Power law decay and self-similar distributions in stadium billiards

Orbits of particles in Hamiltonian systems may spend long times near invariant sets. These orbits, called sticky orbits, can lead to self-similar probability distributions and power law decay. We study problems in stadium-type billiards where the sticky invariant sets consist of orbits that are perpendicular to the straight boundaries of the billiard. We consider the time dependence originating from various initial distributions of the angle of incidence for an ensemble of particles in: the stadium billiard, an open variant of the stadium billiard in which most of the circular wall is removed allowing orbits to leave the billiard, and a quarter stadium billiard in which the stadium is bisected by horizontal and vertical walls with a porous vertical wall. We find that in each of these cases the relaxing distributions are asymptotically self-similar, and that in the open billiards the particle populations exhibit algebraic decay with time. Power law decay exponents are determined for the various situations considered.