Physical Billiards and Open Dynamical Systems

Dissertation Defense
Thursday, July 8, 2021 - 10:00am for 2 hours
Hassan Attarchi – Georgia Institute of Technology – hattarchi@gatech.edu
Hassan Attarchi

This thesis consists of four works in dynamical systems with a focus on billiards. In the first part, we consider open dynamical systems, where there exists at least a ``hole" of positive measure in the phase space which some portion of points in phase space escapes through that hole at each iterate of the dynamical system map. Here, we study the escape rate (a quantity that presents at what rate points in phase space escape through the hole) and various estimations of the escape rate of an open dynamical system. We uncover a reason why the escape rate is faster than expected, which is the convexity of the function defining escape rate. Moreover, exact computations of escape rate and its estimations are present for the skewed tent map and Arnold’s cat map.

In the second part of the thesis, we study physical billiards where the moving particle has a finite nonzero size. In contrast to mathematical billiards where a trajectory is excluded when it hits a corner point of the boundary, in physical billiards reflection of the physical particle (a ball) off a visible corner point is well-defined. Initially, we study properties of such reflections in a physical billiards. Our results confirm that the reflection considered in the literature about physical billiards are indeed no-slip friction-free (elastic) collisions.

In the third part of the thesis, we study physical Ehrenfests' wind-tree models, where we show that physical wind-tree models are dynamically richer than the well-known Lorentz gas model. More precisely, when we replace the point particle by a physical one (a ball), the wind-tree models show a new superdiffusive regimes that never been observed in any other model such as Lorentz gas.

Finally, we prove that typical physical polygonal billiard is hyperbolic at least on a subset of positive measure and therefore has a positive Kolmogorov-Sinai entropy for any positive radius of the moving particle.