We review the use of an exact renormalization group equation in quantum field theory and statistical physics. It describes the dependence of the free energy on an infrared cutoff for the quantum or thermal fluctuations. Non-perturbative solutions follow from approximations to the general form of the coarse-grained free energy or effective average action. They interpolate between the microphysical laws and the complex macroscopic phenomena. Our approach yields a simple unified description for O(N )-symmetric scalar models in two, three or four dimensions, covering in particular the critical phenomena for the second-order phase transitions, including the Kosterlitz-Thouless transition and the critical behavior of polymer chains. We compute the aspects of the critical equation of state which are universal for a large variety of physical systems and establish a direct connection between microphysical and critical quantities for a liquid-gas transition. Universal features of first-order phase transitions are studied in the context of scalar matrix models. We show that the quantitative treatment of coarse graining is essential for a detailed estimate of the nucleation rate. We discuss quantum statistics in thermal equilibrium or thermal quantum field theory with fermions and bosons and we describe the high temperature symmetry restoration in quantum field theories with spontaneous symmetry breaking. In particular we explore chiral symmetry breaking and the high temperature or high density chiral phase transition in quantum chromodynamics using models with effective four-fermion interactions.This work is dedicated to the 60th birthday of Franz Wegner. *

Baryogenesis by heavy-neutrino decay and sphaleron reprocessing of both baryon and lepton number is reconsidered, paying special attention to the flavour structure of the general evolution equations and developing an approximate but sufficiently accurate analytic solution to the prototype evolution equation. Two different models of neutrino masses are examined, based on an Abelian U(1) or a non-Abelian U(2) family symmetry. We show that a consistent picture of baryogenesis can emerge in both cases, although with significant differences. Introduction to the revised versionThis paper contained two different original results:1 We discussed how to include flavour effects in computations of thermal leptogenesis, showing that one must write Boltzmann equations for a 3 × 3 density matrix that describes how the lepton asymmetry is shared between the 3 flavours.2 We gave approximated semi-analytical solutions to the standard Boltzmann equations and presented numerical results in a way that covers all the parameter space systematically.Part 1 remains unchanged. Part 2 has been revised using the corrected Boltzmann equations of [1]. We do not list the changes we made. Final expressions are now simpler.