Fine structure of a one-dimensional discrete point system

We consider a system of $N$ points $x_1<\dots<x_N$ on a segment of the real line. An ideal system (crystal) is a system where all distances between neighbors are the same. Deviation from idealness is characterized by a system of finite differences $\nabla_i^1=x_{i+1}-x_i$, $\nabla_i^{k+1}=\nabla_{i+1}^k-\nabla_i^k$, for all possible $i$ and $k$. We find asymptotic estimates as $N\to\infty$, $k\to\infty$, for a system of points minimizing the potential energy of a Coulomb system in an external field.