Statistical properties of the gravitational force through ordering statistics

TL;DR

The paper analyzes the stochastic Newtonian gravitational force on a test particle in an infinite, homogeneous Poisson gas, showing that the total-force distribution converges to the Holtsmark form in the infinite-limit. By applying order statistics, it derives the exact $n$-th nearest-neighbor distance distribution $w_n(r)$ in $d$-dimensions and the joint densities $w^{(n)}(r_1,\dots,r_n)$, then translates these spatial statistics into force statistics $W_n(F)$, culminating in the Holtsmark distribution for $F$ with a formally divergent variance. Crucially, it demonstrates that the divergent variance of $W_H(F)$ in $d=3$ originates entirely from the nearest neighbor contribution $n=1$, while higher-neighbor contributions yield finite moments, highlighting the local origin of large fluctuations. The work also notes strong correlations among higher-index nearest-neighbor distances and discusses implications for stochastic gravitational fields and potential extensions to finite-size effects and continuous mass distributions.

Abstract

We study the statistical distribution of Newtonian gravitational forces acting on a test particle embedded in an infinite, homogeneous, and uncorrelated random gas of particles. In the limit where both the number of neighboring particles and the confining volume tend to infinity with constant density, this distribution converges to the classic Holtsmark distribution. Our focus here is on the contribution of the nearest particle neighbors to the total Newtonian force. To this end, we derive the joint spatial distribution of the nearest neighbors in arbitrary spatial dimensions, and show that, in three dimensions, the divergence of the variance of the Holtsmark distribution originates entirely from the dominant influence of the nearest neighbor.

Figures (1)

• Figure 1: The Holtsmark distribution in Eq. \ref{['eq:holtsmark']} is represented in dashed line. The $n$-th nearest neighbor gravitational force probability distributions in Eq. \ref{['eq:Wn']} for $n=\{1,2,3,10,100\}$ are represented in colored full-lines.