Transport Theoretical Studies of Some Microscopic and Macroscopic Systems

This dissertation is a report on theoretical transport studies of two systems of vastly different sizes. The first topic is electronic motion in quantum wires. In recent years, it has become possible to fabricate wires that are so small that quantum effects become important. The conduction properties of these wires are quite different than those of macroscopic wires. In this dissertation, we seek to understand scattering effects in quantum wires in a simple way. Some of the existing formalisms for studying transport in quantum wires are reviewed, and one such formalism is applied to calculate conductance in some simple systems. The second topic concerns animals which move in groups, such as flocking birds or schooling fish. Exact analytic calculations of the transport properties of such systems are very difficult because a flock is a system that is far from equilibrium and consists of many interacting particles. We introduce two simplified models of flocking which are amenable to analytic study. The first model consists of a set of overdamped Brownian particles that interact via spring forces. The exact solution for the probability distribution is calculated, and equations of motion for continuous coarse-grained quantities, such as the density, are obtained from the full solution. The second model consists of particles which move in one dimension at constant speed, but which change their directions at random. The flipping rates are constructed in such a way that particles tend to align their directions with each other. The model is solved exactly for one and two particles, the first two moments are obtained, and equations of motion for continuous coarse-grained quantities are written. The model cannot be solved exactly for many particles, but the first and second moments are calculated. Finally, two additional topics are briefly discussed. The first is transport in disordered lattices, and the second is a static magnetic model of flocking.