Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references. Available also from UMI/Dissertation Abstracts International.

Not available / text

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Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 184-188).

Made available in DSpace on 2016-05-17T16:51:02Z (GMT). No. of bitstreams: 0 Previous issue date: 2008-08-29. Added 1 bitstream(s) on 2016-05-17T16:54:20Z : No. of bitstreams: 1 000854764.pdf: 562890 bytes, checksum: 164c6db1a3c04e45b5f0eea9ea15e58e (MD5) / Este trabalho é basicamente uma revisão de alguns aspectos de integrabilidade em duas dimensões discutidos no artigo Renormalization group flows and continual Lie algebras do professor Ioannis Bakas. A idéia é estudar o fluxo do grupo de renormalização das métricas bi-dimensionais nos modelos-'alfa' usando a função beta a 1-loop, e mostrar que elas fornecem análogos contínuos das equações de campo de Toda nas coordenadas conformemente planas do espaço target. Nesta formulção algébrica, a escala logaritmica de comprimento da folha mundo é interpretada como o parâmetro de Dynkin num sistema de raízes de uma álgebra de Lie contínua, denotada por G(d/dt;II), com um kernel de Cartan generalizado anti-simétrico K(t,t') = 'alfa'(t−t'). Usando o formalismo de curvatura zero construiremos a solução geral do fluxo do grupo de renormalização em termos das configurações de campo livre por meio de transformações de Bäcklund. A validade desta solução geral como uma expansão em serie de potência será testada com alguns exemplos especiais que incluim o modelo sausage, as métricas de curvatura constante negativa e a queda de singularidades côonicas / This work is basically a review of some aspect of the integrability in two dimensions discussed in the Prof. Ioannis Bakas's paper called Renormalization group flows and continual Lie algebras. The main idea is to study the renormalization group flow of two-dimensional metrics in sigma models using the one-loop beta function, and demonstrate that they provide a continual analogue of the Toda field equations in conformally flat coordinates in the target space. In this algebraic frame, the logarithm of the world-sheet length scale t is interpreted as Dynkin parameter on the root system of a continual Lie algebra, denoted by G(d/dt;II),witha an ti-symmetric generalized Cartan kernel K(t,t') ='sigmma'(t−t'). Using the zero curvature formalism, we construct a general solution of the renormalization group flow in terms of the free field configurations via B¨acklund transformations. The validity of these general solutions as a power series expansion is verified in some specials examples including the sausage model, the constant negative curvature metrics and the decay of conical singularities

Orientador: Abraham Hirsz Zimerman / Banca: Nathan Jacob Berkovits / Banca: Victor de Oliveira Rivelles / Resumo: Este trabalho é basicamente uma revisão de alguns aspectos de integrabilidade em duas dimensões discutidos no artigo "Renormalization group flows and continual Lie algebras" do professor Ioannis Bakas. A idéia é estudar o fluxo do grupo de renormalização das métricas bi-dimensionais nos modelos-'alfa' usando a função beta a 1-loop, e mostrar que elas fornecem análogos contínuos das equações de campo de Toda nas coordenadas conformemente planas do espaço target. Nesta formulção algébrica, a escala logaritmica de comprimento da folha mundo é interpretada como o parâmetro de Dynkin num sistema de raízes de uma álgebra de Lie contínua, denotada por G(d/dt;II), com um kernel de Cartan generalizado anti-simétrico K(t,t') = 'alfa'(t−t'). Usando o formalismo de curvatura zero construiremos a solução geral do fluxo do grupo de renormalização em termos das configurações de campo livre por meio de transformações de Bäcklund. A validade desta solução geral como uma expansão em serie de potência será testada com alguns exemplos especiais que incluim o modelo "sausage", as métricas de curvatura constante negativa e a queda de singularidades côonicas / Abstract: This work is basically a review of some aspect of the integrability in two dimensions discussed in the Prof. Ioannis Bakas's paper called "Renormalization group flows and continual Lie algebras". The main idea is to study the renormalization group flow of two-dimensional metrics in sigma models using the one-loop beta function, and demonstrate that they provide a continual analogue of the Toda field equations in conformally flat coordinates in the target space. In this algebraic frame, the logarithm of the world-sheet length scale t is interpreted as Dynkin parameter on the root system of a continual Lie algebra, denoted by G(d/dt;II),witha an ti-symmetric generalized Cartan kernel K(t,t') ='sigmma'(t−t'). Using the zero curvature formalism, we construct a general solution of the renormalization group flow in terms of the free field configurations via B¨acklund transformations. The validity of these general solutions as a power series expansion is verified in some specials examples including the sausage model, the constant negative curvature metrics and the decay of conical singularities / Mestre

