We study the fluctuation effects in the seminal cyclic predator-prey model in population dynamics due to Robert May and Warren Leonard both in the zero-dimensional and two-dimensional spatial version. We compute the mean time to extinction of a stable set of coexisting populations driven by large fluctuations. We see that the contribution of large fluctuations to extinction can be captured by a quasi-stationary approximation and the Wentzel–Kramers–Brillouin (WKB) eikonal ansatz. We see that near the Hopf bifurcation, extinctions are fast owing to the flat non-Gaussian distribution whereas away from the bifurcation, extinctions are dominated by large fluctuations of the fat tails of the distribution. We compare our results to Gillespie simulations and a single-species theoretical calculation. In addition, we study the spatio-temporal pattern formation of the stochastic May--Leonard model through the Doi-Peliti coherent state path integral formalism to obtain a coarse-grained Langevin description, i.e. the Complex Ginzburg Landau equation with stochastic noise in one complex field. We see that when one restricts the internal reaction noise to small amplitudes, one can obtain a simple form for the stochastic noise correlations that modify the Complex Ginzburg Landau equation. Finally, we study the effect of coupling a spatially extended May--Leonard model in two dimensions with symmetric predation rates to one with asymmetric rates that is prone to reach extinction. We show that the symmetric region induces otherwise unstable coexistence spiral patterns in the asymmetric May--Leonard lattice. We obtain the stability criterion for this pattern induction as we vary the strength of the extinction inducing asymmetry. This research was sponsored by the Army Research Office and was accomplished under Grant Number W911NF-17-1-0156. / Doctor of Philosophy / In the field of ecology, the cyclic predator-prey patterns in a food web are relevant yet independent to the hierarchical archetype. We study the paradigmatic cyclic May--Leonard model of three species, both analytically and numerically. First, we employ well--established techniques in large-deviation theory to study the extinction of populations induced by large but rare fluctuations. In the zero--dimensional version of the model, we compare the mean time to extinction computed from the theory to numerical simulations. Secondly, we study the stochastic spatial version of the May--Leonard model and show that for values close to the Hopf bifurcation, in the limit of small fluctuations, we can map the coarse-grained description of the model to the Complex Ginsburg Landau Equation, with stochastic noise corrections. Finally, we explore the induction of ecodiversity through spatio-temporal spirals in the asymmetric version of the May--Leonard model, which is otherwise inclined to reach an extinction state. This is accomplished by coupling to a symmetric May-Leonard counterpart on a two-dimensional lattice. The coupled system creates conditions for spiral formation in the asymmetric subsystem, thus precluding extinction.

In this dissertation, I study a nonlinear partial differential equation, the complex Ginzburg-Landau (CGL) equation. I first employed the perturbative field-theoretic renormalization group method to investigate the critical dynamics near the continuous non-equilibrium transition limit in this equation with additive noise. Due to the fact that time translation invariance is broken following a critical quench from a random initial configuration, an independent ``initial-slip'' exponent emerges to describe the crossover temporal window between microscopic time scales and the asymptotic long-time regime. My analytic work shows that to first order in a dimensional expansion with respect to the upper critical dimension, the extracted initial-slip exponent in the complex Ginzburg-Landau equation is identical to that of the equilibrium model A. Subsequently, I studied transient behavior in the CGL through numerical calculations. I developed my own code to numerically solve this partial differential equation on a two-dimensional square lattice with periodic boundary conditions, subject to random initial configurations. Aging phenomena are demonstrated in systems with either focusing and defocusing spiral waves, and the related aging exponents, as well as the auto-correlation exponents, are numerically determined. I also investigated nucleation processes when the system is transiting from a turbulent state to the ``frozen'' state. An extracted finite dimensionless barrier in the deep-quenched case and the exponentially decaying distribution of the nucleation times in the near-transition limit are both suggestive that the dynamical transition observed here is discontinuous. This research is supported by the U. S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering under Award DE-FG02-SC0002308 / Doctor of Philosophy / The complex Ginzburg-Landau equation is one of the most studied nonlinear partial differential equation in the physics community. I study this equation using both analytical and numerical methods. First, I employed the field theory approach to extract the critical initial-slip exponent, which emerges due to the breaking of time translation symmetry and describes the intermediate temporal window between microscopic time scales and the asymptotic long-time regime. I also numerically solved this equation on a two-dimensional square lattice. I studied the scaling behavior in non-equilibrium relaxation processes in situations where defects are interactive but not subject to strong fluctuations. I observed nucleation processes when the system under goes a transition from a strongly fluctuating disordered state to the relatively stable “frozen” state where its dynamics cease. I extracted a finite dimensionless barrier for systems that are quenched deep into the frozen state regime. An exponentially decaying long tail in the nucleation time distribution is found, which suggests a discontinuous transition. This research is supported by the U. S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering under Award DE-FG02-SC0002308.

