Kinetic theory of waves in random media and amelioration of classical chaos.

Persistent Link:
http://hdl.handle.net/10150/186895
Title:
Kinetic theory of waves in random media and amelioration of classical chaos.
Author:
Wolfson, Michael Aaron.
Issue Date:
1994
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 approach to the classical limit of wave mechanics is investigated where, in the classical limit, the dynamical system is nonintegrable and the motion in phase space is chaotic. The problem is cast in the setting of wave propagation in random media, and the fundamental starting point is an idealized stochastic parabolic wave equation (SPE) in two space dimensions with plane wave initial data. The potential is taken to have mean zero, strength ε ≪ 1 fluctuations which are homogeneous, isotropic, and have a single scale. The formal classical limit of the SPE, the parabolic ray equations are inherently non-integrable for any given realization of the potential. For the relative motion of two particles, an advection-diffusion Fokker-Planck equation is derived and shown for small initial separations to exhibit chaotic behavior, characterized by the existence of a positive Lyapunov exponent. It is shown that this physically relates to the exponential proliferation of caustics, or tendrils in phase space. A generalized wave kinetic equation (GWKE) is derived for the evolution in a relative phase space of a mean, two-particle Wigner function which corresponds classically to the advection-diffusion Fokker-Planck equation. The GWKE is analytically examined semi-classically by a novel boundary layer method (called the "extended quantum notch method") which enable the derivation of several important results: First, the "log time" (range) is obtained where semi-classical theory breaks down due to the saturation of caustics, then it is shown that this range is where the normalized intensity fluctuations (scintillation index) approaches unity. Finally, a wave (quantum) manifestation of classical chaos is seen to be the exponential decay of the scintillation index beyond its peak while on approach to saturation.
Type:
text; Dissertation-Reproduction (electronic)
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Applied Mathematics; Graduate College
Degree Grantor:
University of Arizona
Committee Chair:
Levermore, C. David; Tappert, Fred

Full metadata record

DC FieldValue Language
dc.language.isoenen_US
dc.titleKinetic theory of waves in random media and amelioration of classical chaos.en_US
dc.creatorWolfson, Michael Aaron.en_US
dc.contributor.authorWolfson, Michael Aaron.en_US
dc.date.issued1994en_US
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 approach to the classical limit of wave mechanics is investigated where, in the classical limit, the dynamical system is nonintegrable and the motion in phase space is chaotic. The problem is cast in the setting of wave propagation in random media, and the fundamental starting point is an idealized stochastic parabolic wave equation (SPE) in two space dimensions with plane wave initial data. The potential is taken to have mean zero, strength ε ≪ 1 fluctuations which are homogeneous, isotropic, and have a single scale. The formal classical limit of the SPE, the parabolic ray equations are inherently non-integrable for any given realization of the potential. For the relative motion of two particles, an advection-diffusion Fokker-Planck equation is derived and shown for small initial separations to exhibit chaotic behavior, characterized by the existence of a positive Lyapunov exponent. It is shown that this physically relates to the exponential proliferation of caustics, or tendrils in phase space. A generalized wave kinetic equation (GWKE) is derived for the evolution in a relative phase space of a mean, two-particle Wigner function which corresponds classically to the advection-diffusion Fokker-Planck equation. The GWKE is analytically examined semi-classically by a novel boundary layer method (called the "extended quantum notch method") which enable the derivation of several important results: First, the "log time" (range) is obtained where semi-classical theory breaks down due to the saturation of caustics, then it is shown that this range is where the normalized intensity fluctuations (scintillation index) approaches unity. Finally, a wave (quantum) manifestation of classical chaos is seen to be the exponential decay of the scintillation index beyond its peak while on approach to saturation.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineApplied Mathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.chairLevermore, C. Daviden_US
dc.contributor.chairTappert, Freden_US
dc.contributor.committeememberBayly, Bruceen_US
dc.identifier.proquest9517511en_US
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