Tese (doutorado)—Universidade de Brasília, Instituto e Física, 2013. / Submitted by Alaíde Gonçalves dos Santos (alaide@unb.br) on 2014-01-16T12:34:29Z No. of bitstreams: 1 2013_EvandroAlvesRodrigues.pdf: 1158236 bytes, checksum: e0c00b2497fa75c4ea39018e45458a31 (MD5) / Approved for entry into archive by Guimaraes Jacqueline(jacqueline.guimaraes@bce.unb.br) on 2014-02-13T14:41:35Z (GMT) No. of bitstreams: 1 2013_EvandroAlvesRodrigues.pdf: 1158236 bytes, checksum: e0c00b2497fa75c4ea39018e45458a31 (MD5) / Made available in DSpace on 2014-02-13T14:41:35Z (GMT). No. of bitstreams: 1 2013_EvandroAlvesRodrigues.pdf: 1158236 bytes, checksum: e0c00b2497fa75c4ea39018e45458a31 (MD5) / O Cresimento de superfícies fractais é uma área da Física Estatística que adquire grande momento com a popularização de ferramentas computacionais acessíveis, que tornam possível estudar este tipo de sistema por meio de modelos numéricos computacionais. Neste trabalho estudamos as propriedades do modelo de corrosão, com o objetivo de esclarecer algumas das questões em aberto desta grande área de pesquisa, como a existência de dimensão crítica na equação KPZ, descrevemos uma estratégia de obtenção de uma equação que modele a dinâmica da rugosidade e propomos uma forma restrita da relação de Family-Vicsek. _______________________________________________________________________________________ ABSTRACT / Fractal surface growth is one area of Statistical Physics that gains a lot of momentum with the recent popularization of cheap computational tools, that make it possible to study this kind of system through computational models. On this work we study properties of the etching model, to further investigate some of the open questions of this large research area, such as the existence of an upper critical dimension on the KPZ equation. We do describe an strategy to obtain an equation to model the dynamics of the surface rugosity an propose a restricted form of the Family-Vicsek relation.

Tesis para optar al grado de Doctor en Ciencias de la Ingeniería, Mención Modelación Matemática / Las Ecuaciones Estocásticas en Derivadas Parciales (SPDEs por su sigla en inglés) son una herramienta esencial para el análisis de los límites de escalamiento de diversos modelos microscópicos provenientes de otras áreas de las ciencias tales como la física y la química. Este tipo de ecuaciones corresponde a una ecuación en derivadas parciales clásica a la cual se le ha agregado un término de forzamiento externo aleatorio el que suele ser muy irregular; el ejemplo más sencillo es tal vez la Ecuación del Calor Estocástica, de la cual una de sus versiones es estudiada en la presente tesis. En cualquier caso, la irregularidad de este potencial hace que el análisis de las soluciones de estos problemas sea mucho más complicado. En efecto, hay casos en que dichas soluciones sólo pueden ser entendidas en el sentido de las distribuciones. Hay casos más críticos como la ecuación de Kardar--Parisi--Zhang (KPZ) en en una dimensión espacial donde, si bien se puede probar que posee soluciones Hölder, estas no son lo suficientemente regulares para permitir definir uno de los términos no lineales que aparecen en ella. Durante los últimos 20 años se han desarrollado varias técnicas para el análsis de este tipo de ecuaciones, entre las que destacan la teoría de rough paths geométricos de T. Lyons (1998), los rough paths ramificadosde M. Gubinelli (2010), y la más reciente teoría de estructuras de regularidad de M. Hairer (2014) por la que este último obtuvo la medalla Fields en 2014. Aunque diferentes, todas estas técnicas tienen como idea central el concepto de renormalización. En particular, la renormalización de Wick juega un rol esencial en la renormalización en el marco de las estructuras de regularidad. En este trabajo se desarrollan los productos y polinomios de Wick desde un punto de vista algebraico inspirado en el cálculo umbral de G.