In this thesis, we study six Ginzburg-Landau minimization problems in the context of two-dimensional nematic liquid crystals with the intention of finding conditions for the existence of boundary vortices. The first minimization problem consists of the standard Ginzburg-Landau energy on bounded, simply connected domains Ω ⊂ R2 with boundary energy penalizing minimizers who stray from being parallel to some smooth S1-valued boundary function g of degree D ≥ 1. The second and third minimization problems consider the same Ginzburg-Landau energy but now with divergence and curl penalization in the interior and boundary function taken to be g = τ, the positively oriented unit tangent vector to the boundary. The remaining three problems involve minimizing the same energies, but now over the set for which all functions are precisely parallel to the given boundary data (up to a set for which their norms can be zero). These six problems are classified under two categories called the weak and strong orthogonal problems. In each of the six problems, we show that conditions exist for which sequences of minimizers converge to a limiting S1-valued vector field describing an equilibrium configuration for nematic material with defects. In some cases, energy estimates are obtained that show vortices belong to the boundary exclusively and the exact number of these vortices are known. A special case is also studied in the strong orthogonality setting. The analysis here suggests that geometries exist for which boundary vortices may be energetically preferable to interior vortices in the case where interior and boundary vortices have similar energy contributions. / Thesis / Doctor of Philosophy (PhD)

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In quantum mechanics it is not uncommon to find analytically solved problems involvinga degree of math too advanced for most. It is often helpful to use a numerical approachto test solutions and deepen the understanding of such problems. In order to determine the validity of this approach, it is important to examine its accuracy. An exampleof this is the Landau-Zener problem, which is the topic of this thesis. It describes atwo-state quantum mechanical system that is applicable to many real world situations.The numerical method used involves propagating the wave function by calculating thetime evolution operator for numerous time steps. The accuracy using this method wasanalysed by comparing the results with the exact solution with varying parameters. Theconclusion is that the numerical solution does converge toward the known analytical solution. However, it does this with different accuracy, depending on the system parameters.

Les qubits de spin sont des candidats prometteurs pour le traitement de l’information quantique en raison de leurs longs temps de cohérence. Les deux principaux qubits présents dans un système à trois spins ont été démontré au cours des dernières années dans la boîte quantique latérale triple. Le diagramme des niveaux d’énergie de quelques électrons dans la boîte quantique triple est beaucoup plus complexe que son homologue à deux ou à une boîte. Il en résulte des possibilités de fuites hors des qubits ciblés. Dans ce mémoire, nous présenterons une nouvelles technologie pour améliorer le contrôle des états de spin et augmenter le temps de cohérence des qubits. Nous avons effectué des mesures préliminaires sur des échantillons sur lesquels a été incorporé un microaimant. Ce microaimant crée un champ magnétique non-uniforme au niveau des boîtes quantiques qui sera utilisé pour effectuer une rotation de spin et pour améliorer certains types d’oscillations. Nous avons optimisé la forme des géométries afin de créer des gradients de champ magnétique optimaux spécifiquement pour la boîte quantique triple. Différents problèmes ont été encourus et la stratégie que nous avons adoptée pour les régler sera présentée. De plus, nous avons analysé les phénomènes de fuites entre les états quantiques en étudiant la réponse d’un système à trois spins en fonction de différentes impulsions électriques. Nous présentons deux processus d’interférence jamais répertoriés entre les qubits de la boîte quantique triple. Afin d’identifier l’origine de ces interférences, nous avons utilisé leur dépendance en champ magnétique.

