In this thesis, we focus on the steady state properties of two systems which are genuinely out of equilibrium. The first project is an application of dynamic field theory to a specific non equilibrium critical phenomenon, while the second project involves both simulations and analytical calculations. The methods of field theory are used on both these projects. In the first part of this thesis, we investigate a generalization of the well-known field theory for directed percolation (DP). The DP theory is known to describe an evolving population, near extinction. We have coupled this evolving population to an environment with its own nontrivial spatio-temporal dynamics. Here, we consider the special case where the environment follows a simple relaxational (model A) dynamics. We find two marginal couplings with upper critical dimension of four, which couple the two theories in a nontrivial way. While the Wilson-Fisher fixed point remains completely unaffected, a mismatch of time scales destabilizes the usual DP fixed point. Some open questions and future work remain. In the second project, we focus on a simple particle transport model far from equilibrium, namely, the totally asymmetric simple exclusion process (TASEP). While its stationary properties are well studied, many of its dynamic features remain unexplored. Here, we focus on the power spectrum of the total particle occupancy in the system. This quantity exhibits unexpected oscillations in the low density phase. Using standard Monte Carlo simulations and analytic calculations, we probe the dependence of these oscillations on boundary effects, the system size, and the overall particle density. Our simulations are fitted to the predictions of a linearized theory for the fluctuation of the particle density. Two of the fit parameters, namely the diffusion constant and the noise strength, deviate from their naive bare values [6]. In particular, the former increases significantly with the system size. Since this behavior can only be caused by nonlinear effects, we calculate the lowest order corrections in perturbation theory. Several open questions and future work are discussed. / Ph. D.

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Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-06-27T14:21:12Z No. of bitstreams: 1 baltazarjonasribeiromorais.pdf: 448066 bytes, checksum: 970abe2e7caa96e6fe2212d8828ade5e (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-08-07T21:05:33Z (GMT) No. of bitstreams: 1 baltazarjonasribeiromorais.pdf: 448066 bytes, checksum: 970abe2e7caa96e6fe2212d8828ade5e (MD5) / Made available in DSpace on 2017-08-07T21:05:33Z (GMT). No. of bitstreams: 1 baltazarjonasribeiromorais.pdf: 448066 bytes, checksum: 970abe2e7caa96e6fe2212d8828ade5e (MD5) Previous issue date: 2010-04-18 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Na primeira parte deste trabalho, nós obtemos o potencial efetivo para um campo escalar no espaço-tempo curvo, usando dois tipos de regularização cut-off covariante. O primeiro deles é baseado na representação de momento local e coordenandas normais de Riemann, enquanto que o segundo é baseado na representação de tempo próprio de Fock-Scwinger-DeWitt. Nós mostramos, para o exemplo de um campo escalar com auto interação, que ambos os métodos produzem resultados iguais para as divergências. No entanto, o primeiro método fornece informações mais detalhadas com respeito à parte finita. Além disso, nós calculamos também a contribuição, a um loop, de um férmion massivo. Finalmente, discutimos as equações do grupo de renormalização, bem como sua aplicação para teorias de multi-massa. Na segunda parte deste trabalho, usamos a equação para o potencial efetivo previamente obtida e estudamos o tensor energia-momento renormalizado de vácuo. Embora este tensor tenha sido profundamente estudado pela comunidade científica por décadas, notava-se a presença de alguns aspectos duvidosos. Realizamos um estudo sobre a implementação do momento cut-off de maneira covariante. Uma parte qualitativamente nova é o cálculo do tensor energia-momento, no caso da quebra espontânea de simetria. Apesar da complexidade do assunto, mostramos que o resultado final satisfaz as leis de conservação e isso permite controlar bem o resultado final. / In the fist part of this work, we consider derivation of the effective potential for a scalar field in curved space-time within the physical regularization scheme, using two sorts of covariant cut-off regularizations. The first one is based on the local momentum representation and Riemann normal coordinates and the second is operatorial regularization, based on the Fock-Scwinger-DeWitt proper-time representation. We show, on the example of a self-interacting scalar field, that these two methods produce equal results for divergences, but the first one gives more detailed information about the finite part. Furthermore, we calculate the contribution from a massive fermion loop and discuss renormalization group equations and their interpretation for the multi-mass theories. In the second part of the work, we study the renormalized energy-momentum tensor of vacuum. This tensor has been deeply explored many years ago. The main result of these studies was that such a tensor should satisfy the conservation laws which reflects the covariance of the theory in the presence of loop corrections. In view of this general result we address two important questions, namely how to implement the momentum cut-off in a covariant way and whether this general result holds in the theory with Spontaneous Symmetry Breaking. In the last case some new interesting details arise and although the calculations are more involved we show that the final result satisfies the conservation laws.

