Sinai billiard

The Sinai billiard is a dynamical system consisting of a point particle moving freely in a bounded domain with convex obstacles, bouncing off the boundaries elastically. Introduced by Yakov Sinai in 1963, it was the first physically realistic system proved to be ergodic and mixing, establishing that deterministic chaos could produce statistical behavior indistinguishable from randomness.

The Sinai billiard is a cornerstone of Pesin theory and non-uniform hyperbolicity. The convex scatterers create dispersing trajectories — nearby orbits diverge exponentially after each collision — producing the hyperbolic structure that Sinai proved is sufficient for ergodicity. The model has been extended to the Lorentz gas, in which the scatterers are periodically extended rather than bounded.

Despite its physical simplicity, the Sinai billiard exhibits deep mathematical complexity. The singularities in the billiard map — trajectories that graze the obstacles — produce a non-uniform hyperbolicity that resists classical analysis and requires the full machinery of modern dynamical systems theory.