Thesis (Ph. D.)--University of Chicago, Department of Chemistry, December 2002. / Includes bibliographical references. Also available on the Internet.

Thesis (M. Sc.)--University of Canterbury, 2006. / Typescript (photocopy). Includes bibliographical references (p. 111-114). Also available via the World Wide Web.

Orientador: Maria Cristina Batoni Abdalla / Doutor

Numerical Stochastic Perturbation theory is a powerful tool for estimating high-order perturbative expansions in lattice quantum field theory. The standard algorithm based on the Langevin equation, however, suffers from several limitations which in practice restrict the potential of this technique: first of all it is not exact, a sequence of simulations with finer and finer discretization of the relevant equations have to be performed in order to extrapolate away the systematic errors in the results; and, secondly, the numerical simulations suffer from critical slowing down as the continuum limit of the theory is approached. In this thesis I investigate some alternative methods which improve upon the standard approach. In particular, I present a formulation of Numerical Stochastic Perturbation theory based on the Generalised Hybrid Molecular Dynamics algorithm and a study of the recently proposed Instantaneous Stochastic Perturbation Theory. The viability of these methods is investigated in φ4 theory.

Um modelo do tipo Ginzburg-Landau com interações competitivas entre um termo ferromagnético de curto alcance, e outro antiferromagnético de longo alcance (dipolar), é estudado no âmbito de reproduzir a rica fenomenologia de fases moduladas em filmes finos ferromagnéticos. Esse tipo de material pode apresentar ordens intermediárias as de um sólido e um líquido, e suas transições de fase dependem fortemente dos tipos de excitações geométricas que ele suporta. Dentre os estudos teóricos e numéricos, as simulações apresentadas aqui introduzem esse modelo dipolar contínuo como um bom candidato para estudar as fases complexas desse sistema, se mostrando em acordo com resultados experimentais. / A Ginzburg-Landau model with competitive short-range ferromagnetic interactions and long-range (dipolar) antiferromagnetic interactions is studied on the scope to reproduce the rich phenomenology of modulated phases in ferromagnetic thin films. This kind of material can present intermediate order between a solid and a liquid, and its phase transitions depend strongly on the kinds of geometrical excitations that it supports. Amongst the theoretical and numerical studies, the simulations presented here introduce this continuous dipolar model as a good candidate to study the complex phases of this system, showing good agreement with experimental results.

Neste trabalho estudamos os fenômenos observados em filmes magnéticos ultrafinos com anisotropia perpendicular, onde domínios razoavelmente regulares de faixas dominam o regime de baixas temperaturas a campo nulo. Esses domínios magnéticos de faixas constituem uma realização de fases moduladas presentes em inúmeros sistemas físicos, químicos e biológicos, e são resultados da presença de interaçães competindo em diferentes escalas espaciais. No caso de interesse, a competição entre as interações de troca e dipolar levam à estabilidade de uma estrutura de domínios de faixas que possuem ambas ordens translacional anisotrópica e orientacional, semelhantes às encontradas em filmes de cristais líquidos. Através de um modelo escalar de Landau Ginzburg que captura a formação dos domínios de faixas nos filmes magnéticos ultrafinos com anisotropia perpendicular, estudamos o efeito das flutuações térmicas atuando nas escalas de comprimento introduzidas pela competição entre as interações, que, aliadas à baixa dimensionalidade do problema, estabelecem fases ordenadas de baixa temperatura com ordem de quase longo alcance, onde defeitos topológicos exercem um papel fundamental. Introduzimos uma técnica de simulação de Langevin para o modelo de Landau Ginzburg, através da qual obtivemos resultados de equilíbrio determinando a natureza das fases de baixa temperatura. Confirmamos junto aos resultados experimentais a estabilidade de uma fase esmética, associada à quebra de simetria translacional, em baixas temperaturas. Entre essa fase e a fase isotrópica, encontramos resultados que apontam a estabilidade da fase nemática, associada à quebra de simetria orientacional, que é prevista teoricamente mas não foi ainda observada experimentalmente. A simulação de Langevin introduzida aqui se mostrou capaz de reproduzir fenômenos como a dependência da largura das faixas com a temperatura e o perfil das paredes de domínio, assim como flutuações térmicas e defeitos topológicos das faixas, muito próximos aos observados experimentalmente. / ln this work we study the phenomena observed in ultrathin magnetic films with perpendicular anisotropy, in which stripe domains with reasonable regularity dominate the low temperature regime under zero external applied field. These stripe magnetic domains are a manifestation of modulated phases present in a large number of physical, chemical and biological systems, and are the result of the presence of interactions competing in different spacial scales. In the case we are interested in, the competition between the exchange and dipolar interactions stabilize a stripe domain structure that have both translacional anisotropic and orientacional orders, similar to those found in liquid crystal films. Through a scalar Landau Ginzburg model that captures the stripe domain formation in ultrathin magnetic films with perpendicular anisotropy, we study the effect of thermal fluctuations acting in the length scales introduced by the competition of the interactions, that, together with the low dimensionality of the problem, estabilize low temperature ordered phases with quasi-long-range order, where topological defects play a fundamental role. We introduce here a Langevin simulation technique to the Landau Ginzburg model, through which we obtain equilibrium results determining the nature of the low temperature phases. We confirm, in agreement with experimental observations, the stability of a smectic phase, related to the break of translational symmetry. Between this phase and the isotropic phase, we find results that point to the stability of the nematic phase, related to the break of orientational symmetry, that is predicted theoretically but was not observed experimentally. The Langevin simulation introduced here is capable to reproduce some of the phenomena, like the stripe domain width temperature dependence and the domain wall profile, as well as stripe thermal fluctuations and topological defects, very close to those observed experimentally.

