This is a two-part thesis concerning the motion of a testparticle in a bath. In part one we use an expansion of the operatorPLeit(1-P)LLP to shape the Zwanzig equation into a generalizedFokker-Planck equation which involves a diffusion tensor dependingon the test particle's momentum and the time.
In part two the resultant equation is studied in some detailfor the case of test particle motion in a weakly coupled Lorentz Gas.The diffusion tensor for this system is considered. Some of itsproperties are calculated; it is computed explicitly for the caseof a Gaussian potential of interaction.
The equation for the test particle distribution function canbe put into the form of an inhomogeneous Schroedinger equation.The term corresponding to the potential energy in the Schroedingerequation is considered. Its structure is studied, and some of itssimplest features are used to find the Green's function in thelimiting situations of low density and long time.
