Made available in DSpace on 2017-07-21T19:25:57Z (GMT). No. of bitstreams: 1 Romeu Szmoski.pdf: 12436289 bytes, checksum: 6e754552b30a293cd49e5368fbb3bfd0 (MD5) Previous issue date: 2009-03-03 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The Brownian motion is one important topic of the non-equilibrium statistical mechanics and it is related to many natural phenomena. The first observations and theories on this motion were essential for understand the microscopic behavior of the nature and its influence on macroscopics observables. In this dissertation, we studied the dynamics of a system composed of several interacting Brownian particle from the point of view of coupled maps lattices. We use a map with a direct correlation to the abovementioned motion and we employ four different kinds of couplings in order to represent several ways of interaction among the particles. Using nonlinear dynamics tools, we observe the situations in which the particles velocities synchronize or show a tendency to the synchronized state. We also obtain algebrics expressions for the Lyapunov spectra of lattices with regular couplings whose interactions decays with distance as a power-law and we raise two hypotheses about Lyapunov exponents of a lattice with the coupling probability decreasing with the distance, as follows: the exponents of this lattice converge to the exponents of the lattice whose interactions decay with the distance in agreement to a power-law when the number of particles is very large; and the Lyapunov exponents of this lattice are given by the sum of the probabilities products of the each coupling matrix by eigenvalues of these matrixes. The values obtained for the Lyapunov exponents by means of the expressions deducted are in agreement with those obtained by numerical approximations techniques. Regarding distributions of the velocities of the particles, we observed that occur an aproximation to a Gaussian distribuition when the intensity of the coupling tends to its maximum. / O movimento browniano e um dos assuntos mais intrigantes da mecanica estatıstica de nao-equilıbrio e explica uma serie de fenomenos observados na natureza. As primeiras observaçoes a respeito deste movimento e as teorias propostas para descreve-lo foram fundamentais para entender o comportamento microscópico da natureza e a influência deste sobre observáveis macroscópicos. Nesta dissertação, estudamos a dinâmica de um sistema composto por várias partículas brownianas interagentes a partir de modelos de redes de mapas acoplados. Utilizamos um mapa que possui uma correspondência física direta com o movimento mencionado e empregamos quatro formas distintas de acoplamentos a fim de representar as várias formas de interação entre as partículas. Por meio de ferramentas da dinâmica não ao linear, observamos as situações em que as velocidades das partículas sincronizam ou tendem para o estado sincronizado. Também em obtivemos expressões exatas para determinar os expoentes de Lyapunov das redes com acoplamentos regulares cujas interações decaem com a distância segundo uma lei de potência e levantamos duas hipóteses sobre os expoentes de Lyapunov de uma rede com probabilidade de acoplamento decaindo com a distância, a saber: que os expoentes desta rede convergem para os expoentes da rede cujas interações decaem com a distância segundo uma lei de potência quando o número de partículas é muito grande; e que os expoentes de Lyapunov desta rede são dados pela soma dos produtos da probabilidade de ocorrer cada matriz de acoplamento pelos respectivos autovalores destas matrizes. Os valores obtidos para os expoentes de Lyapunov por meio das expressões deduzidas mostraram-se em acordo com aqueles obtidos por técnicas de aproximações numéricas. Em relação às distribuições das velocidades das partículas, observamos que elas se aproximam de uma gaussiana quando a intensidade do acoplamento tende a seu valor máximo.