Made available in DSpace on 2016-01-13T13:27:50Z (GMT). No. of bitstreams: 0 Previous issue date: 1988. Added 1 bitstream(s) on 2016-01-13T13:31:36Z : No. of bitstreams: 1 000036077.pdf: 4483670 bytes, checksum: 03892faf78b15a06dda178fbff397aad (MD5)

Um modelo do tipo Ginzburg-Landau com interações competitivas entre um termo ferromagnético de curto alcance, e outro antiferromagnético de longo alcance (dipolar), é estudado no âmbito de reproduzir a rica fenomenologia de fases moduladas em filmes finos ferromagnéticos. Esse tipo de material pode apresentar ordens intermediárias as de um sólido e um líquido, e suas transições de fase dependem fortemente dos tipos de excitações geométricas que ele suporta. Dentre os estudos teóricos e numéricos, as simulações apresentadas aqui introduzem esse modelo dipolar contínuo como um bom candidato para estudar as fases complexas desse sistema, se mostrando em acordo com resultados experimentais. / A Ginzburg-Landau model with competitive short-range ferromagnetic interactions and long-range (dipolar) antiferromagnetic interactions is studied on the scope to reproduce the rich phenomenology of modulated phases in ferromagnetic thin films. This kind of material can present intermediate order between a solid and a liquid, and its phase transitions depend strongly on the kinds of geometrical excitations that it supports. Amongst the theoretical and numerical studies, the simulations presented here introduce this continuous dipolar model as a good candidate to study the complex phases of this system, showing good agreement with experimental results.

