Contents -- Normalization of Fourier integrals -- Notation for Chapters 1-4 and 7-10 -- Notation for Chapters 5, 6, and 11 -- I: WHAT IS RENORMALIZATION? -- 1 The bedrock problem: why we need renormalization methods -- 1.1 Some practical matters -- 1.2 Quasi-particles and renormalization -- 1.3 Solving the relevant differential equations by perturbation theory -- 1.4 Quantum field theory: a first look -- 1.5 What is the renormalization group? -- 1.6 Discrete dynamical systems: recursion relations and fixed points -- 1.7 Revision of statistical mechanics -- 1.8 Exercises -- 2 Easy applications of Renormalization Group to simple models -- 2.1 A one-dimensional magnet -- 2.2 Two-dimensional percolation -- 2.3 A two-dimensional magnet -- 2.4 Exercises -- 3 Mean-field theories for simple models -- 3.1 The Weiss theory of ferromagnetism -- 3.2 The Debye-Hückel theory of the electron gas -- 3.3 Macroscopic mean-field theory: the Landau model for phase transitions -- 3.4 Exercises -- II: RENORMALIZED PERTURBATION THEORIES -- 4 Perturbation theory using a control parameter -- 4.1 High-temperature expansions -- 4.2 Application to a one-dimensional magnet -- 4.3 Low-density expansions -- 4.4 Application to a "slightly imperfect" gas -- 4.5 The Van der Waals equation -- 4.6 The Debye-Hückel theory revisited -- 4.7 Exercises -- 5 Classical nonlinear systems driven by random noise -- 5.1 The generic equation of motion -- 5.2 The moment closure problem -- 5.3 The pair-correlation tensor -- 5.4 The zero-order "model" system -- 5.5 A toy version of the equation of motion -- 5.6 Perturbation expansion of the toy equation of motion -- 5.7 Renormalized transport equations for the correlation function -- 5.8 Reversion of power series -- 5.9 Formulation in Wyld diagrams -- 6 Application of renormalized perturbation theories to turbulence and related problems.

6.1 The real and idealized versions of the turbulence problem -- 6.2 Two turbulence theories: the DIA and LET equations -- 6.3 Theoretical results: free decay of turbulence -- 6.4 Theoretical results: stationary turbulence -- 6.5 Detailed energy balance in wave number -- 6.6 Application to other systems -- III: RENORMALIZATION GROUP (RG) -- 7 Setting the scene: critical phenomena -- 7.1 Some background material on critical phenomena -- 7.2 Theoretical models -- 7.3 Scaling behavior -- 7.4 Linear response theory -- 7.5 Serious mean-field theory -- 7.6 Mean-field critical exponents α, β, γ, and δ for the Ising model -- 7.7 The remaining mean-field critical exponents for the Ising model -- 7.8 Validity of mean-field theory -- 7.9 Upper critical dimension -- 7.10 Exercises -- 8 Real-space Renormalization Group -- 8.1 A general statement of the RG transformation -- 8.2 RG transformation of the Hamiltonian and its fixed points -- 8.3 Relations between critical exponents from RG -- 8.4 Applications of the linearized RGT -- 8.5 Exercises -- 9 Momentum-space Renormalization Group -- 9.1 Overview of this chapter -- 9.2 Statistical field theory -- 9.3 Renormalization group transformation in wave number space -- 9.4 Scaling dimension: anomalous and normal -- 9.5 Restatement of our objectives: numerical calculation of the critical exponents -- 9.6 The Gaussian zero-order model -- 9.7 Partition function for the Gaussian model -- 9.8 Correlation functions -- 9.9 Fixed points for the Gaussian model -- 9.10 Ginsburg-Landau (GL) theory -- 9.11 Exercises -- 10 Field-theoretic Renormalization Group -- 10.1 Preliminary remarks -- 10.2 The Ginsburg-Landau model as a quantum field theory -- 10.3 Infrared and ultraviolet divergences -- 10.4 Renormalization invariance -- 10.5 Perturbation theory in x-space -- 10.6 Perturbation expansion in x-space.

10.7 Perturbation expansion in k-space -- 10.8 The UV divergence and renormalization -- 10.9 The IR divergence and the ∊ -expansion -- 10.10 The pictorial significance of Feynman diagrams -- 11 Dynamical Renormalization Group applied to classical nonlinear system -- 11.1 The dynamical RG algorithm -- 11.2 Application to the Navier-Stokes equation -- 11.3 Application of RG to stirred fluid motion with asymptotic freedom as k → 0 -- 11.4 Relevance of RG to the large-eddy simulation of turbulence -- 11.5 The conditional average at large wave numbers -- 11.6 Application of RG to turbulence at large wave numbers -- IV: APPENDICES -- A: Statistical ensembles -- A.1 Statistical specification of the N-body assembly -- A.2 The basic postulates of equilibrium statistical mechanics -- A.3 Ensemble of assemblies in energy contact -- A.4 Entropy of an assembly in an ensemble -- A.5 Principle of maximum entropy -- A.6 Variational method for the most probable distribution -- B: From statistical mechanics to thermodynamics -- B.1 The canonical ensemble -- B.2 Overview and summary -- C: Exact solutions in one and two dimensions -- C.1 The one-dimensional Ising model -- C.2 Bond percolation in d = 2 -- D: Quantum treatment of the Hamiltonian N-body assembly -- D.1 The density matrix [sub(ρmn)] -- D.2 Properties of the density matrix -- D.3 Density operator for the canonical ensemble -- E: Generalization of the Bogoliubov variational method to a spatially varying magnetic field -- References -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Z.