This thesis focusses on the properties of, and relationships between, several fundamental objects arising from critical physical models. In particular, we consider Schramm--Loewner evolutions, the Gaussian free field, Liouville quantum gravity and the Brownian continuum random tree. We begin by considering branching diffusions in a bounded domain $D\subset$ $R^{d}$, in which particles are killed upon hitting the boundary $\partial D$. It is known that such a system displays a phase transition in the branching rate: if it exceeds a critical value, the population will no longer become extinct almost surely. We prove that at criticality, under mild assumptions on the branching mechanism and diffusion, the genealogical tree associated with the process will converge to the Brownian CRT. Next, we move on to study Gaussian multiplicative chaos. This is the rigorous framework that allows one to make sense of random measures built from rough Gaussian fields, and again there is a parameter associated with the model in which a phase transition occurs. We prove a uniqueness and convergence result for approximations to these measures at criticality. From this point onwards we restrict our attention to two-dimensional models. First, we give an alternative, ``non-Gaussian" construction of Liouville quantum gravity (a special case of Gaussian multiplicative chaos associated with the 2-dimensional Gaussian free field), that is motivated by the theory of multiplicative cascades. We prove that the Liouville (GMC) measures associated with the Gaussian free field can be approximated using certain sequences of ``local sets" of the field. This is a particularly natural construction as it is both local and conformally invariant. It includes the case of nested CLE$_{4}$, when it is coupled with the GFF as its set of ``level lines". Finally, we consider this level line coupling more closely, now when it is between SLE$_{4}$ and the GFF. We prove that level lines can be defined for the GFF with a wide range of boundary conditions, and are given by SLE$_{4}$-type curves. As a consequence, we extend the definition of SLE$_{4}(\rho)$ to the case of a continuum of force points.

O estudo apresenta um modelo de precificação de derivativos financeiros baseado em processos de mistura de movimentos brownianos. A partir de uma modelagem probabilística, são apresentados ajustes ao modelo tradicional de Black-Scholes-Merton para contemplar situações em que o retorno do ativo-objeto não segue uma distribuição normal. O trabalho discute ainda um mecanismo de estimação de parâmetros da mistura de normais. O resultado da pesquisa possibilita a análise de preço de opções em situações mais gerais. / The study presents a model for pricing financial derivatives based on a mixture of Brownian motion processes. From a probabilistic modeling, the research focuses on adjustments to the traditional Black- Scholes- Merton model to address situations where the return of the underlying asset does not follow a normal distribution. The paper also discusses a mechanism to estimate parameters of a mixture of normal distributions. The result of the study allows an analysis of option price in more general situations.

Doctor of philosophy (Ph D) / This thesis deals with numerical investigations on some rheological problems of fibre suspensions: the fibre level simulation of non-dilute fibre suspensions in shear flow; the numerical simulation of complex fibre suspension flows and simulating the particle motion in viscoelastic flows. These are challenging problems in rheology. Two numerical approaches were developed for simulating non-dilute fibre suspensions from the fibre level. The first is based on a model that accounts for full hydrodynamic interactions between fibres, which are approximately calculated as a superposition of the long-range and short-range hydrodynamic interactions. The long-range one is approximated by using slender body theory and includes infinite particle interactions. The short-range one is approximated in terms of the normal lubrication forces between close neighbouring fibres. The second is based on a model that accounts only for short-range interactions, which comprise the lubrication forces and normal contact and friction forces. These two methods were applied to simulate the microstructure evolution and rheological properties of non-dilute fibre suspensions. The Brownian configuration method was combined with the highly stable finite element method to simulate the complex flow of fibre suspensions. The method is stable and robust, and can provide both micro and macro information. It does not require any closure approximations in calculating the fibre stress tensor and is more efficient and variance reduction, compared to CONNFFESSITT, for example. The flow of fibre suspensions past a sphere in a tube and the shear induced fibre migration were successfully simulated using this method The completed double layer boundary element method was extended to viscoelastic flow cases. A point-wise solver was developed to solve the constitutive equation point by point and the fixed least square method was employed to interpolate and differentiate data locally. The method avoids volume meshing and only requires the boundary mesh on particle surfaces and data points in the flow domain. A sphere settling in the Oldroyd-B fluid and a prolate spheroid rotating in shear flow of the Oldroyd-B fluid were simulated. Based on the simulated orbit of a prolate spheroid in shear flow, a constitutive model for the weakly viscoelastic fibre suspensions was proposed and its predictions were compared with some available experimental results. All simulated results are in general agreement with experimental and other numerical results reported in literature. This indicates that these numerical methods are useful tools in rheological research.

