Shrooms!

The cover story of the March issue of Notices of the American Mathematical Society (the AMS's analog of Physics Today) comes from my work. The actual cover is a portion of a phase portrait of a particular mushroom billiard (to be explained shortly). This is my third cover story overall, but my first in about 4.5 years. One of the others was also for an expository paper and the other was for a pair of research articles that appeared back-to-back.

The first important thing here is to define and discuss mathematical billiards, which come in both classical and quantum varieties. In the classical version, a particle (usually but not always a point particle) collides elastically inside (or sometimes outside) a boundary of some shape, and the properties one gets (whether it's chaotic or not, for example) depend fundamentally on the shape. In the language of classical mechanics, this system has two degrees of freedom, so whether or not a region is chaotic depends on whether anything besides energy is a constant of motion there. The quantum version is essentially a particle-in-a-box, but with funky properties coming from the weird box shapes. (Quantum billiards are fundamental to the subject of quantum chaos, on which I did my Ph.D. thesis.)

Anyway, billiards can either be fully "integrable" (non-chaotic, loosely speaking---though this is not entirely accurate and is certainly not a definition), fully chaotic---either via a direct dispersing mechanism, as in the Sinai billiard (which consists of an exterior square boundary and interior circular one), or a net defocusing mechanism after initial focusing, as in the Bunimovich stadium (which consists of two parallel line segments with circular caps), or mixed (with some integrable regions and some chaotic regions). (The previous sentence is stylistically poor, but I hope you can parse it.) Bunimovich was my postdoc advisor at Georgia Tech, by the way.

A mushroom billiard is a generalization of the stadium billiard. Behavior is regular for trajectories that stay in the cap (because circular billiards are integrable) and chaotic for all trajectories (except for a set of measure zero) that enter the stem. The expository article in question discusses both mushroom billiards and generalized mushroom billiards. Billiards of very precise geometries can---and have been---built in laboratories, by the way. Cold atoms probably give the best control in terms of geometry, but several other systems have been used to do this as well. Also, the infamous question of "Can you hear the shape of a drum?" is a quantum billiard problem (because one again has the Helmholtz equation with homogeneous Dirichlet boundary conditions).

You can find more details in the expository article and an about-to-appear research article of mine (I'll post an entry on this once the published version gets posted online, which might actually be any day now), as well as the references in the expository article, of course.

The plots in the article (and on the cover) were created using a GUI billiard simulator for Matlab that I had one of my students designed. (I subsequently supervised this student's work on scientific problems with the GUI and on some multiple-particle billiards that didn't use the GUI but had some similarities in its code.) Technically, figure 1 was based on a pair of those plots rather than being produced from the GUI per se.