Force distribution in a randomly perturbed lattice of identical particles with $1/r^2$ pair interaction

TL;DR

This paper analyzes the statistical distribution of forces on particles in a perturbed lattice with $1/r^2$ interactions, deriving analytical expressions and validating them with simulations, relevant for gravitational and plasma systems.

Contribution

It extends Chandrasekhar's method to perturbed lattices, providing a comprehensive analytical framework for force distributions across all perturbation scales.

Findings

Derived an expression for force variance in small displacement regime.

Obtained an approximate force distribution valid from lattice to Poisson limit.

Confirmed analytical results with high-precision numerical simulations.

Abstract

We study the statistics of the force felt by a particle in the class of spatially correlated distribution of identical point-like particles, interacting via a $1/r^2$ pair force (i.e. gravitational or Coulomb), and obtained by randomly perturbing an infinite perfect lattice. In the first part we specify the conditions under which the force on a particle is a well defined stochastic quantity. We then study the small displacements approximation, giving both the limitations of its validity, and, when it is valid, an expression for the force variance. In the second part of the paper we extend to this class of particle distributions the method introduced by Chandrasekhar to study the force probability density function in the homogeneous Poisson particle distribution. In this way we can derive an approximate expression for the probability distribution of the force over the full range of perturbations of the lattice, i.e., from very small (compared to the lattice spacing) to very large where the Poisson limit is recovered. We show in particular the qualitative change in the large-force tail of the force distribution between these two limits. Excellent accuracy of our analytic results is found on detailed comparison with results from numerical simulations. These results provide basic statistical information about the fluctuations of the interactions (i) of the masses in self-gravitating systems like those encountered in the context of cosmological N-body simulations, and (ii) of the charges in the ordered phase of the One Component Plasma.