#### Date of Award

1995

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Physics and Astronomy

#### First Advisor

Robert F. O'Connell

#### Abstract

In Chapter I of this dissertation, we present a pedagogical introduction to the basic concepts of stochastic theory and review the progress made as well as the outstanding unsolved problems in the field. For the rest of this dissertation, we use a generalized quantum Langevin equation (GLE) approach to investigate various properties of quantum dissipative systems. In Chapter II, calculations for the displacement and random force correlation functions for Brownian motion are generalized to the case of an arbitrary heat bath for a damped harmonic quantum oscillator. The mean square displacement of such an oscillator is then evaluated for both Ohmic and blackbody radiation heat baths, to determine the effects of many parameters on the localization of the oscillator. In Chapter III, the formalism is extended to the Brownian motion of a charged particle in an external magnetic field as well as in a potential. The influence of the magnetic field on the memory function and random force is determined, with the blackbody radiation heat bath analyzed as a special case. For a charged harmonic oscillator, the generalized susceptibility is obtained, which enables us to derive the symmetrized position correlation functions using the fluctuation-dissipation theorem. In addition, we obtain the free energy of the system by generalizing the "remarkable formula" of Ford, Lewis, and O'Connell. Explicit calculations are performed for Ohmic and blackbody radiation heat baths. Furthermore, the effect of dissipation on the localization of the oscillator in an Ohmic heat bath at zero temperature is shown to differ qualitatively from that without the magnetic field. Finally, we formulate retarded Green's functions and symmetrized position correlation functions for the oscillator, reach some general conclusions, and make explicit calculations for the Ohmic heat bath. For the special case of Brownian motion at both zero and nonzero temperatures, we prove two general asymptotic relations between the retarded Green's functions and the displacement correlation functions, which we use to evaluate the long-time behaviors of the latter from those of the former, for both the Ohmic heat bath and a rather general class of heat baths discussed extensively in the literature.

#### Recommended Citation

Li, Xiao Liang, "Investigation in the Theory of Stochastic Processes." (1995). *LSU Historical Dissertations and Theses*. 5967.

https://digitalcommons.lsu.edu/gradschool_disstheses/5967

#### Pages

145

#### DOI

10.31390/gradschool_disstheses.5967