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Rough Billiards

Peter Rudzis
Monday, November 22, 2021 - 2:30pm to 3:30pm
Zoom Meeting ID: 992 8713 1654

Abstract: Stochastic billiards is the study of billiards systems where particles are reflected randomly from the boundary of the billiard. Rough billiards is concerned with the special case in which the random law of reflection arises from a geometric microstructure on the boundary.  A basic example to have in mind is a ray of light reflecting from a rough, matte surface. More complicated examples are obtained by considering certain frictional collisions between rough rigid bodies.  In this talk, we will first introduce rough billiards, illustrating the concept with a number of examples. We will then present a characterization of rough reflection laws originally due to Plakhov and later obtained independently by Angel, Burdzy, and Sheffield. This theorem says that a random reflection law is a rough reflection law if and only if the law is symmetric with respect to a certain well-known invariant measure from billiards theory.  Second, we will introduce a billiards model for collisions between a free-moving rough disk and a fixed rough wall. Here we also obtain a characterization theorem, saying that a collision law for the disk-wall system is a rough collision law if and only if it is symmetric with respect to the invariant measure and it conserves a certain quantity associated with ``rolling collisions.'' We will also describe a method for constructing rough collision laws.  Rough collision laws are in one-to-one correspondence with rough reflection laws via a mapping obtained by ``foreshortening'' the geometric microstructure on the boundary.

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