Linear and nonlinear stochastic differential equations with applications

Novel analytical nonperturbative techniques are developed in the area of nonlinear and linear stochastic differential equations and applications are considered to a variety of physical problems. First, a method is introduced for deriving first- and second-order moment equations for a general class of stochastic nonlinear equations by performing a renormalization at the level of the second moment. These general results, when specialized to the weak-coupling limit, lead to a complete set of closed equations for the first two moments within the framework of an approximation corresponding to the direct-interaction approximation. Additional restrictions result in a self-consistent set of equations for the first two moments in the stochastic quasi-linear approximation. The technique is illustrated by considering two specific nonlinear physical random problems: model hydrodynamic and Vlasov-plasma turbulence. The equations for the phenomenon of hydrodynamic turbulence are examined in more detail at the level of the quasi-linear approximation, which is valid for small turbulence Reynolds numbers. Closed form solutions are found for the equations governing the random fluctuations of the velocity field under the assumption of special time-dependent, uniform or sheared, mean flow profiles. Constant, transient and oscillatory flows are considered. The smoothing approximation for solving linear stochastic differential equations is applied to several specific physical problems. The problem of a randomly perturbed quantum mechanical harmonic oscillator is investigated first using the wave kinetic technique. The equations for the ensemble average of the Wigner distribution function are defined within the framework of the smoothing approximation. Special attention is paid to the so-called long-time Markovian approximation, where the discrete nature of the quantum mechanical oscillator is explicitly visible. For special statistics of the random perturbative potential, the dependence of physical observables on time is examined in detail. As a last example of the application of the stochastic techniques, the diffusion of a scalar quantity in the presence of a turbulent fluid is investigated. An equation corresponding to the smoothing approximation is obtained, and its asymptotic long-time version is examined for the cases of zero-mean flow and linearly sheared mean flow.

Name thesis:degree_name

Ph. D.

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Engineering Science and Mechanics

Department dc:contributor.department

Engineering Science and Mechanics

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

1977

Author dc:creator

• Stasiak, Wojciech Boguslaw

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language.iso

en

Dc Identifier Other

etd-07092010-020011

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/38739