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Equilibration, thermalization and scrambling appear to be intimately related, however their exact relation is unknown. As presented in the introduction these fields are at their current stages still quite popular and new directions are appearing regularly. We first put the fields into focus in the introduction and then the following chapters present three manuscripts as my contributions to these fields during my Master of Science project. Further introductions and background are presented in the manuscripts and my contributions are summarized at the beginning of each chapter. / Thesis / Master of Science (MSc)

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by Hon-chor Lee. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaf 56). / by Hon-chor Lee. / Introduction --- p.1 / Series expansion for the conductivity of a linear random resistor network / Chapter 1. --- Introduction --- p.4 / Chapter 2. --- Comparison of the EMA with symbolic simulations in 2D --- p.5 / Chapter 3. --- Comparison of the EMA with Bergman and Kantor's findings --- p.6 / Chapter 4. --- Conclusion --- p.7 / Current moments of linear random resistor network / Chapter 1. --- Introduction --- p.10 / Chapter 2. --- Review of the definition of current moment --- p.10 / Chapter 3. --- Tremblay et. al.'s findings and symbolic simulation of current moments --- p.11 / Chapter 4. --- Conclusion --- p.13 / Effective medium theory for strongly nonlinear composites: comparison with numerical simulations / Chapter 1. --- Introduction --- p.15 / Chapter 2. --- Variational principles --- p.16 / Chapter 3. --- Formalism of EMA --- p.17 / Chapter 4. --- Comparison with numerical simulations --- p.19 / Chapter 5. --- Discussion --- p.21 / Percolative conduction in two-component strongly nonlinear composites / Chapter 1. --- Introduction --- p.25 / Chapter 2. --- Spherical inclusions --- p.25 / Chapter 3. --- Effective medium approximation in the vicinity of the percolation threshold --- p.27 / Chapter 4. --- Acknowledgment --- p.29 / Percolation Effects in Two Component Strongly Nonlinear Composites: Universal Scaling Behaviours / Chapter 1. --- Introduction --- p.31 / Chapter 2. --- General Scaling Relations for Two-Component Composites --- p.33 / Chapter 3. --- Estimate of Critical Exponents --- p.35 / Chapter 4. --- Numerical Simulations --- p.38 / Chapter 5. --- Discussions and Conclusions --- p.40 / Chapter 6. --- Acknowledgment --- p.40 / Improved effective medium theory for strongly nonlinear composites / Chapter 1. --- Introduction --- p.46 / Chapter 2. --- Formalism of Improved EMA --- p.47 / Chapter 3. --- Comparisons with numerical simulations and HS bound --- p.50 / Chapter 4. --- Discussions --- p.51 / Conclusion --- p.54

Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 80-87).

Thesis (Ph. D.)--University of Washington, 2003. / Vita. Includes bibliographical references (leaves 99-114).

Interactions are what gives us the knowledge of the world around us. Interactions on all levels may fundamentally be seen as an exchange of information and a possible response of the same. Whether it is an electron in an electrical field or a handsome dude in a bar responding to a flirtation---interactions make things happen. In this sense we can see that objects without the capability of interacting with each other also are invisible to each other. Chains of pairwise interacting entities can serve as mediators of indirect interactions between objects. Nonetheless, in the limit of no interactions, we get into a philosophical debate whether we actually may consider anything to exist since it can not be detected in any way. Interactions between matter tend to be organized and show a hierarchical structure in which smaller sub-systems can be seen as parts of a bigger system, which in turn might be a smaller part of an even bigger system. This is reflected by the fact that we have sciences that successfully study specific interactions between objects or matter---physics, chemistry, biology, ecology, sociology,... What happens in a situation where all length scales are important? How does the structure of the underlying network of interactions affect the dynamical properties of a system? What network structures do we find and how are they created? This thesis is a physicist's view of collective dynamics, from superconductors to social systems and navigation in city street networks.

The main aim of this work is to study percolation theory on regular two dimensional / Thesis (M.Sc.)-University of Natal, Pietermaritzburg, 2002.

Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2006. / Includes bibliographical references (leaves 53-54). Also available in electronic version.

Orientador: Mário Noboru Tamashiro / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-08-31T00:17:06Z (GMT). No. of bitstreams: 1 Ercule_Adinei_M.pdf: 1649654 bytes, checksum: a32bd621f02831c6f06d0618847ad1a9 (MD5) Previous issue date: 2016 / Resumo: Estudamos através da solução do modelo de Blume¿Emery¿Griffiths na rede de Bethe, o efeito da coordenação da rede no comportamento multicrítico do modelo ferromagnético (J > 0) na região de acoplamentos biquadráticos repulsivos, analisando o caso particular K/J = ?3,5. O estudo foi feito para duas coordenações diferentes (z = 4, 6) e a campo nulo (H = 0). Durante a análise do mapeamento surgiram órbitas de período 2 e por esse motivo foi feita uma divisão da rede em duas sub-redes interpenetrantes. Nesta região o diagrama de fases apresentou quatro fases termodinâmicas distintas: paramagnética, ferromagnética, quadrupolar alternada (ou antiquadrupolar) e ferrimagnética. Além disso, o diagrama de fases apresentou regiões de coestabilidade numérica que foram investigadas pelo cálculo da energia livre. A expressão analítica da energia livre foi obtida através do engenhoso método de Gujrati. Para a coordenação z = 4 a transição entre as fases ferrimagnética e quadrupolar alternada é sempre contínua, enquanto a transição das fases ferromagnética e ferrimagnética é descontínua a baixas temperaturas, tornando-se contínua para temperaturas acima do ponto tricrítico. Por outro lado, para a coordenação z = 6 a transição entre as fases ferromagnética e ferrimagnética é sempre contínua. No entanto, a transição entre as fases ferrimagnética e quadrupolar alternada é descontínua para temperaturas abaixo do ponto tricrítico e contínua para temperaturas maiores, em concordância com os diagramas de fases obtidos na aproximação de campo médio / Abstract: We studied through the solution of the Blume¿Emery¿Griffiths model on the Bethe lattice, the effect of the lattice coordination on the multicritical behavior of the ferromagnetic model (J > 0) in the region of repulsive biquadratic couplings, analyzing the particular case K/J = ?3,5. The study was performed for two different lattice coordinations (z = 4, 6) and zero magnetic field (H = 0). During the mapping analysis, 2-cicle orbits arose and therefore a division of the lattice into two interpenetrating sublattices was implemented. In this region the phase diagram displayed four distinct thermodynamic phases: paramagnetic, ferromagnetic, staggered quadrupolar and ferrimagnetic. In addition, the phase diagram displayed regions of numerical co-stability, that were investigated by calculating the free energy. The analytical free-energy expression was obtained by the inventive Gujrati¿s method. For coordination z = 4 the transition between the ferrimagnetic and the stag- gered quadrupolar phases is always continuous, while the transition of the ferromagnetic and ferrimagnetic phases is discontinuous at low temperatures, turning into continuous for temperatures above the tricritical point. On the other hand, for coordination z = 6 the transition between the ferromagnetic and the ferrimagnetic phases is always continuous. However, the transition between the ferrimagnetic and the staggered quadrupolar phases is discontinuous for temperatures below the tricritical point and continuous for higher temperatures, in agreement with the phase diagrams obtained in the mean-field approximation / Mestrado / Física / Mestre em Física / 132988/2014-3 / CNPQ