Suppose you launch $n$ point-particles on 
distinct reflecting nonperiodic [billiard trajectories][1] 
inside a convex polygon. Assume they all have the same speed.
Define an <em>$\epsilon$-cluster</em> 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][2]_
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!

<b>Addendum</b>. 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!


  [1]: https://en.wikipedia.org/wiki/Dynamical_billiards
  [2]: https://en.wikipedia.org/wiki/Poincaré_recurrence_theorem