Neste trabalho estudamos os fenômenos observados em filmes magnéticos ultrafinos com anisotropia perpendicular, onde domínios razoavelmente regulares de faixas dominam o regime de baixas temperaturas a campo nulo. Esses domínios magnéticos de faixas constituem uma realização de fases moduladas presentes em inúmeros sistemas físicos, químicos e biológicos, e são resultados da presença de interaçães competindo em diferentes escalas espaciais. No caso de interesse, a competição entre as interações de troca e dipolar levam à estabilidade de uma estrutura de domínios de faixas que possuem ambas ordens translacional anisotrópica e orientacional, semelhantes às encontradas em filmes de cristais líquidos. Através de um modelo escalar de Landau Ginzburg que captura a formação dos domínios de faixas nos filmes magnéticos ultrafinos com anisotropia perpendicular, estudamos o efeito das flutuações térmicas atuando nas escalas de comprimento introduzidas pela competição entre as interações, que, aliadas à baixa dimensionalidade do problema, estabelecem fases ordenadas de baixa temperatura com ordem de quase longo alcance, onde defeitos topológicos exercem um papel fundamental. Introduzimos uma técnica de simulação de Langevin para o modelo de Landau Ginzburg, através da qual obtivemos resultados de equilíbrio determinando a natureza das fases de baixa temperatura. Confirmamos junto aos resultados experimentais a estabilidade de uma fase esmética, associada à quebra de simetria translacional, em baixas temperaturas. Entre essa fase e a fase isotrópica, encontramos resultados que apontam a estabilidade da fase nemática, associada à quebra de simetria orientacional, que é prevista teoricamente mas não foi ainda observada experimentalmente. A simulação de Langevin introduzida aqui se mostrou capaz de reproduzir fenômenos como a dependência da largura das faixas com a temperatura e o perfil das paredes de domínio, assim como flutuações térmicas e defeitos topológicos das faixas, muito próximos aos observados experimentalmente. / ln this work we study the phenomena observed in ultrathin magnetic films with perpendicular anisotropy, in which stripe domains with reasonable regularity dominate the low temperature regime under zero external applied field. These stripe magnetic domains are a manifestation of modulated phases present in a large number of physical, chemical and biological systems, and are the result of the presence of interactions competing in different spacial scales. In the case we are interested in, the competition between the exchange and dipolar interactions stabilize a stripe domain structure that have both translacional anisotropic and orientacional orders, similar to those found in liquid crystal films. Through a scalar Landau Ginzburg model that captures the stripe domain formation in ultrathin magnetic films with perpendicular anisotropy, we study the effect of thermal fluctuations acting in the length scales introduced by the competition of the interactions, that, together with the low dimensionality of the problem, estabilize low temperature ordered phases with quasi-long-range order, where topological defects play a fundamental role. We introduce here a Langevin simulation technique to the Landau Ginzburg model, through which we obtain equilibrium results determining the nature of the low temperature phases. We confirm, in agreement with experimental observations, the stability of a smectic phase, related to the break of translational symmetry. Between this phase and the isotropic phase, we find results that point to the stability of the nematic phase, related to the break of orientational symmetry, that is predicted theoretically but was not observed experimentally. The Langevin simulation introduced here is capable to reproduce some of the phenomena, like the stripe domain width temperature dependence and the domain wall profile, as well as stripe thermal fluctuations and topological defects, very close to those observed experimentally.

This thesis addresses the sampling problem in a high-dimensional space, i.e., the computation of averages with respect to a defined probability density that is a function of many variables. Such sampling problems arise in many application areas, including molecular dynamics, multiscale models, and Bayesian sampling techniques used in emerging machine learning applications. Of particular interest are thermostat techniques, in the setting of a stochastic-dynamical system, that preserve the canonical Gibbs ensemble defined by an exponentiated energy function. In this thesis we explore theory, algorithms, and numerous applications in this setting. We begin by comparing numerical methods for particle-based models. The class of methods considered includes dissipative particle dynamics (DPD) as well as a newly proposed stochastic pairwise Nosé-Hoover-Langevin (PNHL) method. Splitting methods are developed and studied in terms of their thermodynamic accuracy, two-point correlation functions, and convergence. When computational efficiency is measured by the ratio of thermodynamic accuracy to CPU time, we report significant advantages in simulation for the PNHL method compared to popular alternative schemes in the low-friction regime, without degradation of convergence rate. We propose a pairwise adaptive Langevin (PAdL) thermostat that fully captures the dynamics of DPD and thus can be directly applied in the setting of momentum-conserving simulation. These methods are potentially valuable for nonequilibrium simulation of physical systems. We again report substantial improvements in both equilibrium and nonequilibrium simulations compared to popular schemes in the literature. We also discuss the proper treatment of the Lees-Edwards boundary conditions, an essential part of modelling shear flow. We also study numerical methods for sampling probability measures in high dimension where the underlying model is only approximately identified with a gradient system. These methods are important in multiscale modelling and in the design of new machine learning algorithms for inference and parameterization for large datasets, challenges which are increasingly important in "big data" applications. In addition to providing a more comprehensive discussion of the foundations of these methods, we propose a new numerical method for the adaptive Langevin/stochastic gradient Nosé-Hoover thermostat that achieves a dramatic improvement in numerical efficiency over the most popular stochastic gradient methods reported in the literature. We demonstrate that the newly established method inherits a superconvergence property (fourth order convergence to the invariant measure for configurational quantities) recently demonstrated in the setting of Langevin dynamics. Furthermore, we propose a covariance-controlled adaptive Langevin (CCAdL) thermostat that can effectively dissipate parameter-dependent noise while maintaining a desired target distribution. The proposed method achieves a substantial speedup over popular alternative schemes for large-scale machine learning applications.