-C. Rota. También se explora la teoría general de losrough paths en general y su versión ramificada en particular, probándose nuevos resultados en la dirección de incorporar un análogo de la renormalización de Wick existente en las estructuras de regularidad. Por último, se estudia el modelo de polímero semidiscreto multicapas introducido por I. Corwin and A. Hammond (2014) para el cual se prueba la convergencia de su función de partición hacia la "solución" de la Ecuación del Calor Estocástica multicapas definida por N. O'Connell y J. Warren (2011) algunos años antes. Cabe destacar que al momento de redacción de esta tesis no existen resultados que permitan interpretar este proceso en el continuo como la solución de una SPDE singular como en el caso de la ecuación de KPZ, lo que ha sido una de las principales fuentes de inspiración para este trabajo. / CONICYT/Doctorado Nacional/2013-21130733 CMM - Conicyt PIA AFB170001

Made available in DSpace on 2014-12-17T15:15:01Z (GMT). No. of bitstreams: 1 SelmaRV_DISSERT.pdf: 2852447 bytes, checksum: 2b9488e49475092749d00972ff6c30bf (MD5) Previous issue date: 1994-05-27 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / Consideramos um vidro de spins misto definido em termos de dois grupos constitu?dos respectivamente, por vari?veis de tipo XY e planares discretas ( rel?gio com p-estados); apenas intera??es entre grupos distintos, do tipo alcance infinito, s?o permitidas. Um par?metro de anisotropia Dp que privilegia p dire??es no plano ? introduzido para as vari?veis XY, de tal forma a nos permitir uma interpola??o entre os modelos XY (Dp = 0 ; p = ∞ ) e rel?gio p-estados (Dp = ∞). Analisamos o procedimento de quebra da simetria entre r?plicas, onde diferentes tipos de fun??es par?metro de ordem s?o estudadas

The asymptotic scaling properties of conservative algorithms for parallel discrete-event simulations (e.g.: for spatially distributed parallel simulations of dynamic Monte Carlo for spin systems) of one-dimensional systems with system size $L$ is studied. The particular case studied here is the case of one or two elements assigned to each processor element. The previously studied case of one element per processor is reviewed, and the two elements per processor case is presented. The key concept is a simulated time horizon which is an evolving non equilibrium surface, specific for the particular algorithm. It is shown that the flat-substrate initial condition is responsible for the existence of an initial non-scaling regime. Various methods to deal with this non-scaling regime are documented, both the final successful method and unsuccessful attempts. The width of this time horizon relates to desynchronization in the system of processors. Universal properties of the conservative time horizon are derived by constructing a distribution of the interface width at saturation.

Lattice Monte Carlo methods are used to investigate far from and out-of-equilibrium systems, including surface growth, spin systems and solid mixtures. Applications range from the determination of universal growth or aging behaviors to palpable systems, where coarsening of nanocomposites or self-organization of functional nanostructures are of interest. Such studies require observations of large systems over long times scales, to allow structures to grow over orders of magnitude, which necessitates massively parallel simulations. This work addresses the problem of parallel processing introducing correlations in Monte Carlo updates and proposes a virtually correlation-free domain decomposition scheme to solve it. The effect of correlations on scaling and dynamical properties of surface growth systems and related lattice gases is investigated further by comparing results obtained by correlation-free and intrinsically correlated but highly efficient simulations using a stochastic cellular automaton (SCA). Efficient massively parallel implementations on graphics processing units (GPUs) were developed, which enable large-scale simulations leading to unprecedented precision in the final results. The primary subject of study is the Kardar–Parisi–Zhang (KPZ) surface growth in (2 + 1) dimensions, which is simulated using a dimer lattice gas and the restricted solid-on-solid model (RSOS) model. Using extensive simulations, conjectures