Billiard dynamics for multiple balls

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.

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.