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We explore a path integral approach to Darcy flow through a stochastic permeable medium. In one dimension, Darcy's law can be solved exactly. We give a derivation of the path integral used to obtain the Darcy pressure statistics. We also outline the computational setup for the conventional finite-volume method and the implementation of a stochastic field generator. We provide a detailed user's guide to the calculation of path integrals on a lattice, including an explicit computational setup and corresponding pseudocode. The higher-dimensional form of Darcy's law lacks an analytic solution. We show that the simulated annealing algorithm provides a viable alternative to simulating a path integral for Darcy's law. We compare the results for the path integral and simulated annealing methods to those for the finite-volume method. All comparisons pass a Kolmogorov-Smirnov test at the 95% confidence level. We discuss log-normal and Gaussian fits to the pressure statistics. Finally, we make a number of suggestions for future work, such as the use of the renormalization group and the extension of Darcy's law to multiphase flow.

In this thesis, we propose a simplified approach to the lattice dynamics of crystals by operating within the framework of local normal modes, i.e, the normal modes of the isolated unit cell. We do this by transforming the Cartesian coordinates into the normal modes of the unit cell by using group theoretic techniques. The dynamical matrix obtained with this method is different to the one obtained through standard methods, but has the same eigenvalues and eigenvectors. Using perturbation theory, we treat the off-diagonal elements as perturbations and proceed in calculating the phonon dispersions. In doing so, we obtain approximate analytical expressions and accurate numerical solutions to phonon dispersion without explicitly solving the eigenvalue problem. This method provides insight into the symmetry analysis of the phonon modes, and opens a pathway for studying phase transitions in molecular solids and Jahn-Teller systems. A Jahn-Teller molecule is one which is not stable and distorts to a lower-symmetry geometry due to the electronic degeneracy of the highest occupied molecular orbital. If several such Jahn-Teller molecules interact with each other, they can undergo a cooperative phase transition. We use our results of the local normal mode analysis to develop a model of cooperative Jahn-Teller phase transitions. We then apply our model to a system of interacting icosahedral Jahn-Teller clusters. Since atoms tend to form clusters of icosahedral symmetry in the amorphous and liquid phases, our results serve as a paradigm for liquid-solid and glass transitions. This is a new approach which has been taken in establishing a relationship between lattice dynamics, cooperative Jahn-Teller effect and certain types of phase transitions.