsive interactions (at a given number density of the system) and to Lennard-Jones-type potentials. For these cases a number of theorems have been proved rigorously. The theorems are stated in the Introduction and the proofs are given in subsequent sections.