regard- ing growth, autocorrelation and autoresponse properties are tested and new precise numerical predictions for several universal parameters are made.:1. Introduction 1.1. Motivations and Goals 1.2. Overview 2. Methods and Models 2.1. Estimation of Scaling Exponents and Error Margins 2.2. From Continuum- to Atomistic Models 2.3. Models for Phase Ordering and Nanostructure Evolution 2.3.1. The Kinetic Metropolis Lattice Monte-Carlo Method 2.3.2. The Potts Model 2.4. The Kardar–Parisi–Zhang and Edwards–Wilkinson Universality Classes 2.4.0.1. Physical Aging 2.4.1. The Octahedron Model 2.4.2. The Restricted Solid on Solid Model 3. Parallel Implementation: Towards Large-Scale Simulations 3.1. Parallel Architectures and Programming Models 3.1.1. CPU 3.1.2. GPU 3.1.3. Heterogeneous Parallelism and MPI 3.1.4. Bit-Coding of Lattice Sites 3.2. Domain Decomposition for Stochastic Lattice Models 3.2.1. DD for Asynchronous Updates 3.2.1.1. Dead border (DB) 3.2.1.2. Double tiling (DT) 3.2.1.3. DT DD with random origin (DTr) 3.2.1.4. Implementation 3.2.2. Second DD Layer on GPUs 3.2.2.1. Single-Hit DT 3.2.2.2. Single-Hit dead border (DB) 3.2.2.3. DD Parameters for the Octahedron Model 3.2.3. Performance 3.3. Lattice Level DD: Stochastic Cellular Automaton 3.3.1. Local Approach for the Octahedron Model 3.3.2. Non-Local Approach for the Octahedron Model 3.3.2.1. Bit-Vectorized GPU Implementation 3.3.3. Performance of SCA Implementations 3.4. The Multi-Surface Coding Approach 3.4.0.1. Vectorization 3.4.0.2. Scalar Updates 3.4.0.3. Domain Decomposition 3.4.1. Implementation: SkyMC 3.4.1.1. 2d Restricted Solid on Solid Model 3.4.1.2. 2d and 3d Potts Model 3.4.1.3. Sequential CPU Reference 3.4.2. SkyMC Benchmarks 3.5. Measurements 3.5.0.1. Measurement Intervals 3.5.0.2. Measuring using Heterogeneous Resources 4. Monte-Carlo Investigation of the Kardar–Parisi–Zhang Universality Class 4.1. Evolution of Surface Roughness 4.1.1. Comparison of Parallel Implementations of the Octahedron Model 4.1.1.1. The Growth Regime 4.1.1.2. Distribution of Interface Heights in the Growth Regime 4.1.1.3. KPZ Ansatz for the Growth Regime 4.1.1.4. The Steady State 4.1.2. Investigations using RSOS 4.1.2.1. The Growth Regime 4.1.2.2. The Steady State 4.1.2.3. Consistency of Fine-Size Scaling with Respect to DD 4.1.3. Results for Growth Phase and Steady State 4.2. Autocorrelation Functions 4.2.1. Comparison of DD Methods for RS Dynamics 4.2.1.1. Device-Layer DD 4.2.1.2. Block-Layer DD 4.2.2. Autocorrelation Properties under RS Dynamics 4.2.3. Autocorrelation Properties under SCA Dynamics 4.2.3.1. Autocorrelation of Heights 4.2.3.2. Autocorrelation of Slopes 4.2.4. Autocorrelation in the SCA Steady State 4.2.5. Autocorrelation in the EW Case under SCA 4.2.5.1. Autocorrelation of Heights 4.2.5.2. Autocorrelations of Slopes 4.3. Autoresponse Functions 4.3.1. Autoresponse Properties 4.3.1.1. Autoresponse of Heights 4.3.1.2. Autoresponse of Slopes 4.3.1.3. Self-Averaging 4.4. Summary 5. Further Topics 5.1. Investigations of the Potts Model 5.1.1. Testing Results from the Parallel Implementations 5.1.2. Domain Growth in Disordered Potts Models 5.2. Local Scale Invariance in KPZ Surface Growth 6. Conclusions and Outlook Acknowledgements A. Coding Details A.1. Bit-Coding A.2. Packing and Unpacking Signed Integers A.3. Random Number Generation / Gitter-Monte-Carlo-Methoden werden zur Untersuchung von Systemen wie Oberflächenwachstum, Spinsystemen oder gemischten Feststoffen verwendet, welche fern eines Gleichgewichtes bleiben oder zu einem streben. Die Anwendungen reichen von der Bestimmung universellen Wachstums- und Alterungsverhaltens hin zu konkreten Systemen, in denen die Reifung von Nanokompositmaterialien oder die Selbstorganisation von funktionalen Nanostrukturen von Interesse sind. In solchen Studien müssen große Systemen über lange Zeiträume betrachtet werden, um Strukturwachstum über mehrere Größenordnungen zu erlauben. Dies erfordert massivparallele Simulationen. Diese Arbeit adressiert das Problem, dass parallele Verarbeitung Korrelationen in Monte-Carlo-Updates verursachen und entwickelt