Kinetic theories for stochastic models of liquids with highly cooperative dynamics

Liquids are often characterized by the highly correlated motion of constituent particles. Understanding the dynamics of particle motion can give deep insight into both microscopic and macroscopic properties of liquids, including the diffusive properties of labeled particles and macroscopic transport coefficients. In many dense fluids, correlated particle motion develops primarily as a result of caging phenomena, in which the motion of each particle is severely constrained by the presence of other nearby particles. Characterizing the dynamics of caging is of paramount importance in understanding the properties of many liquids, including atomic, polymeric, and colloidal fluids.;In this thesis, to gain insight into caging phenomena in real liquids, we theoretically investigate two stochastic lattice models of liquids in which caging effects are important: a one-dimensional, single occupancy lattice gas and a lattice model of rigid rod polymers. Particles in both models undergo random motion on a lattice subject to the constraint that they cannot pass through one another. To characterize particle dynamics in these models, we develop a diagrammatic kinetic theory to study time correlation functions of density fluctuations. Correlation functions are represented as infinite sums of diagrams, and various approximations are constructed by evaluating subsets of the diagrams with certain physically or mathematically motivated features. The primary function of interest, the trace correlation function, is the time-dependent conditional probability distribution of the position of a labeled particle given its position at an earlier time.;For each model, we derive diagrammatic series for the memory function and irreducible memory function. After showing that mean field approximations fail to provide an adequate description of the correlated particle dynamics, we develop approximations to the irreducible memory function motivated by the mode coupling theory of supercooled liquids. Mode coupling theory is commonly regarded as a theory of caging, and testing the efficacy of mode coupling-like theories on models in which caging effects are important is a major goal of this work. Using both analytical and numerical methods, we solve the mode coupling-like equations self-consistently for the trace correlation function. Comparing the results with simulation data demonstrates that the theories are successful at describing, both qualitatively and quantitatively, many features of the models. A physical interpretation of the diagrams contributing to the theory provides additional insight into the dynamics of caging and may suggest how to develop theories of caging in more complicated and realistic models of liquids.