Scale invariant forces in 1d shuffled lattices

TL;DR

This paper provides an exact analysis of the probability density function of forces in 1D perturbed lattices, revealing Gaussian or Lévy-like tails depending on particle proximity constraints, with tail exponents linked to interaction laws.

Contribution

It offers a detailed, exact characterization of the force distribution in 1D perturbed lattices, connecting tail behavior to interaction laws and perturbation constraints.

Findings

Gaussian-like force distribution when close particle pairs are forbidden

Lévy-like power law tail when close pairs are allowed

Tail exponent matches the interaction law exponent

Abstract

In this paper we present a detailed and exact study of the probability density function $P(F)$ of the total force $F$ acting on a point particle belonging to a perturbed lattice of identical point sources of a power law pair interaction. The main results concern the large $F$ tail of $P(F)$ for which two cases are mainly distinguished: (i) Gaussian-like fast decreasing $P(F)$ for lattice with perturbations forbidding any pair of particles to be found arbitrarily close to one each other; (ii) L\'evy-like power law decreasing $P(F)$ when this possibility is instead permitted. It is important to note that in the second case the exponent of the power law tail of $P(F)$ is the same for all perturbation (apart from very singular cases), and is in an one to one correspondence with the exponent characterizing the behavior of the pair interaction with the distance between the two particles.