La thèse de doctorat comporte trois parties rédigées en anglais. Les deux premières parties correspondent principalement à l'étude de l'énergie de l'état fondamental. La dernière partie est consacrée à l'analyse de l'effet de pinning dans la supraconductivité.Dans une première partie de cette thèse, nous considérons la fonctionnelle de Ginzburg -Landau avec un champ magnétique variable appliqué dans un domaine borné et régulier de dimension 2. Nous déterminons le comportement asymptotique du paramètre d'ordre dans le régime o\`u le paramètre de Ginzburg-Landau et le champ magnétique sont grands et de même ordre. Comme conséquence, nous montrons que le paramètre d'ordre est localisé asymptotiquement dans la région où le profil du champ magnétique appliqué est petit.Dans une autre partie, nous considérons la fonctionnelle de Ginzburg -Landau avec un champ magnétique variable appliqué dans un domaine borné et régulier de dimension 2. Le profil du champ magnétique appliqué varie régulièrement et peut s'annuler exactement à l'ordre 1 le long d'une courbe. En supposant que la l'intensité du champ magnétique appliqué varie entre deux échelles caractéristiques, et que le paramètre de Ginzburg- Landau tend vers l'infini, nous déterminons une formule asymptotique précise pour minimiser l'énergie et montrer que les minimiseurs de l'énergie ont des vortex. Nous mettons en évidence que la présence d'un champ magnétique variable implique que la distribution de la vorticité dans l'échantillon n'est pas uniforme.Dans la dernière partie, nous étudions l'énergie de Ginzburg-Landau d'un supraconducteur avec un champ magnétique variable et un terme de pinning dans un domaine borné et régulier de dimension 2. En supposant que le paramètre de Ginzburg-Landau et l'intensité du champ magnétique sont grands et de même ordre, nous déterminons une formule asymptotique précise pour l'énergie. De plus, nous discutons l'existence des solutions non-triviales et déterminons le comportement asymptotique du troisième champ critique de la supraconductivité. / The PHD thesis has three parts, the first and the second part correpond mainly to study the groundstate energy, the last one being devoted to the analysis of the pinning effect in superconductivity.In a first part of this thesis, we consider the Ginzburg-Landau functional with a variable applied magnetic field in a bounded and smooth two-dimensional domain. We determine an accurate asymptotic formula for the minimizing energy when the Ginzburg-Landau parameter and the magnetic field are large and of the same order. As a consequence, it is shown how bulk superconductivity decreases in average as the applied magnetic field increases.In another part, we consider the Ginzburg-Landau functional with a variable applied magnetic field in a bounded and smooth two-dimensional domain. The profile of the applied magnetic field varies smoothly and is allowed to vanish non-degenerately along a curve. Assuming that the strength of the applied magnetic field varies between two characteristic scales, and that the Ginzburg-Landau parameter tends to , we determine an accurate asymptotic formula for the minimizing energy and show that the energy minimizers have vortices. The new aspect in the presence of variable magnetic field is that the distribution of vortices in the sample is not uniform.In the final part, we study the Ginzburg-Landau energy of a superconductor with a variable magnetic field and a pinning term in a bounded and smooth two-dimensional domain . Supposing that the Ginzburg-Landau parameter and the intensity of magnetic field are large and of the same order, we determine an accurate asymptotic formula for the minimizing energy. Also, we discuss the existence of non-trivial solutions and prove an asymptotics of the third critical field.