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We present and explore a new formulation of perturbative QCD based not on the renormalised coupling but on the dimensional transmutation parameter of the theory and the property of asymptotic scaling. The approach yields a continued function, the iterated function being that involved in the solution of the two-loop β-function equation. In the so-called large-b limit the continued function reduces to a continued fraction and the successive approximants are diagonal Padé approximants. We investigate numerically the convergence of successive approximants using the leading-b approximation, motivated by renormalons, to model the all-orders result. We consider the Adler D-function of vacuum polarisation, the Polarised Bjorken and Gross-Llewellyn Smith sum rules, the (unpolarised) Bjorken sum rule, and the Minkowskian quantities R(_r) and the R-ratio of e(^+)e(^-) annihilation. In contrast to diagonal Fade approximants the truncated continued function method gives remarkably stable large-order approximants in cases where infra-red renormalon effects are important. We also use the new approach to determine the QCD fundamental parameters from the R(_r) and the R-ratio measurements, where we find Ā(^(3))(_MS)=516±48 MeV (which yields a(_s)(µ=m(_r))=0.360(^+0.021)(_=0.020)), and Ā(^(5))(_MS)=299(^+6)(_-7) MeV (which yields a(_s)(µ=m(_zo)=0.1218±0.0004), respectively. The evolution of the former value to the m(_zo) energy results in a(_s)(µ= m(_zo)) = 0.123 ± 0.002. These values are in line with other determinations available in the literature. We implement the Complete Renormalisation Group Improvement (CORGI) scheme throughout all the calculations. We report on how the mathematical concept of Stieltjes series can be used to assess the convergence of Padé approximants of perturbative series. We find that the combinations of UV renormalons which occur in perturbative QCD may or may not be Stieltjes series depending on the renormalisation scheme used.

Renormalization group analysis is a useful tool for studying critical behaviour of stochastic systems. In this thesis, field-theoretic renormalization group will be applied to the scalar model representing directed percolation, known as Gribov model, in presence of the random velocity field. Turbulent mixing will be modelled by the compressible form of stochastic Navier-Stokes equation where the compressibility is described by an additional field related to the density. The task will be to find corresponding scaling properties.

This thesis begins with the fundamentals of conformal field theory in three dimensions. The general properties of the conformal bootstrap are then reviewed. The three-dimensional Ising model is presented from the perspective of the renormalization group, after which the conformal field theory aspect at the critical point is discussed. Finally, the bootstrap programme is applied to the three-dimensional Ising model using numerical techniques, and the results analysed.

We demonstrate a renormalisation group improved formulation of QCD perturbation theory. At next-to-leading order (NLO) and beyond this permits a direct extraction of the QCD dimensional transmutation parameter, A(_ms) that typifies the one parameter freedom of the theory in the limit of massless quarks. We apply this to a variety of experimental data on e(^+)e" jet observables at NLO. We take into consideration data from PETRA, PEP, TRISTAN, SLC and LEP 1 and 2. In this procedure there is no need to mention, let alone to arbitrarily vary, the unphysical renormalization scale µ, and one avoids the spurious and meaningless "theoretical error" associated with standard a(_8) determinations. An attempt is made to estimate the importance of uncalculated next-to-NLO and higher order perturbative corrections, and power corrections, by studying the scatter in the values of ∆(_MS) obtained for different observables. We also consider large infrared logarithm resummations in these jet observables and present results for the particular cases of the four-jet rate to a next-to-leading logarithm approximation and the distributions for the four-jet variables, "light hemisphere mass" and "narrow jet broadening" to a next-to-next-to-leading logarithm approximation in the perturbative expansion. We apply a simple power correction to these variables and obtain remarkably good fits to the data.