Geometrical Formulation of Renormalization-Group Method as an Asymptotic Analysis: With Applications to Derivation of Causal Fluid Dynamics

Description

This book presents a comprehensive account of the  renormalization-group (RG) method and its extension, the doublet scheme, in a geometrical point of view. It extract long timescale macroscopic/mesoscopic dynamics from microscopic equations in an intuitively understandable way rather than in a mathematically rigorous manner and introduces readers to a mathematically elementary, but useful and widely applicable technique for analyzing asymptotic solutions in mathematical models of nature. The book begins with the basic notion of the RG theory, including its connection with the separation of scales. Then it formulates the RG method as a construction method of envelopes of the naive perturbative solutions containing secular terms, and then demonstrates the formulation in various types of evolution equations. Lastly, it describes successful physical examples, such as stochastic and transport phenomena including second-order relativistic as well as nonrelativistic fluid dynamics with causality and  transport phenomena in cold atoms, with extensive numerical expositions of transport coefficients and relaxation times. Requiring only an undergraduate-level understanding of physics and mathematics, the book clearly describes the notions and mathematical techniques with a wealth of examples. It is a unique and can be enlightening resource for readers who feel mystified by renormalization theory in quantum field theory. Preface Acknowledgements Contents Acronyms 1 Introduction: Reduction of Dynamics, Notion of Effective Theories, and Renormalization Groups 1.1 Reduction of Dynamics of a Simple Equation and the Notion of Effective Theory 1.2 Notion of Effective Theories and Renormalization Group in Physical Sciences 1.3 The Renormalization Group Method in Global and Asymptotic Analysis 1.4 Derivation of Stochastic Equations and Fluid Dynamic Limit of Boltzmann Equation Part I Geometrical Formulation of Renormalization-Group Method and Its Extention for Global and Asymptotic Analysis with Examples 2 Naïve Perturbation Method for Solving Ordinary Differential Equations and Notion of Secular Terms 2.1 Introduction 2.2 A Simple Example: Damped Oscillator 2.3 Motion of a Particle in an Anharmonic Potential: Duffing Equation 2.3.1 Exact Solution of Duffing Equation 2.3.2 Naïve Perturbation Theory Applied to Duffing Equation 2.4 van der Pol Equation 2.5 Concluding Remarks 3 Conventional Resummation Methods for Differential Equations 3.1 Introduction 3.2 Solvability Condition of Linear Equations and Appearance … 3.3 Solvability Condition of Linear Differential Equations with Hermitian Operator 3.4 Lindstedt-Poincaré Method: Duffing Equation Revisited 3.5 Krylov-Bogoliubov-Mitropolsky Method 3.5.1 Generalities 3.5.2 Damped Oscillator 3.5.3 Duffing Equation 3.5.4 The van der Pol Equation 3.6 Multiple-Scale Method 3.6.1 Duffing Equation 3.6.2 Bifurcation in the Lorenz Model 4 Renormalization Group Method for Global Analysis: A Geometrical Formulation and Simple Examples 4.1 Introduction 4.2 Classical Theory of Envelopes and Its Adaptation for Global Analysis … 4.2.1 Envelope Curve in Two-Dimensional Space 4.2.2 Envelope Curves/Trajectories in n-Dimensional Space 4.2.3 Adaptation of the Envelope Theory in a Form Applicable to Dynamical Equations 4.3 Damped Oscillator in RG Method 4.3.1 Treatment as a Second-Order Differential Equation for Single Dependent Variable x 4.3.2 Treatment of Damped Oscillator as a System of First-Order Equations 4.4 RG/E