Title: Steep billiard potentials and the Boltzmann ergodic hypothesis
Speaker: Professor Vered Rom-Kedar
Speaker Info: Weizmann Institute of Science
The behavior of a point particle traveling with a constant speed in a region D, undergoing elastic collisions at the region's boundary, is known as the billiard problem. In many applications (e.g. molecular dynamics, cold atoms optical traps, microwaves in the semi-classical limit), the billiard's flow is a simplified model which imitates the conservative motion of a particle in a smooth steep potential $V_\epsilon$ , which, in the small $\epsilon$ limit, becomes a hard-wall potential. Indeed, one of the underlying assumptions of Boltzman ergodic hypothesis is that molecules behave like hard spheres.Date: Tuesday, November 28, 2006
We study rigorously this steep potential limit for arbitrary geometry and dimension; on one hand, for regular reflections, under some natural assumptions on the potentials, we provide the asymptotic expansion of the smooth solutions in terms of auxiliary billiard approximations, with error estimates which are small and have small derivatives. On the other hand, in two dimensions, we prove that tangent periodic orbits and corner polygons produce stability islands in arbitrary geometry, even in the dispersing case for which the billiards are mixing. Partial generalizations to the n dimensional case are emerging: we proved recently that linearly stable periodic orbits appear in arbitrarily steep smooth power-law potentials which limit to specific n dimensional dispersing Sinai billiards. Thus, smooth dispersing arbitrarily steep potentials may produce non ergodic flows in arbitrary large dimension.
Joint work with A. Rapoport and D. Turaev.