Gravitational force in an infinite one-dimensional Poisson distribution

TL;DR

This paper analyzes the statistical properties of gravitational forces in an infinite one-dimensional Poisson particle distribution, revealing divergence issues, renormalization solutions, and well-defined force differences relevant for clustering dynamics.

Contribution

It provides an exact analytic form of the force PDF, discusses divergence and renormalization, and examines force difference properties in a one-dimensional Poisson system.

Findings

Force PDF is ill-defined in the infinite-range limit without renormalization.

Renormalization yields a Gaussian force distribution.

Force differences are well-defined and relevant for clustering analysis.

Abstract

We consider the statistical properties of the gravitational field F in an infinite one-dimensional homogeneous Poisson distribution of particles, using an exponential cut-off of the pair interaction to control and study the divergences which arise. Deriving an exact analytic expression for the probability density function (PDF) P(F), we show that it is badly defined in the limit in which the well known Holtzmark distribution is obtained in the analogous three-dimensional case. A well defined P(F) may, however, be obtained in the infinite range limit by an appropriate renormalization of the coupling strength, giving a Gaussian form. Calculating the spatial correlation properties we show that this latter procedure has a trivial physical meaning. Finally we calculate the PDF and correlation properties of differences of forces (at separate spatial points), which are well defined without any renormalization. We explain that the convergence of these quantities is in fact sufficient to allow a physically meaningful infinite system limit to be defined for the clustering dynamics from Poissonian initial conditions.