Suppose $n$ air molecules (infinitesimal points) are bouncing around in a unit $d$-dimensional cube, with perfectly elastic wall collisions. Let $k=n^{\frac{1}{d}}$. For example, in 3D, $d=3$, with $n=64$, $k=4$. In analogy with the lonely runner conjecture, say that a molecule $m$ is lonely if it is at least distance $\frac{1}{k}$ from every other molecule, $\frac{1}{4}$ in the example.

I am not entirely certain this $\frac{1}{k}$ distance is the appropriate generalization, but let me proceed nevertheless.

Q1. Assuming the molecules start at random positions, and have distinct speeds and distinct, irrational-multiples-of-$\pi$ initial direction vector angles w.r.t. the cube edges—and so never collide—is it the case that each molecule is lonely at some time? Or more generally, what is the largest distance $\lambda$ for which is it guaranteed that each molecule is at least $\lambda$ from every other molecule at some time?

Q2. And an obverse question: Under the same assumptions, what is the smallest subcube volume $v$ in which it is guaranteed that all molecules will at some time simultaneously reside?

Perhaps $v=(\frac{1}{k-1})^d$ is the value to beat for Q2?

These questions lay behind this recent question. I was wondering about the chance that all air molecules would spontaneously cluster in a corner of a room. :-) I am unclear on whether the circle in the lonely runner conjecture, and reflecting cubes above, are sufficiently analogous to transfer insights from one to the other.