Quantitative Derivation of Effective Evolution Equations for the Dynamics of Bose-Einstein Condensates

This thesis proves certain results concerning an important question in non-equilibrium quantum statistical mechanics which is the derivation of effective evolution equations approximating the dynamics of a system of large number of bosons initially at equilibrium (ground state at very low temperatures). The dynamics of such systems are governed by the time-dependent linear many-body Schroedinger equation from which it is typically difficult to extract useful information due to the number of particles being large. We will study quantitatively (i.e. with explicit bounds on the error) how a suitable one particle non-linear Schroedinger equation arises in the mean field limit as number of particles N → ∞ and how the appropriate corrections to the mean field will provide better approximations of the exact dynamics. In the first part of this thesis we consider the evolution of N bosons, where N is large, with two-body interactions of the form N³ᵝv(Nᵝ⋅), 0≤β≤1. The parameter β measures the strength and the range of interactions. We compare the exact evolution with an approximation which considers the evolution of a mean field coupled with an appropriate description of pair excitations, see [18,19] by Grillakis-Machedon-Margetis. We extend the results for 0 ≤ β < 1/3 in [19, 20] to the case of β < 1/2 and obtain an error bound of the form p(t)/Nᵅ, where α>0 and p(t) is a polynomial, which implies a specific rate of convergence as N → ∞. In the second part, utilizing estimates of the type discussed in the first part, we compare the exact evolution with the mean field approximation in the sense of marginals. We prove that the exact evolution is close to the approximate in trace norm for times of the order o(1)√N compared to log(o(1)N) as obtained in Chen-Lee-Schlein [6] for the Hartree evolution. Estimates of similar type are obtained for stronger interactions as well.

Angler Perception and Population Dynamics of the Northern Snakehead (Channa argus) in the Potomac River & Tributaries

Our research sought to address the extent to which the northern snakehead (Channa argus), an invasive fish species, represents a threat to the Potomac River ecosystem. The first goal of our research was to survey the perceptions and opinions of recreational anglers on the effects of the snakehead population in the Potomac River ecosystem. To determine angler perceptions, we created and administered 113 surveys from June – September 2014 at recreational boat ramps along the Potomac River. Our surveys were designed to expand information collected during previous surveys conducted by the U.S. Fish and Wildlife Service. Our results indicated recreational anglers perceive that abundances and catch rates of target species, specifically largemouth bass, have declined since snakehead became established in the river. The second goal of our research was to determine the genetic diversity and potential of the snakehead population to expand in the Potomac River. We hypothesized that the effective genetic population size would be much less than the census size of the snakehead population in the Potomac River. We collected tissue samples (fin clippings) from 79 snakehead collected in a recreational tournament held between Fort Washington and Wilson’s Landing, MD on the Potomac River and from electrofishing sampling conducted by the Maryland Department of Natural Resources in Pomonkey Creek, a tributary of the Potomac River. DNA was extracted from the tissue samples and scored for 12 microsatellite markers, which had previously been identified for Potomac River snakehead. Microsatellite allele frequency data were recorded and analyzed in the software programs GenAlEx and NeEstimator to estimate heterozygosity and effective genetic population size. Resampling simulations indicated that the number of microsatellites and the number of fish analyzed provided sufficient precision. Simulations indicated that the effective population size estimate would expect to stabilize for samples > 70 individual snakehead. Based on a sample of 79 fish scored for 12 microsatellites, we calculated an Ne of 15.3 individuals. This is substantially smaller than both the sample size and estimated population size. We conclude that genetic diversity in the snakehead population in the Potomac River is low because the population has yet to recover from a genetic bottleneck associated with a founder effect due to their recent introduction into the system.

