| In this dissertation, I present my work on the statistical study of highly correlated systems in three fields of science: ecology, microbial ecology and physics.; I propose an explanation for how the highly correlated distribution of species individuals, and an abundance distribution commonly observed in ecological systems, give rise to a power law dependence between a given area and the number of unique species it harbors. This is one of the oldest known ecological patterns: the power-law Species Area Rule.; As a natural extension of my studies in ecology, I have undertaken both theoretical research and field work in the developing field of microbial ecology. In particular, I participated in a multidisciplinary study of the impact of microbes on the formation of macroscopic calcium carbonate terraces at Yellowstone National Park Hot Springs. I have used ecological techniques to characterize the biodiversity of our study site and developed a new bootstrap method for extracting abundance information from clone libraries. This has singled out the most abundant microorganisms and paved the way for future studies of the possible non-passive role of microorganisms in carbonate precipitation.; The third part of my thesis uses statistical techniques to explore the correlations in rotating Bose-Einstein condensates. I have used finite difference techniques to solve the Gross-Pitaevskii equation in order to obtain the structure of a vortex in a lattice. Surprisingly, I have found that, in order to understand this structure, it is necessary to add a correction to the Gross-Pitaevskii equation which introduces a dependence on the particle scattering length.; I have also used Path Integral Monte Carlo techniques to explore the limit of rapid rotations, where the Gross-Pitaevskii equation is no longer valid. Interestingly, the Gross-Pitaevskii equation seems to be valid for much higher densities than expected if properly renormalized. I show that, in accord with the prediction of the Gross-Pitaevskii equations, a scale invariant vortex lattice state is found at high rotations. |
