In this thesis we investigate properties of equilibrium and non-equilibrium systems by means of renormalization group (RG) analysis. In the study of the d−dimensional Coulomb gas we have formulated a continuum model from the underlying hyper-cubic lattice and employed the irreducible differential formulation of the Wilson RG. We have identified a Thouless-Kosterletz transition in d = 2 and found no non-trivial fixed points for d > 2. As an example of a non-equilibrium system, we have investigated properties of quasi-neutral plasmas which are governed by stochastic magnetohydrodynamic (MHD) equations. The present method is based upon the Martin-Siggia-Rose field-theory formulation of stochastic dynamics. We develop a diagrammatic representation for the theory and carry out a momentum-shell RG of Wilson-Kadanoff type. An infinite set of diagrams is identified which are marginal in the RG sense. We have shown, in accordance with previous literature, that the same problem arises for the randomly-forced Navier-Stokes equation. The problem of marginal variables can be suppressed by working near equilibrium, where stochastic forcing represents thermal fluctuations.In a similar manner we have considered regimes when MHD equations are subject either to kinetic or magnetic forcing only. In such models the macroscopic limit can be taken such that all marginal terms are irrelevant and the dynamics is governed by linear equations. Furthermore, non-trivial fixed points are identified in such regimes and limiting values of either kinematic viscosity or magnetic diffusivity are derived. A consistent description of MHD dynamics far from equilibrium is still absent. We highlight some of the aspects of the functional integral formulation with regards to the symmetries of the system and propose possible ways in which the system can be studied non-pertubatively. 1

The properties of quasi-neutral fluids governed by stochastic magnetohydrodynamic (MHD) equations are investigated with a method based on the diagrammatic representation of the Martin-Siggia-Rose field-theoretic formulation of stochastic dynamics. We carry out a Wilson-Kadanoff momentum-shell renormalization-group (RG) analysis and identify an infinite set of diagrams which are marginal in the RG sense. We show, in agreement with previous work, that the same problem arises for the randomly stirred Navier-Stokes equation. The marginal variables can be suppressed by working near equilibrium, where stochastic forcing represents thermal fluctuations. In a similar manner, we consider regimes where the MHD equations are subject either to kinetic or magnetic forcing only. In such models the macroscopic limit can be taken such that all marginal terms are irrelevant and the dynamics is governed by linear equations. Non-trivial fixed points are identified and limiting values of either the kinematic viscosity or magnetic diffusivity are derived. A consistent description of MHD far from equilibrium is still absent, though we highlight several aspects of the functional integral formulation with regard to the symmetries of the system and propose possible ways to study this system non-perturbatively.

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