<p>This thesis describes water proton and deuterium relaxation processes, as seen by Nuclear Magnetic Resonance (NMR) spectroscopy, using Brownian Dynamics (BD) or Molecular Dynamics (MD) simulations. The MD simulations reveal new detailed information about the dynamics and order of water molecules outside of a lipid bilayer. This is very important information in order to fully understand deuterium NMR measurements in lipid bilayer systems, which require an advanced analysis, because of the complicated water motion (such as tumbling and self-diffusion). The BD simulation methods are combined with the powerful Stochastic Liouville Equation (SLE) in its Langevin form (SLEL) to give new insight into both ¹H₂O and ²H₂O relaxation. The new simulation techniques which combine BD and SLEL can give important new information in cases where other methods do not apply. The deuterium relaxation is described in the context of a water/lipid interface and is in a very elegant way combined with the simulation of diffusion on curved surfaces developed by our research group. ¹H₂O spin-lattice relaxation is described for paramagneticsystems. With this we mean systems with paramagnetic transition metal ions or complexes, that are dissolved into a water solvent. The theoretical description of such systems are quite well investigated but such systems are not yet fully understood. An important consequence of the Paramagnetic Relaxation Enhancement (PRE) calculations when using the SLEL approach combined with BD simulations is that we obtain the electron correlation functions, which describe the relaxation of the paramagnetic electron spins. This means for example that it is also straight forward to generate Electron Spin Resonance (ESR) lineshapes.</p>

This thesis describes water proton and deuterium relaxation processes, as seen by Nuclear Magnetic Resonance (NMR) spectroscopy, using Brownian Dynamics (BD) or Molecular Dynamics (MD) simulations. The MD simulations reveal new detailed information about the dynamics and order of water molecules outside of a lipid bilayer. This is very important information in order to fully understand deuterium NMR measurements in lipid bilayer systems, which require an advanced analysis, because of the complicated water motion (such as tumbling and self-diffusion). The BD simulation methods are combined with the powerful Stochastic Liouville Equation (SLE) in its Langevin form (SLEL) to give new insight into both 1H2O and 2H2O relaxation. The new simulation techniques which combine BD and SLEL can give important new information in cases where other methods do not apply. The deuterium relaxation is described in the context of a water/lipid interface and is in a very elegant way combined with the simulation of diffusion on curved surfaces developed by our research group. 1H2O spin-lattice relaxation is described for paramagneticsystems. With this we mean systems with paramagnetic transition metal ions or complexes, that are dissolved into a water solvent. The theoretical description of such systems are quite well investigated but such systems are not yet fully understood. An important consequence of the Paramagnetic Relaxation Enhancement (PRE) calculations when using the SLEL approach combined with BD simulations is that we obtain the electron correlation functions, which describe the relaxation of the paramagnetic electron spins. This means for example that it is also straight forward to generate Electron Spin Resonance (ESR) lineshapes.

Coating colors are applied to the base paper in order to maximize the performance of the end product. Coating colors are complex colloidal systems, mainly consisting of water, binders, and pigments. To understand the behavior of colloidal suspensions, an understanding of the interactions between its components is essential.

Relaxation models are developed using numerical solutions of the Stochastic Liouville Equation of motion. Simplified descriptions such as the stochastic master equation is described in the context of fluorescence depolarisation experiments. Redfield theory is used in order to describe NMR relaxation in bicontinuous phases. The stochastic fluctuations in the relaxation models are accounted for using Brownian Dynamics simulation technique. A novel approach to quantitatively analyse fluorescence depolarisation experiments and to determine intramolecular distances is presented. A new Brownian Dynamics simulation technique is developed in order to characterize translational diffusion along the water lipid interface of bicontinuous cubic phases.

