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Abstract

The standard technique for the solution of kinetic theory and transport problems usually involves the expansion of the velocity distribution function (VDF) in some suitable set of basis functions. For steady problems, the Boltzmann equation (BE) for the distri-bution function is then reduced to algebraic equations for the expansion coefficients. For time dependent problems, the B E is reduced to ordinary differential equations. The main objective with this solution technique is to calculate the transport coefficients which can be expressed in terms of the lower order expansion coefficients. Recent experiments on different physical systems are capable of probing the details of the velocity distribution functions. The main objective of this thesis is to develop accurate and efficient numerical methods for the BE with the aim to calculate the distribution function. For an ensemble of ions dilutely dispersed in a background of neutrals under the influence of an applied electric field, the exact solution of the Bhatnagar, Gross, and Krook (BGK) collision model is used to study different numerical solutions of the BE. In this study, the main theme of the thesis is presented. In place of the traditional expansion techniques, a discretization of the VDF is proposed. The discretization is based upon a set of polynomial basis functions that yield the optimum convergence of the solution. For electrons dilutely dispersed in a background of neutrals, the BE is approximated by a Fokker-Planck equation (FPE). The one-dimensional F P E has been used to describe many different phenomena in chemistry, physics and engineering. A detailed comparison of the eigenvalues and eigenfunctions of the FP operator is carried out with different numerical methods including a finite difference (FD) scheme, Lagrange interpolation (LI) method, the Quadrature Discretization Method (QDM) with speed points and the QDM with Davydov points. The QDM involves the discretization of the distribution at a set of points that coincide with the points in a quadrature that is based upon some weight function. The Davydov distribution is the steady distribution for electrons in a gas moderator at some given electric field strength. The time dependent solutions of the FP are obtained with the different energy discretizations applied in conjunction with an appropriate time stepping procedure. The QDM with Davydov points provides a more rapid convergence of the eigenvalues and eigenfunctions relative to the other methods used. The relaxation of a nonequilibrium distribution of electrons in a mixture of CCI4 with either Ar or Ne is studied. The electron collisions in this analysis include elastic collisions between electrons and CCI4 and the inert gas, vibrationally inelastic collisions between electrons and CCI4, as well as the electron attachment reaction with CCI4. The time dependent electron energy distribution function is determined from the BE and the energy relaxation times are determined. The coupling of the thermalization process and the attachment process is discussed in detail. The results from the calculations are compared with experimental studies, and the methodology of the experimental reduction of the data is examined in detail. In addition, the nature of the spectrum of the Boltzmann collision operator for realistic ion-atom interactions is studied with a discretization of the integral collision operator. The singular nature of the collision operator is considered in detail with a novel interpolation technique. The eigenvalues and eigenfunctions for the Henyey-Greenstein (HG) model differential cross section are calculated. The relaxation of anisotropic ion distri-butions with and without an applied electric field is studied.