Fine structure of one-dimensional discrete point system

TL;DR

This paper investigates the fine structure of fixed configurations in a one-dimensional discrete Coulomb system, analyzing the asymptotic behavior of finite differences as the number of points grows.

Contribution

It provides a detailed asymptotic analysis of finite differences in fixed configurations of a 1D Coulomb point system, using classical finite difference theory.

Findings

Asymptotic behavior of finite differences characterized

Results applicable to large N and l regimes

Enhanced understanding of fixed point structures in Coulomb systems

Abstract

We consider the system of $N$ points on the segment of the real line with the nearest-neighbor Coulomb repulsive interaction and external force $F$. For the fixed points of such systems (fixed configurations) we study the asymptotics (in $N$ and $l$) of finite differences of order $l$. Classical theory of finite differences is extensively used.