Spatial and Temporal Dynamics of Larval Atlantic Menhaden on the East Coast of the United States

Atlantic Menhaden Brevoortia tyrannus is a commercially and ecologically important forage fish abundant on the Atlantic Coast of the United States. We conducted spatial and temporal analyses of larval Atlantic Menhaden using data collected from two large-scale ichthyoplankton programs during 1977-1987 and 1999-2013 to construct indices of larval abundance and survival over time, evaluate how environmental factors affect early life survival, and examine how larvae are distributed in space to gain knowledge on spawning and larval dispersal. Over time, we found larval abundance to increase, while early life survival declined. Coastal temperature, wind speed, and Atlantic Multidecadal Oscillation were found to potentially explain some of this decline in survival. Over both periods, we found evidence spawning predominantly occurs near shore, from New York to North Carolina, increasing in intensity southwards. While the general spatial patterns were consistent, we observed some localized variation and overall expansion of occupied area by larvae.

Nonlinear Optics and Carrier Dynamics in Nanostructured and Two-Dimensional Materials

Understanding and measuring the interaction of light with sub-wavelength structures and atomically thin materials is of critical importance for the development of next generation photonic devices.  One approach to achieve the desired optical properties in a material is to manipulate its mesoscopic structure or its composition in order to affect the properties of the light-matter interaction.  There has been tremendous recent interest in so called two-dimensional materials, consisting of only a single to a few layers of atoms arranged in a planar sheet.  These materials have demonstrated great promise as a platform for studying unique phenomena arising from the low-dimensionality of the material and for developing new types of devices based on these effects.  A thorough investigation of the optical and electronic properties of these new materials is essential to realizing their potential.  In this work we present studies that explore the nonlinear optical properties and carrier dynamics in nanoporous silicon waveguides, two-dimensional graphite (graphene), and atomically thin black phosphorus. We first present an investigation of the nonlinear response of nanoporous silicon optical waveguides using a novel pump-probe method. A two-frequency heterodyne technique is developed in order to measure the pump-induced transient change in phase and intensity in a single measurement. The experimental data reveal a characteristic material response time and temporally resolved intensity and phase behavior matching a physical model dominated by free-carrier effects that are significantly stronger and faster than those observed in traditional silicon-based waveguides.  These results shed light on the large optical nonlinearity observed in nanoporous silicon and demonstrate a new measurement technique for heterodyne pump-probe spectroscopy. Next we explore the optical properties of low-doped graphene in the terahertz spectral regime, where both intraband and interband effects play a significant role. Probing the graphene at intermediate photon energies enables the investigation of the nonlinear optical properties in the graphene as its electron system is heated by the intense pump pulse. By simultaneously measuring the reflected and transmitted terahertz light, a precise determination of the pump-induced change in absorption can be made. We observe that as the intensity of the terahertz radiation is increased, the optical properties of the graphene change from interband, semiconductor-like absorption, to a more metallic behavior with increased intraband processes. This transition reveals itself in our measurements as an increase in the terahertz transmission through the graphene at low fluence, followed by a decrease in transmission and the onset of a large, photo-induced reflection as fluence is increased.  A hybrid optical-thermodynamic model successfully describes our observations and predicts this transition will persist across mid- and far-infrared frequencies.  This study further demonstrates the important role that reflection plays since the absorption saturation intensity (an important figure of merit for graphene-based saturable absorbers) can be underestimated if only the transmitted light is considered. These findings are expected to contribute to the development of new optoelectronic devices designed to operate in the mid- and far-infrared frequency range.  Lastly we discuss recent work with black phosphorus, a two-dimensional material that has recently attracted interest due to its high mobility and direct, configurable band gap (300 meV to 2eV), depending on the number of atomic layers comprising the sample. In this work we examine the pump-induced change in optical transmission of mechanically exfoliated black phosphorus flakes using a two-color optical pump-probe measurement. The time-resolved data reveal a fast pump-induced transparency accompanied by a slower absorption that we attribute to Pauli blocking and free-carrier absorption, respectively. Polarization studies show that these effects are also highly anisotropic - underscoring the importance of crystal orientation in the design of optical devices based on this material. We conclude our discussion of black phosphorus with a study that employs this material as the active element in a photoconductive detector capable of gigahertz class detection at room temperature for mid-infrared frequencies.