eine praktisch korrelationsfreie Domänenzerlegungsmethode, um es zu lösen. Der Einfluss von Korrelationen auf Skalierungs- und dynamische Eigenschaften von Oberflächenwachtums- sowie verwandten Gittergassystemen wird weitergehend durch den Vergleich von Ergebnissen aus korrelationsfreien und intrinsisch korrelierten Simulationen mit einem stochastischen zellulären Automaten untersucht. Effiziente massiv parallele Implementationen auf Grafikkarten wurden entwickelt, welche großskalige Simulationen und damit präzedenzlos genaue Ergebnisse ermöglichen. Das primäre Studienobjekt ist das (2 + 1)-dimensionale Kardar–Parisi–Zhang- Oberflächenwachstum, welches durch ein Dimer-Gittergas und das Kim-Kosterlitz-Modell simuliert wird. Durch massive Simulationen werden Thesen über Wachstums-, Autokorrelations- und Antworteigenschaften getestet und neue, präzise numerische Vorhersagen zu einigen universellen Parametern getroffen.:1. Introduction 1.1. Motivations and Goals 1.2. Overview 2. Methods and Models 2.1. Estimation of Scaling Exponents and Error Margins 2.2. From Continuum- to Atomistic Models 2.3. Models for Phase Ordering and Nanostructure Evolution 2.3.1. The Kinetic Metropolis Lattice Monte-Carlo Method 2.3.2. The Potts Model 2.4. The Kardar–Parisi–Zhang and Edwards–Wilkinson Universality Classes 2.4.0.1. Physical Aging 2.4.1. The Octahedron Model 2.4.2. The Restricted Solid on Solid Model 3. Parallel Implementation: Towards Large-Scale Simulations 3.1. Parallel Architectures and Programming Models 3.1.1. CPU 3.1.2. GPU 3.1.3. Heterogeneous Parallelism and MPI 3.1.4. Bit-Coding of Lattice Sites 3.2. Domain Decomposition for Stochastic Lattice Models 3.2.1. DD for Asynchronous Updates 3.2.1.1. Dead border (DB) 3.2.1.2. Double tiling (DT) 3.2.1.3. DT DD with random origin (DTr) 3.2.1.4. Implementation 3.2.2. Second DD Layer on GPUs 3.2.2.1. Single-Hit DT 3.2.2.2. Single-Hit dead border (DB) 3.2.2.3. DD Parameters for the Octahedron Model 3.2.3. Performance 3.3. Lattice Level DD: Stochastic Cellular Automaton 3.3.1. Local Approach for the Octahedron Model 3.3.2. Non-Local Approach for the Octahedron Model 3.3.2.1. Bit-Vectorized GPU Implementation 3.3.3. Performance of SCA Implementations 3.4. The Multi-Surface Coding Approach 3.4.0.1. Vectorization 3.4.0.2. Scalar Updates 3.4.0.3. Domain Decomposition 3.4.1. Implementation: SkyMC 3.4.1.1. 2d Restricted Solid on Solid Model 3.4.1.2. 2d and 3d Potts Model 3.4.1.3. Sequential CPU Reference 3.4.2. SkyMC Benchmarks 3.5. Measurements 3.5.0.1. Measurement Intervals 3.5.0.2. Measuring using Heterogeneous Resources 4. Monte-Carlo Investigation of the Kardar–Parisi–Zhang Universality Class 4.1. Evolution of Surface Roughness 4.1.1. Comparison of Parallel Implementations of the Octahedron Model 4.1.1.1. The Growth Regime 4.1.1.2. Distribution of Interface Heights in the Growth Regime 4.1.1.3. KPZ Ansatz for the Growth Regime 4.1.1.4. The Steady State 4.1.2. Investigations using RSOS 4.1.2.1. The Growth Regime 4.1.2.2. The Steady State 4.1.2.3. Consistency of Fine-Size Scaling with Respect to DD 4.1.3. Results for Growth Phase and Steady State 4.2. Autocorrelation Functions 4.2.1. Comparison of DD Methods for RS Dynamics 4.2.1.1. Device-Layer DD 4.2.1.2. Block-Layer DD 4.2.2. Autocorrelation Properties under RS Dynamics 4.2.3. Autocorrelation Properties under SCA Dynamics 4.2.3.1. Autocorrelation of Heights 4.2.3.2. Autocorrelation of Slopes 4.2.4. Autocorrelation in the SCA Steady State 4.2.5. Autocorrelation in the EW Case under SCA 4.2.5.1. Autocorrelation of Heights 4.2.5.2. Autocorrelations of Slopes 4.3. Autoresponse Functions 4.3.1. Autoresponse Properties 4.3.1.1. Autoresponse of Heights 4.3.1.2. Autoresponse of Slopes 4.3.1.3. Self-Averaging 4.4. Summary 5. Further Topics 5.1. Investigations of the Potts Model 5.1.1. Testing Results from the Parallel Implementations 5.1.2. Domain Growth in Disordered Potts Models 5.2. Local Scale Invariance in KPZ Surface Growth 6. Conclusions and Outlook Acknowledgements A. Coding Details A.1. Bit-Coding A.2. Packing and Unpacking Signed Integers A.3. Random Number Generation