Statistics of the gravitational force in various dimensions of space: from Gaussian to Levy laws

TL;DR

This paper explores how the statistical distribution of gravitational forces varies across different spatial dimensions, revealing a transition from Levy to Gaussian laws and identifying critical behaviors in two dimensions.

Contribution

It generalizes the distribution of gravitational forces for various dimensions and inhomogeneous systems with power-law density and force profiles.

Findings

In 3D, the force follows a Levy Holtsmark distribution.

In 2D, the distribution is a marginal Gaussian, marking a critical dimension.

In 1D, the force distribution is Bernoulli, approaching Gaussian for large N.

Abstract

We discuss the distribution of the gravitational force created by a Poissonian distribution of field sources (stars, galaxies,...) in different dimensions of space d. In d=3, it is given by a Levy law called the Holtsmark distribution. It presents an algebraic tail for large fluctuations due to the contribution of the nearest neighbor. In d=2, it is given by a marginal Gaussian distribution intermediate between Gaussian and Levy laws. In d=1, it is exactly given by the Bernouilli distribution (for any particle number N) which becomes Gaussian for N>>1. Therefore, the dimension d=2 is critical regarding the statistics of the gravitational force. We generalize these results for inhomogeneous systems with arbitrary power-law density profile and arbitrary power-law force in a d-dimensional universe.