For the latter half of the last century renormalization methods have played an important part in tackling problems in fundamental physics and in providing a deeper understanding of systems with many interacting scales or degrees of freedom with strong coupling. The study of turbulence is no exception, and this thesis presents an investigation of renormalization techniques available in the study of the statistical theory of homogeneous and isotropic turbulence. The thesis consists of two parts which assess the two main renormalization approaches available in modeling turbulence. In particular we will be focusing on the renormalization procedures developed by McComb and others. The first part of this thesis will discuss Renormalization Group (RG) approaches to turbulence, with a focus on applications to reduce the degrees of freedom in a large-eddy simulation. The RG methods as applied to classical dynamical systems will be reviewed in the context of the Navier-Stokes equations describing fluid flow. This will be followed by introducing a functional based formalism of a conditional average first introduced by McComb, Roberts and Watt [Phys. Rev A 45, 3507 (1992)] as a tool for averaging out degrees of freedom needed in an RG calculation. This conditional average is then used in a formal RG calculation applied to the Navier-Stokes equations, originally done by McComb and Watt [Phys. Rev. A 46, 4797 (1992)], and later revised by Mc- Comb and Johnston [Physica A 292, 346 (2001)]. A correction to the summing of the time-integral detailed in the latter work is shown to introduce an extra viscous life-time term to the denominator of the increment to the renormalized viscosity and is shown to have a negligible effect in the numerical calculations. We follow this study by outlining some problems with the previous approach. In particular it is shown that a cross-term representing the interaction between high and low wavenumber modes which was neglected in the previous studies on the grounds that it does not contribute to energy dissipation, does in fact contribute significantly. A heuristic method is then put forward to include the effects of this term in the RG calculation. This leads to results which agree qualitatively with numerical calculations of eddy-viscosities. We finish this part of the thesis with an application of the RG method to the modeling of a passive scalar advected by a turbulent velocity field. The second part of this thesis will begin by reviewing Eulerian renormalized perturbation theory attempts in closing the infinite moment hierarchy introduced by averaging the Navier-Stokes equations. This is followed by presenting a new formulation of the local energy transfer theory (LET) of McComb et. al. [J. Fluid Mech. 245, 279 (1992)] which resolves some problems of previous derivations. In particular we show by the introduction of time-ordering that some previous problems with the exponential representation of the correlator can be overcome. Furthermore, we show that the singularity in the LET propagator equation cancels by way of a counter-term. We end this study by introducing a single-time Markovian closure based on LET which, unlike other Markovian closures, does not rely on any arbitrary parameters being introduced in the theory.

O comportamento critico de sistemas unidimensionais de spin do tipo Z(N) na ausência de campos magnéticos, é estudado sob a luz da teoria do grupo de renormalização. Os modelos são resolvidos exatamente pelo método da matriz de transferência e expressões para as funções de correlação e susceptibilidade (a campo zero) por si tio são também calculadas. As transformações do grupo de renormalização são efetuadas através de um traço parcial na função de partição, obtendo- se um conjunto de relações de recorrência que podem ser escritas de maneira simples para qualquer valor inteiro do fator de reescala espacial, mediante o uso de campos de escala convenientes. Tirando vantagem de um ponto fixo inteiramente atrativo, calculamos uma expressão para a energia livre por sitio, exata para T ¢ O. Analisamos o comportamento de nossos modelos no espaço de parâmetros, onde identificamos em particular as ~s ferro e antiferromagnéticas. O problema de correções às previsões de escala em termos de campos de escala não lineares é discutido. Aventamos também a possibilidade de calcular os auto valores da matriz de transferência através dos campos não lineares / In this work we study the criticai behaviour of one dimensional Z(N) spin systems in zero magnetic fields, using the approach of the renormalization group (RG) theory. The models are solved by the transfer matrix method and expressions for the correlation functions and zero field susceptibility per site are found. The RG transformations are carried out via a partial trace over the partition function and one obtains a set of recursion relations which, with the use of a convenient set of scaling fields, are written out in a simple manner for any integer value of the spatial rescaling factor. Using a totaly attractive fixed point we calculate an expression for the free energy per site, valid exactly for non zero values of the temperature. We analyse the behaviour of our models in the space of parameters, identifying in particular ferro and antiferromagnetic regions. The problem of corrections to scaling in terms of nonlinear scaling fields is discussed and a possibility of finding the eigen values of the transfer matrix from such fields is contemplated

