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!