One-particle and few-particle billiards

My paper, One-particle and few-particle billiards (joint with Steven Lansel and Leonid Bunimovich), was just published in the March 2006 issue of Chaos.

Here's the abstract:

We study the dynamics of one-particle and few-particle billiard systems in containers of various shapes. In few-particle systems, the particles collide elastically both against the boundary and against each other. In the one-particle case, we investigate the formation and destruction of resonance islands in generalized mushroom billiards, which are a recently discovered class of Hamiltonian systems with mixed regular-chaotic dynamics. In the few-particle case, we compare the dynamics in container geometries whose counterpart one-particle billiards are integrable, chaotic, and mixed. One of our findings is that two-, three-, and four-particle billiards confined to containers with integrable one-particle counterparts inherit some integrals of motion and exhibit a regular partition of phase space into ergodic components of positive measure. Therefore, the shape of a container matters not only for noninteracting particles but also for interacting particles.


Steven Lansel was one of my undergraduate research students at Georgia Tech. He's now in Stanford's Ph.D. program in EE. (Interestingly, I found out yesterday that one of the Tech writers went to high school with Lansel.) Steven is a very sharp guy and would have excelled at any undergrad school in the country.

Leonid Bunimovich is Mr. Billiard, or at least one of them (Yasha Sinai is the other one). A very important model in the field---the stadium billiard geometry, which is one of the two standard ones in experimental studies of billiard systems---is named after him.

I discussed billiards a bit before in my post about my expository article (with Lansel) on mushroom billiards. I've included a couple of the same wikipedia links that explain why billiard systems are very interesting objects to study (both mathematically and physically).

In the present article, we start off by looking at some generalizations of mushroom billiards, a system with mixed regular-chaotic dynamics in which one can control precisely which parts are regular and which are chaotic as well as the fraction of volume in phase space given by regular and chaotic components. This makes a more precise analysis possible here than for basically all other mixed dynamical systems, and obtaining a better understanding of mixed systems is extremely important--for example, one can use such systems to study transitions from one chaotic region to another as the system's parameters are varied (there are numerous studies of transition from order to chaos but very few from one chaotic regime to another, because it's typically very difficult to undertake such studies). Because one knows which regions are chaotic and which are not, this also provides a geometry in which mixed dynamics has increased experimental relevance---basically, the fact that one has large regular regions amidst chaotic seas rather than the very small islands of regular behavior that might get washed out in experimental situations. The "next frontier" (as it were) is to examine quantum mushroom billiards (and generalizations thereof) to achieve a better understanding of the quantization of mixed systems. (There is currently very little understanding of them.)

Perhaps the most important results on this paper concern few-particle billiards, in which the confined particles collide both against the confining walls and against each other. There are tons of papers on one-particle hard-ball gases (i.e., billiards) and many-particle hard-ball gases (thousands or millions for numerical studies and infinitely many for analytical work). In the latter case, one can invoke thermodynamics, some aspects of which (ergodicity) some (possibly most?) people suspected would actually already hold even with as few as two interacting particles. This paper uses numerical simulations to demonstrate that simply isn't true. While two-particle billiards are fully chaotic, there remain signatures of regular behavior in appropriate container shapes, so this whole "silent consensus" of arbitrarily stating that an interacting particle system has to be ergodic simply isn't always true. An important message, then, is that people can't just invoke ergodicity and ignore the shape of the container for interacting particles.