(Georgia Institute of Technology, 2009-05-18)
Grigo, Alexander
In this thesis we consider mathematical problems related to different aspects of hard sphere systems.
In the first part we study planar billiards, which arise in the context of hard sphere systems when only one or two spheres are present. In particular we investigate the possibility of elliptic periodic orbits in the general construction of hyperbolic billiards. We show that if non-absolutely focusing components are present there can be elliptic periodic orbits with arbitrarily long free paths. Furthermore, we show that smooth stadium like billiards have elliptic periodic orbits for a large range of separation distances.
In the second part we consider hard sphere systems with a large number of particles, which we model by the Boltzmann equation. We develop a new approach to derive hydrodynamic limits, which is based on classical methods of geometric singular perturbation theory of ordinary differential equations. This method provides new geometric and dynamical interpretations of hydrodynamic limits, in particular, for the of the dissipative Boltzmann equation.
(Georgia Institute of Technology, 2008-11-11)
Yurchenko, Aleksey
The first part of this work deals with open dynamical systems. A natural question of how the survival probability
depends upon a position of a hole was seemingly never addresses in the theory of open dynamical systems. We found
that this dependency could be very essential. The main results are related to the holes with equal sizes
(measure) in the phase space of strongly chaotic maps. Take in each hole a periodic point of minimal period.
Then the faster escape occurs through the hole where this minimal period assumes its maximal value. The results
are valid for all finite times (starting with the minimal period), which is unusual in dynamical systems theory
where typically statements are asymptotic when time tends to infinity. It seems obvious that the bigger the hole
is the bigger is the escape through that hole. Our results demonstrate that generally it is not true, and that
specific features of the dynamics may play a role comparable to the size of the hole.
In the second part we consider some classes of cellular automata called Deterministic Walks in Random
Environments on Z^1. At first we deal with the system with constant rigidity and Markovian distribution
of scatterers on Z^1. It is shown that these systems have essentially the same properties as DWRE on
Z^1 with constant rigidity and independently distributed scatterers. Lastly, we consider a system with
non-constant rigidity (so called process of aging) and independent distribution of scatterers. Asymptotic laws
for the dynamics of perturbations propagating in such environments with aging are obtained.
(Georgia Institute of Technology, 2003-11-26)
Kreslavskiy, Dmitry Michael
The present work consists of three parts. In the first part (chapters III and IV), the dynamics of Lorentz lattice gases (LLG) on graphs is analyzed. We study the fixed scatterer model on finite graphs. A tight bound is established on the size of the orbit for arbitrary graphs, and the model is shown to perform a depth-first search on trees. Rigidity models on trees are also considered, and the size of the resulting orbit is established. In the second part (chapter V), we give a complete description of dynamics for LLG on the one-dimensional integer lattice, with a particular interest in showing that these models are not capable of universal computation. Some statistical properties of these models are also analyzed. In the third part (chapter VI) we attempt to partition a pool of workers into teams that will function as independent TSS lines. Such partitioning may be aimed to make sure that all groups work at approximately the same rate. Alternatively, we may seek to maximize the rate of convergence of the corresponding dynamical systems to their fixed points with optimal production at the fastest rate. The first problem is shown to be NP-hard. For the second problem, a solution for splitting into pairs is given, and it is also shown that this solution is not valid for partitioning into teams composed of more than two workers.