These introductory notes are about functional renormalization group equations and some of their applications. It is emphasised that the applicability of this method extends well beyond critical systems, it actually provides us a general purpose algorithm to solve strongly coupled quantum eld theories. The renormalization group equation of F. Wegner and A. Houghton is shown to resum the loop-expansion. Another version, due to J. Polchinski, is obtained by the method of collective coordinates and can be used for the resummation of the perturbation series. The genuinely non-perturbative evolution equation is obtained by a manner reminiscent of the Schwinger-Dyson equations. Two variants of this scheme are presented where the scale which determines the order of the successive elimination of the modes is extracted from external and internal spaces. The renormalization of composite operators is discussed brie®y as an alternative way to arrive at the renormalization group equation. The scaling laws and xed points are considered from local and global points of view. Instability induced renormalization and new scaling laws are shown to occur in the symmetry broken phase of the scalar theory. The ®attening of the e¬ective potential of a compact variable is demonstrated in case of the sine-Gordon model. Finally, a manifestly gauge invariant evolution equation is given for QED.

With the goal of deriving dissipative hydrodynamics from an action, we study classical actions for open systems, which follow from the generic structure of effective actions in the Schwinger-Keldysh Closed-Time-Path formalism with two time axes and a doubling of degrees of freedom. The central structural feature of such effective actions is the coupling between degrees of freedom on the two time axes. This reflects the fact that from an effective field theory point of view, dissipation is the loss of energy of the low-energy hydrodynamical degrees of freedom to the integrated-out, UV degrees of freedom of the environment. The dynamics of only the hydrodynamical modes may therefore not posses a conserved stress-energy tensor. After a general discussion of the CTP effective actions, we use the variational principle to derive the energy-momentum balance equation for a dissipative fluid from an effective Goldstone action of the long-range hydrodynamical modes. Despite the absence of conserved energy and momentum, we show that we can construct the first-order dissipative stressenergy tensor and derive the Navier-Stokes equations near hydrodynamical equilibrium. The shear viscosity is shown to vanish in the classical theory under consideration, while the bulk viscosity is determined by the form of the effective action. We also discuss the thermodynamics of the system and analyse the entropy production. CONTENTS

It is pointed out that models with condensates have nontrivial renormalization group flow on the tree level. The infinitesimal form of the tree level renormalization group equation is obtained and solved numerically for the φ 4 model in the symmetry broken phase. We find an attractive infrared fixed point that eliminates the metastable region and reproduces the Maxwell construction.We have two systematic nonperturbative methods to handle multi-particle or quantum systems, the saddle point approximation and the renormalization group. Our goal in this letter is to combine these two apparently independent approximation methods in order to obtain a better understanding of the instabilities and first order phase transitions.We start with the path integral,

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