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I am interested to learn to what extent results on billiards in polygons have been extended to multiple balls. Assume the balls have equal radii and the same mass, the same initial speed, and all collisions are perfectly elastic, as depicted in this* image:
           Wikipedia image
To be specific, let me start with just the square table. For single particle billiards, it is well known that (1) a trajectory of rational slope that avoids the corners is periodic, and (2) a trajectory of irrational slope that avoids the corners will be "uniformly distributed" in the sense that it spends equal times in equal areas.

Are there analogous results for billiard systems of $n>1$ balls within a square?

Perhaps it is necessary to make some assumption concerning the size of the ball radii and the box dimensions?
           3 balls
The literature I've seen on billiard dynamics does not explore this territory. Likely there are results in the literature, in which case pointers would be appreciated. Thanks!


*This impressive Wikipedia image by A.Greg was once the "Picture of the Day."
Addendum. Following Steve Huntsman's keyword hint, I retrieved Hard Ball Systems and the Lorentz Gas, which indeed contains many fascinating results. I will mention two:

  • In a chapter by Murphy & Cohen, an easy but pleasing result: For $n$ hard spheres moving in unbounded Euclidean space, with any combination of masses, there exist initial conditions such that $\binom{n}{2}$ collisions occur.
  • In a chapter by Burago, Ferleger, & Krononenko, a difficult-to-establish bound: The maximal number of collisions that may occur for $n$ balls moving in a simply connected Riemannian space of nonpositive sectional curvature never exceeds $$(400 n^2 \max{/}\min)^{2 n^4} \;,$$ where $\max{/}\min$ is the largest ratio of masses. Note there is no dependence on the radii. Matters are (or were in 2000) unclear without the nonpositive curvature assumption.
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You are probably aware of this, but even so: en.wikipedia.org/wiki/Dynamical_billiards#Sinai_billiard –  Steve Huntsman May 25 '11 at 12:33
    
Thanks, Steve! The "Sinai billiard" seems to answer my specific question for $n=2$. –  Joseph O'Rourke May 25 '11 at 13:22
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For $n >2$ try looking at "Lorentz gas". –  Steve Huntsman May 25 '11 at 13:38
    
@Steve: Great lead! Hard Ball Systems and the Lorentz Gas. I will retrieve that collection from Interlibrary Loan. springer.com/mathematics/probability/book/978-3-540-67620-1 –  Joseph O'Rourke May 25 '11 at 14:29
    
If memory serves, Matt Kahle math.ias.edu/~mkahle has some results concerning your last comment, about the phase transition that occurs when the billiard balls get too large. See for instance his blog article matthewkahle.wordpress.com/2010/02/22/many-markov-components –  Theo Johnson-Freyd May 25 '11 at 14:46
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2 Answers

up vote 7 down vote accepted

Despite the popularity of this question, the only good answer that it can ever have is to point to the vast literature of hard sphere interactions. If you go to Google scholar and ask for "hard sphere interaction" with quotes, then you will see thousands of papers, mostly in physics but some in dynamical systems. Also "lorentz gas", again with quotes, matches thousands of papers. Of course you can't expect very many explicit orbit results that you sometimes get for one circle. Much of the rigorous attention is concerned with proving ergodicity without intermediate conjectures. This is still apparently open in many cases in which you would expect it to be true. Anyway, the literature is so vast that I don't think that you can reasonably expect a survey of it as an MO answer.

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Thanks, Greg! When I asked this last May, I was ignorant of the literature and even the right search terms (e.g., Lorentz gas). Now I am much better informed, thanks the replies I received. –  Joseph O'Rourke Oct 29 '11 at 17:07
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Yes, so I meant to say that basically the question was answered in the comments --- that's why I made the answer CW. –  Greg Kuperberg Oct 29 '11 at 21:01
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The literature on hard sphere interactions is indeed vast, though I think it is appropriate to single out Simanyi's recent completion of the proof of the Boltzmann-Sinai ergodic hypothsis: Nonlinearity 26 1703 (2013).

In my view this question specifically addresses multi-particle billiards in a fixed container, about which much less is known. Chernov's paper "The work of Dmitry Dolgopyat on physical models with moving particles" mentions only three results: Simanyi showed in 1999 that two identical particles in a recangular box is ergodic in any dimension. Chernov and Dolgopyat investigate a two-particle model of Brownian motion (ie with one heavy and one light particle) in a 193-page volume of the Memoirs of the AMS in 2009. The question of ergodicity of more than two particles in a rectangular box is open.

The third result is Bunimovich et al, Commun Math Phys 146 357-396 (1992) which considers many particles in a dispersing container (ie each part of the boundary is concave) which can touch each other but are each confined to their own region. Models of this type have been taken up much later in the physics literature to study heat conduction, see Gaspard and Gilbert, Phys. Rev. Lett. 101 020601 (2008).

A few other, mostly numerical, papers on few-particle billiards are: Chaos 16 013129 (2006), Chaos 18 013127 (2008), Chaos 23 013123 (2013). These are interesting models and results, but I think it's fair to say that the study of two (let alone more) particle billiards is still in its infancy.

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