Mode Coupling in Statistical Mechanics.

In this thesis, a general mode coupling formalism is developed and applied to simple and granular flow hydrodynamic fluid systems. The mode coupling theory is then applied to several different problems in equilibrium and nonequilibrium statistical mechanics. First we examine N particle density equilibrium correlations and obtain a series of equations which allows the generalized transport coefficients to be obtained exactly in the thermodynamic limit up to arbitrary order in the wavevector, frequency and mode coupling parameters. We then apply the mode coupling formalism to equilibrium tagged particle correlation functions in simple fluids. The mode coupling formalism and N ordering approximation scheme allow a series for the generalized diffusion constant to be obtained which is exact in the thermodynamic limit. Finally, we apply the mode coupling formalism to nonequilibrium inelastic granular flow systems and derive nonlinear equations of motion for equal time multilinear densities as well as linear density correlations by applying projection operator techniques. An effective time-displacement operator for the granular distribution function is derived by exploiting the fact that each granular particle has many interacting internal degrees of freedom which remain at equilibrium and provide a sink for the translational relative momenta of the inelastic granular system. The effective time-displacement operator is then used to derive the nonlinear hydrodynamic equations and an equation which permits the equal time multilinear correlation functions to be obtained for the granular system. The solutions to the linearized equations are also analyzed in different regimes comparing the additional terms due to the inelasticity of collisions with the magnitude of the gradients of the system. Using these results, we examine equal time bilinear hydrodynamic density correlation functions in a steady state system characterized by a small, linear shear flow and calculate these quantities to leading order in the wavevectors, mode coupling parameter and magnitude of the shear flow. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617 -253-5668; Fax 617-253-1690.) (Abstract shortened by UMI.).