Suppose you launch $n$ point-particles on
distinct reflecting nonperiodic billiard trajectories
inside a convex polygon. Assume they all have the same speed.
Define an *$\epsilon$-cluster* as a configuration of the particles
in which they all simultaneously lie within a disk of radius $\epsilon$.

It is my understanding that *Poincaré's Recurrence Theorem*
implies that at some time after launch, the particles
will form an $\epsilon$-cluster somewhere.
(Please correct me if I am wrong here, in which case the remainder
is moot.)
Picturesquely, if I sit in my office long enough, all the air molecules
will cluster into a corner of the room. :-)

The reason I specify that the trajectories be *distinct* is to
exclude
the particles being shot in a stream all on the same trajectory.
The reason I specify *nonperiodic* is to exclude sending
the particles on parallel periodic trajectories whose length
ratios are rational, in which case no clustering need occur.
My question is:

How long must one wait for an $\epsilon$-cluster to occur?

Essentially I am seeking a quantitative version of Poincaré's Recurrence Theorem, quantitative enough to actually make a calculation. I would like to put a number of years to the air-molecule example (air molecules move perhaps 700 mph or 300 m/s). It could serve as a useful pedagogical anecdote. I found a beautiful paper that should help me answer this question:

Benoit Saussol,
"An Introduction to Quantitative Poincaré Recurrence in Dynamical Systems,"
*Reviews in Mathematical Physics*, Volume 21, Issue 08, pp. 949-979 (2009).

But I am having difficulty making the leap from the abstract theorems to an explicit calculation. Any help or additional pointers would be appreciated!

**Addendum**. Vaughn's analysis, although leaving a few loose ends (as he notes), largely
answers my question. Thanks to all for the astute comments and responses!