Dynamical Properties of a Generalized Collision Rule for Multi-Particle Systems

Persistent Link:
http://hdl.handle.net/10150/315858
Title:
Dynamical Properties of a Generalized Collision Rule for Multi-Particle Systems
Author:
Dinius, Joseph
Issue Date:
2014
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
The theoretical basis for the Lyapunov exponents of continuous- and discrete-time dynamical systems is developed, with the inclusion of the statement and proof of the Multiplicative Ergodic Theorem of Oseledec. The numerical challenges and algorithms to approximate Lyapunov exponents and vectors are described, with multiple illustrative examples. A novel generalized impulsive collision rule is derived for particle systems interacting pairwise. This collision rule is constructed to address the question of whether or not the quantitative measures of chaos (e.g. Lyapunov exponents and Kolmogorov-Sinai entropy) can be reduced in these systems. Major results from previous studies of hard-disk systems, which interact via elastic collisions, are summarized and used as a framework for the study of the generalized collision rule. Numerical comparisons between the elastic and new generalized rules reveal many qualitatively different features between the two rules. Chaos reduction in the new rule through appropriate parameter choice is demonstrated, but not without affecting the structural properties of the Lyapunov spectra (e.g. symmetry and conjugate-pairing) and of the tangent space decomposition (e.g. hyperbolicity and domination of the Oseledec splitting). A novel measure of the degree of domination of the Oseledec splitting is developed for assessing the impact of fluctuations in the local Lyapunov exponents on the observation of coherent structures in perturbation vectors corresponding to slowly growing (or contracting) modes. The qualitatively different features observed between the dynamics of generalized and elastic collisions are discussed in the context of numerical simulations. Source code and complete descriptions for the simulation models used are provided.
Type:
text; Electronic Dissertation
Keywords:
Dynamical systems; Lyapunov exponents; Statistical mechanics; Applied Mathematics; chaos
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Applied Mathematics
Degree Grantor:
University of Arizona
Advisor:
Lega, Joceline

Full metadata record

DC FieldValue Language
dc.language.isoen_USen
dc.titleDynamical Properties of a Generalized Collision Rule for Multi-Particle Systemsen_US
dc.creatorDinius, Josephen_US
dc.contributor.authorDinius, Josephen_US
dc.date.issued2014en
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractThe theoretical basis for the Lyapunov exponents of continuous- and discrete-time dynamical systems is developed, with the inclusion of the statement and proof of the Multiplicative Ergodic Theorem of Oseledec. The numerical challenges and algorithms to approximate Lyapunov exponents and vectors are described, with multiple illustrative examples. A novel generalized impulsive collision rule is derived for particle systems interacting pairwise. This collision rule is constructed to address the question of whether or not the quantitative measures of chaos (e.g. Lyapunov exponents and Kolmogorov-Sinai entropy) can be reduced in these systems. Major results from previous studies of hard-disk systems, which interact via elastic collisions, are summarized and used as a framework for the study of the generalized collision rule. Numerical comparisons between the elastic and new generalized rules reveal many qualitatively different features between the two rules. Chaos reduction in the new rule through appropriate parameter choice is demonstrated, but not without affecting the structural properties of the Lyapunov spectra (e.g. symmetry and conjugate-pairing) and of the tangent space decomposition (e.g. hyperbolicity and domination of the Oseledec splitting). A novel measure of the degree of domination of the Oseledec splitting is developed for assessing the impact of fluctuations in the local Lyapunov exponents on the observation of coherent structures in perturbation vectors corresponding to slowly growing (or contracting) modes. The qualitatively different features observed between the dynamics of generalized and elastic collisions are discussed in the context of numerical simulations. Source code and complete descriptions for the simulation models used are provided.en_US
dc.typetexten
dc.typeElectronic Dissertationen
dc.subjectDynamical systemsen_US
dc.subjectLyapunov exponentsen_US
dc.subjectStatistical mechanicsen_US
dc.subjectApplied Mathematicsen_US
dc.subjectchaosen_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.disciplineApplied Mathematicsen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorLega, Jocelineen_US
dc.contributor.committeememberLega, Jocelineen_US
dc.contributor.committeememberFlaschka, Hermannen_US
dc.contributor.committeememberSanfelice, Ricardoen_US
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