Suppose there are $N$ unit-mass particles whose initial positions are uniformly distributed in a unit-radius disk. Each particle is assigned a randomly oriented initial velocity vector of length $1$ (green below). Then the particles act upon one another via inverse-square gravity.
[![NBodyRand8][1]][1]
Dots show initial positions inside unit disk. Green vectors: initial velocity; red vectors: final velocity.
Q. What is the probability that $k \ge 1$ of the $N$ particles remain within a disk of some radius $R \ge 1$ forever?
In the illustration above, $N=8$ and $R=3$. (But I do not trust my crude simulations.)
I am wondering if the answer is: zero, independent of $k$ and $R$ and the gravitational constant? Perhaps the answer differs for points in $\mathbb{R}^2$ vs. points in $\mathbb{R}^3$ (or $\mathbb{R}^d$, $d > 3$)?