Title:
Physical Billiards and Open Dynamical Systems

dc.contributor.advisor Bunimovich, Leonid
dc.contributor.author Attarchi, Hassan
dc.contributor.committeeMember Cvitanovic, Predrag
dc.contributor.committeeMember Liu, Yingjie
dc.contributor.committeeMember Yao, Yao
dc.contributor.committeeMember Bonetto, Federico
dc.contributor.department Mathematics
dc.date.accessioned 2021-09-15T15:42:22Z
dc.date.available 2021-09-15T15:42:22Z
dc.date.created 2021-08
dc.date.issued 2021-07-14
dc.date.submitted August 2021
dc.date.updated 2021-09-15T15:42:22Z
dc.description.abstract 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.
dc.description.degree Ph.D.
dc.format.mimetype application/pdf
dc.identifier.uri http://hdl.handle.net/1853/65048
dc.language.iso en_US
dc.publisher Georgia Institute of Technology
dc.subject Physical Billiards
dc.subject Open Dynamical Systems
dc.subject Escape Rate
dc.subject Wind-Tree Models
dc.title Physical Billiards and Open Dynamical Systems
dc.type Text
dc.type.genre Dissertation
dspace.entity.type Publication
local.contributor.advisor Bunimovich, Leonid
local.contributor.corporatename College of Sciences
local.contributor.corporatename School of Mathematics
relation.isAdvisorOfPublication 8385d52f-b627-4839-8603-2683ea2daa55
relation.isOrgUnitOfPublication 85042be6-2d68-4e07-b384-e1f908fae48a
relation.isOrgUnitOfPublication 84e5d930-8c17-4e24-96cc-63f5ab63da69
thesis.degree.level Doctoral
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