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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.

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 $\mathbb{R}^d$, $d > 3$)?

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 $v_i$ of length $1$ (green below). Added: Robert Israel's incisive comment suggests that I should also stipulate that $\sum_i v_i = 0$. Then the particles act upon one another via inverse-square gravity.

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 $\mathbb{R}^d$, $d > 3$)?

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.

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 $R=3$.

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 $\mathbb{R}^d$, $d > 3$)?

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.

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 $\mathbb{R}^d$, $d > 3$)?