Kinematics, equilibrium, and shape in Hamiltonian systems: the "LAB" effect.

We demonstrate that a system of a finite number of particles in a container with a "typical" shape does not get mixed in the sense that in an (asymptotic) steady state distribution the particles are not equally likely to occupy (almost) any position inside the container. Our incomplete mixing result should be contrasted with the complete mixing of noninteracting or elastically colliding particles in a box, which does occur because the box is not a "typical" shape. The effect we describe does not disappear in the limit of infinitely many particles. However, it is unlikely to be observed in nature but rather in laboratory (the "LAB" effect) experiments. Ideal gas in a container, which is a product of billiards, serves as a transparent model to understand the "LAB" (look at billiards) effect.