The paper is a survey devoted to planar hyperbolic billiards with emphasis on their applications in statistical physics. Let \(\mathcal D\) be a bounded domain on a plane or a two-torus with piecewise smooth boundary. A billiard system in \(\mathcal D\) is generated by a single particle moving freely inside \(\mathcal D\) with specular reflections off the boundary \(\partial \mathcal D\). The phase space of a billiard is a three-dimensional manifold \(\Omega\) with the corresponding flow \(\Phi ^t:\Omega\to\Omega\). The space of all collision points forms a two-dimensional cross-section \(\mathcal M\subset\Omega\) with the corresponding return map \(\mathcal F:\mathcal M\to\mathcal M\). The billiard is hyperbolic if the flow \(\Phi ^t\) and the map \(\mathcal F\) have nonzero Lyapunov exponents.
The survey encompasses the following items:
(1) statistical properties of dispersing billiards (Sinai billiards);
(2) systems with slow mixing rates, i.e. hyperbolic billiards with nonuniform expansion and contraction rates (billiards with convex outward boundary components, semidispersing billiards, where the boundary is convex inward, but at some points its curvature vanishes, dispersing billiards with cusps);
(3) billiard-related models of mathematical physics (billiards in the presence of external forces) with discussion of transport coefficients and their dependence on parameters;
(4) multiparticle systems (billiards with interacting particles);
(5) billiards with infinite volume.
The paper contains statement and discussion of about twenty problems of primary importance on the matter.
For the entire collection see [Zbl 1095.00005].