Phase transitions in the one-dimensional Coulomb medium

A Coulomb medium is a system of $N$ charged particles of equal charge on an interval with nearest-neighbor Coulomb interaction and constant external electric field. We show that, asymptotically as $N\to\infty$, stable configurations have four possible phases of the particle density depending on the external field, which is assumed to be a function of $N$. Moreover, we find these phases explicitly.