by Siu Wing Hon. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references. / by Siu Wing Hon. / Acknowledgement --- p.i / Abstract --- p.ii / Publication List --- p.iv / Chapter 1 --- Introduction --- p.1 / References --- p.6 / Chapter 2 --- Real-Space Renormalization Group (RG) Theory in Electrical Conduction --- p.9 / Chapter 2.1 --- Scale Invariance --- p.10 / Chapter 2.2 --- Critical Exponents --- p.14 / Chapter 2.3 --- Alternative View-Point of RG Theory --- p.15 / References --- p.18 / Chapter 3 --- "Weakly Nonlinear Composites: Critical Behavior, Flicker Noise and Crossover Behavior" --- p.19 / Chapter 3.1 --- Introduction --- p.19 / Chapter 3.2 --- Formalism --- p.20 / Chapter 3.3 --- Critical Exponents by RG Method --- p.22 / Chapter 3.4 --- Connection to Flicker Noise Problem and Crossover Behavior --- p.25 / Chapter 3.5 --- Discussions and Conclusions --- p.27 / References --- p.28 / Chapter 4 --- Critical Behavior of Strongly Nonlinear Composites --- p.30 / Chapter 4.1 --- Introduction --- p.30 / Chapter 4.2 --- Formalism --- p.31 / Chapter 4.3 --- Applications of RG Theory to Strongly Nonlinear Composites --- p.32 / Chapter 4.4 --- Connections with Links-Nodes-Blobs picture --- p.36 / Chapter 4.5 --- Discussions and Conclusions --- p.39 / References --- p.41 / Chapter 5 --- "Enhanced Nonlinear Response of Superconductor-Normal-conductor Composite Wires, Strips and Rods" --- p.43 / Chapter 5.1 --- Introduction --- p.43 / Chapter 5.2 --- Formalism --- p.45 / Chapter 5.3 --- Linear and Nonlinear Responses of Composite Wires --- p.46 / Chapter 5.4 --- Linear and Nonlinear Response of Composite Strips --- p.49 / Chapter 5.5 --- Linear and Nonlinear Responses of Composite Rods --- p.56 / Chapter 5.6 --- Scaling Behaviors --- p.59 / Chapter 5.7 --- Discussions and Conclusions --- p.63 / References --- p.64 / Chapter 6 --- Renormalized Effective Medium Theory for Weakly Nonlinear Composites --- p.66 / Chapter 6.1 --- Introduction --- p.66 / Chapter 6.2 --- Weakly Nonlinear Conductance Network --- p.69 / Chapter 6.3 --- Simulation --- p.70 / Chapter 6.4 --- Effective Medium Approximation --- p.76 / Chapter 6.5 --- Renormalized Effective Medium Approximation --- p.79 / Chapter 6.6 --- Discussion and Conclusions --- p.81 / References --- p.83 / Chapter 7 --- Conclusions --- p.86 / Chapter A --- Derivation of Voltage-Summation Formulas --- p.88 / Chapter B --- Effective Linear and Nonlinear Response of 2 x 2 cell --- p.92 / Chapter C --- Duality Symmetry in 2D Network --- p.97 / Chapter D --- Derivation of Effective-Medium Approximation --- p.99