Essays on Uncertainty, Imperfect Information, and Investment Dynamics

Understanding how imperfect information affects firms' investment decision helps answer important questions in economics, such as how we may better measure economic uncertainty; how firms' forecasts would affect their decision-making when their beliefs are not backed by economic fundamentals; and how important are the business cycle impacts of changes in firms' productivity uncertainty in an environment of incomplete information. This dissertation provides a synthetic answer to all these questions, both empirically and theoretically. The first chapter, provides empirical evidence to demonstrate that survey-based forecast dispersion identifies a distinctive type of second moment shocks different from the canonical volatility shocks to productivity, i.e. uncertainty shocks. Such forecast disagreement disturbances can affect the distribution of firm-level beliefs regardless of whether or not belief changes are backed by changes in economic fundamentals. At the aggregate level, innovations that increase the dispersion of firms' forecasts lead to persistent declines in aggregate investment and output, which are followed by a slow recovery. On the contrary, the larger dispersion of future firm-specific productivity innovations, the standard way to measure economic uncertainty, delivers the ``wait and see" effect, such that aggregate investment experiences a sharp decline, followed by a quick rebound, and then overshoots. At the firm level, data uncovers that more productive firms increase investments given rises in productivity dispersion for the future, whereas investments drop when firms disagree more about the well-being of their future business conditions. These findings challenge the view that the dispersion of the firms' heterogeneous beliefs captures the concept of economic uncertainty, defined by a model of uncertainty shocks. The second chapter presents a general equilibrium model of heterogeneous firms subject to the real productivity uncertainty shocks and informational disagreement shocks. As firms cannot perfectly disentangle aggregate from idiosyncratic productivity because of imperfect information, information quality thus drives the wedge of difference between the unobserved productivity fundamentals, and the firms' beliefs about how productive they are. Distribution of the firms' beliefs is no longer perfectly aligned with the distribution of firm-level productivity across firms. This model not only explains why, at the macro and micro level, disagreement shocks are different from uncertainty shocks, as documented in Chapter 1, but helps reconcile a key challenge faced by the standard framework to study economic uncertainty: a trade-off between sizable business cycle effects due to changes in uncertainty, and the right amount of pro-cyclicality of firm-level investment rate dispersion, as measured by its correlation with the output cycles.

POSITIVE FILTERED P_N METHOD FOR LINEAR TRANSPORT EQUATIONS AND THE ASSOCIATED OPTIMIZATION ALGORITHM

We propose a positive, accurate moment closure for linear kinetic transport equations based on a filtered spherical harmonic (FP_N) expansion in the angular variable. The FP_N moment equations are accurate approximations to linear kinetic equations, but they are known to suffer from the occurrence of unphysical, negative particle concentrations. The new positive filtered P_N (FP_N+) closure is developed to address this issue. The FP_N+ closure approximates the kinetic distribution by a spherical harmonic expansion that is non-negative on a finite, predetermined set of quadrature points. With an appropriate numerical PDE solver, the FP_N+ closure generates particle concentrations that are guaranteed to be non-negative. Under an additional, mild regularity assumption, we prove that as the moment order tends to infinity, the FP_N+ approximation converges, in the L2 sense, at the same rate as the FP_N approximation; numerical tests suggest that this assumption may not be necessary. By numerical experiments on the challenging line source benchmark problem, we confirm that the FP_N+ method indeed produces accurate and non-negative solutions. To apply the FP_N+ closure on problems at large temporal-spatial scales, we develop a positive asymptotic preserving (AP) numerical PDE solver. We prove that the propose AP scheme maintains stability and accuracy with standard mesh sizes at large temporal-spatial scales, while, for generic numerical schemes, excessive refinements on temporal-spatial meshes are required. We also show that the proposed scheme preserves positivity of the particle concentration, under some time step restriction. Numerical results confirm that the proposed AP scheme is capable for solving linear transport equations at large temporal-spatial scales, for which a generic scheme could fail. Constrained optimization problems are involved in the formulation of the FP_N+ closure to enforce non-negativity of the FP_N+ approximation on the set of quadrature points. These optimization problems can be written as strictly convex quadratic programs (CQPs) with a large number of inequality constraints. To efficiently solve the CQPs, we propose a constraint-reduced variant of a Mehrotra-predictor-corrector algorithm, with a novel constraint selection rule. We prove that, under appropriate assumptions, the proposed optimization algorithm converges globally to the solution at a locally q-quadratic rate. We test the algorithm on randomly generated problems, and the numerical results indicate that the combination of the proposed algorithm and the constraint selection rule outperforms other compared constraint-reduced algorithms, especially for problems with many more inequality constraints than variables.

