Why to study billiards?

Since Birkhoff mentioned billiards in order to illustrate a theorem on the existence of periodic orbits, various versions of the problem have been used to model quite different physical problems:

• Systems with impacts: systems of colliding particles are isomorphic to billiards. The simplest example of three particles in a line, interacting with some potential and making elastic collisions, is equivalent to a billiard in a wedge with a force field generated by the potential. The sides of the wedge correspond to double collisions, and the vertex to triple collisions. Systems with more dimensions or particles correspond to billiards in higher dimensional domains.

• Ergodic theory: the analogy between systems of colliding particles and billiards led Sinai to model a gas by a billiard with circular scatters, in order to justify the ergodic hypothesis of Equilibrium Statistical Mechanics. The Sinai billiard turns out to have strong chaotic properties. This result has been generalised to certain more realistic models of gases, at sufficiently low density, but is probably not true for any gas.

• Transport phenomena: billiards with scatterers and electromagnetic fields are used to study transport properties (such as normal and Hall conductivity) in the Lorenz gas.

• Quantum chaos and mesoscopic systems: the quantum version of the billiard problem consists in solving the Laplace-Poisson equation in the considered domain. In the semiclassical limit, there are strong connections between the eigenvalues of this problem and the orbits of the corresponding classical billiard, involving sometimes problems of number theory. These models are used to describe certain physical systems of nanoscopic size.