This thesis consists of two papers devoted to the asymptotics of random matrix ensembles and measure valued stochastic processes which can be considered as generalizations of the Gaussian unitary ensemble (GUE) of Hermitian matrices H=A+A†, where the entries of A are independent identically distributed (iid) centered complex Gaussian random variables. In the first paper, a system of interacting diffusing particles on the real line is studied; special cases include the eigenvalue dynamics of matrix-valued Ornstein-Uhlenbeck processes (Dyson's Brownian motion). It is known that the empirical measure process converges weakly to a deterministic measure-valued function and that the appropriately rescaled fluctuations around this limit converge weakly to a Gaussian distribution-valued process. For a large class of analytic test functions, explicit formulae are derived for the mean and covariance functionals of this fluctuation process. The second paper concerns a family of random matrix ensembles interpolating between the GUE and the Ginibre ensemble of n x n matrices with iid centered complex Gaussian entries. The asymptotic spectral distribution in these models is uniform in an ellipse in the complex plane, which collapses to an interval of the real line as the degree of non-Hermiticity diminishes. Scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum, and it is shown that a non-trivial transition occurs between Poisson and Airy point process statistics when the ratio of the axes of the supporting ellipse is of order n -1/3. / Denna avhandling består av två vetenskapliga artiklar som handlar om gränsvärdessatser för slumpmatriser och måttvärda stokastiska processer. De modeller som studeras kan betraktas som generaliseringar av den gaussiska unitära ensembeln (GUE) av hermiteska n x n-matriser H=A+A†, där A är en matris vars element är oberoende, likafördelade, centrerade, komplexa normalfördelade stokastiska variabler. I artikel I betraktas ett system av växelverkande diffunderande partiklar på reella linjen, vissa specialfall av denna modell kan tolkas som egenvärdesdynamiken för matrisvärda Ornstein-Uhlenbeck-processer (Dysons brownska rörelse). Sedan tidigare är det känt att den empiriska måttprocessen konvergerar svagt mot en deterministisk måttvärd funktion och att fluktuationerna runt denna gräns, i lämplig skalning, konvergerer svagt mot en distributionsvärd gaussisk process. För en stor klass av analytiska testfunktioner härleds explicita formler för medelvärdes- och kovariansfunktionalerna för denna fluktuationsprocess. Artikel II behandlar en familj av slumpmatrisensembler som interpolerar mellan GUE och Ginibre-ensembeln, bestående av matriser A som ovan. För denna modell är egenvärdena komplexa och asymptotiskt likformigt fördelade i en ellips i komplexa planet. Skalningsgränsvärdessatser för egenvärdet med maximal realdel och för egenvärdespunktprocessen kring detta visas för ett allmänt val av interpolationsparametern i modellen. Då förhållandet mellan axlarna i den asymptotiska ellipsen är av storleksordning n-1/3 uppträder en övergångsfas mellan Airypunktprocess- och Poissonprocessbeteendena, typiska för GUE respektive Ginibre-ensembeln. / QC 20100705

This thesis is concerned with simulation output analysis. In particular, we are inter- ested in estimating the variance parameter of a steady-state output process. The estimation of the variance parameter has immediate applications in problems involving (i) the precision of the sample mean as a point estimator for the steady-state mean and #956;X, and (ii) confidence intervals for and #956;X. The thesis focuses on new variance estimators arising from Schrubens method of standardized time series (STS). The main idea behind STS is to let such series converge to Brownian bridge processes; then their properties are used to derive estimators for the variance parameter. Following an idea from Shorack and Wellner, we study different levels of folded Brownian bridges. A folded Brownian bridge is obtained from the standard Brownian bridge process by folding it down the middle and then stretching it so that it spans the interval [0,1]. We formulate the folded STS, and deduce a simplified expression for it. Similarly, we define the weighted area under the folded Brownian bridge, and we obtain its asymptotic properties and distribution. We study the square of the weighted area under the folded STS (known as the folded area estimator ) and the weighted area under the square of the folded STS (known as the folded Cram??von Mises, or CvM, estimator) as estimators of the variance parameter of a stationary time series. In order to obtain results on the bias of the estimators, we provide a complete finite-sample analysis based on the mean-square error of the given estimators. Weights yielding first-order unbiased estimators are found in the area and CvM cases. Finally, we perform Monte Carlo simulations to test the efficacy of the new estimators on a test bed of stationary stochastic processes, including the first-order moving average and autoregressive processes and the waiting time process in a single-server Markovian queuing system.