The Impact of Motor Learning on Motor Behavior and Cortical Dynamics in a Complex Stressful Social Environment

An economy of effort is a core characteristic of highly skilled motor performance often described as being effortless or automatic. Electroencephalographic (EEG) evaluation of cortical activity in elite performers has consistently revealed a reduction in extraneous associative cortical activity and an enhancement of task-relevant cortical processes. However, this has only been demonstrated under what are essentially practice-like conditions. Recently it has been shown that cerebral cortical activity becomes less efficient when performance occurs in a stressful, complex social environment. This dissertation examines the impact of motor skill training or practice on the EEG cortical dynamics that underlie performance in a stressful, complex social environment. Sixteen ROTC cadets participated in head-to-head pistol shooting competitions before and after completing nine sessions of skill training over three weeks. Spectral power increased in the theta frequency band and decreased in the low alpha frequency band after skill training. EEG Coherence increased in the left frontal region and decreased in the left temporal region after the practice intervention. These suggest a refinement of cerebral cortical dynamics with a reduction of task extraneous processing in the left frontal region and an enhancement of task related processing in the left temporal region consistent with the skill level reached by participants. Partitioning performance into ‘best’ and ‘worst’ based on shot score revealed that deliberate practice appears to optimize cerebral cortical activity of ‘best’ performances which are accompanied by a reduction in task-specific processes reflected by increased high-alpha power, while ‘worst’ performances are characterized by an inappropriate reduction in task-specific processing resulting in a loss of focus reflected by higher high-alpha power after training when compared to ‘best’ performances. Together, these studies demonstrate the power of experience afforded by practice, as a controllable factor, to promote resilience of cerebral cortical efficiency in complex environments.

PHYTOPLANKTON AND NUTRIENT DYNAMICS WITH A FOCUS ON NITROGEN FORM IN THE ANACOSTIA RIVER, IN WASHINGTON, D.C. AND WEST LAKE, IN HANGZHOU, CHINA

Nutrient loading has been linked with severe water quality impairment, ranging from hypoxia to increased frequency of harmful algal blooms (HABs), loss of fisheries, and changes in biodiversity. Waters around the globe are experiencing deleterious effects of eutrophication; however, the relative amount of nitrogen (N) and phosphorus (P) reaching these waters is not changing proportionately, with high N loads increasingly enriched in chemically-reduced N forms. Research involving two urban freshwater and nutrient enriched systems, the Anacostia River, USA, a tributary of the Potomac River feeding into the Chesapeake Bay, and West Lake, Hangzhou, Zhejiang Province, China, was conducted to assess the response of phytoplankton communities to changing N-form and N/P-ratios. Field observations involving the characterization of ambient phytoplankton communities and N-forms, as well as experimental (nutrient enrichment) manipulations were used to understand shifts in phytoplankton community composition with increasing NH4+ loads. In both locations, a >2-fold increase in ambient NH4+:NO3- ratios was followed by a shift in the phytoplankton community, with diatoms giving way to chlorophytes and cyanobacteria. Enrichment experiments mirrored this, in that samples enriched with NH4+ lead to increased abundance of chlorophytes and cyanobacteria. This work shows that in both of these systems experiencing nutrient enrichment that NH4+ supports communities dominated by more chlorophytes and cyanobacteria than other phytoplankton groups.