Analysis of a Boundary-Layer Problem Without … 4.5 The van der Pol Equation in RG Method 4.6 Jump Phenomenon in Forced Duffing Equation 4.7 Proof of a Global Validness of the Envelope Function … 5 RG Method for Asymptotic Analysis with Reduction of Dynamics: An Elementary Construction of Attractive/Invariant Manifold 5.1 Introduction 5.2 Non-perturbative RG/E Equation for Reduction of Dynamics 5.3 Perturbative RG/E Equation 5.4 Invariant/Attractive Manifold and Renormalizability 5.5 Example I: A Generic System with the Linear … 5.5.1 Generic Model that Admits an Attractive/Invariant Manifold 5.5.2 First-Order Analysis 5.5.3 Second-Order Analysis 5.6 Example II: The Case With the Generic System … 5.6.1 Preliminaries for a Linear Operator with Two-Dimensional Jordan Cell 5.6.2 Perturbative Construction of the Attractive/Invariant Manifold 5.7 Concluding Remarks 6 Miscellaneous Examples of Reduction of Dynamics 6.1 Introduction 6.2 RG/E Analysis of a Bifurcation in The Lorenz Model 6.3 RG/E Analysis of the Brusselator with a Diffusion … 6.3.1 The Model Equation 6.3.2 Linear Stability Analysis 6.3.3 Perturbative Expansion with the Diffusion Term 6.3.4 The Reduced Dynamics and Invariant Manifold 6.4 Example with a Jordan Cell I: Extended Takens Equation 6.5 Example with a Jordan Cell II … 7 RG Method Applied to Stochastic Equations 7.1 Introduction 7.2 Langevin Equation: Simple Examples 7.3 RG/E Derivation of Fokker-Planck Equation from a Generic Langevin Equation 7.3.1 A Generic Langevin Equation with a Multiplicative Noise 7.3.2 The RG/E Derivation of the Fokker-Planck Equation 7.4 Adiabatic Elimination of Fast Variables in Fokker-Planck Equation 7.4.1 Perturbative Expansion in the Case of a Strong Friction 7.4.2 The Eigenvalue Problem of 0 7.4.3 The Solution to the Perturbative Equations 7.4.4 Application of the RG/E Equation 7.4.5 Smoluchowski Equation with Corrections 7.4.6 Simple Examples 7.5 Concluding Remarks 8 RG/E Derivation of Dissipative Fluid Dynamics from Classical Non-relativistic Boltzmann Equation 8.1 Introduction: Fluid Dynamics as Asymptotic Slow Dynamics of Boltzmann Equation 8.2 Basics of Non-relativistic Classical Boltzmann Equation 8.3 Asymptotic Analysis and Dynamical Reduction of Boltzmann Equation in RG Method 8.3.1 Preliminaries and Set Up 8.3.2 Analysis of Unperturbed Solution 8.3.3 First-Order Equation 8.3.4 Spectral Analysis of Collision Operator L 8.3.5 Solution to First-Order Equation 8.3.6 Second-Order Solution 8.3.7 Application of RG/E Equation and Construction of a Global Solution 8.4 Reduction of RG/E Equation To fluid Dynamic Equation … 8.5 Summary 9 A General Theory for Constructing Mesoscopic Dynamics: Doublet Scheme in RG Method 9.1 Introduction 9.2 General Formulation 9.2.1 Preliminaries 9.2.2 Construction of the Approximate Solution Around Arbitrary Time 9.2.3 First-Order Solution and Introduction of the Doublet Scheme 9.2.4 Second-Order Analysis 9.2.5 RG Improvement of Perturbative Expansion 9.2.6 Reduction of RG/E Equation to Simpler Form 9.2.7 Transition of the Mesoscopic Dynamics to the Slow Dynamics in Asymptotic Regime 9.3 An Example: Mesoscopic Dynamics of the Lorenz Model Part II RG/E Derivation of Second-Order Relativistic and Non-relativistic Dissipative Fluid Dynamics 10 Introduction to Relativistic Dissipative Fluid Dynamics and Its Derivation from Relativistic Boltzmann Equation by Chapman-Enskog and Fourteen-Moment Methods 10.1 Basics of Relativistic Dissipative Fluid Dynamics 10.2 Basics of Relativistic Boltzmann Equation with Quantum Statistics 10.3 Review of Conventional Methods to Derive Relativistic Dissipative … 10.3.1 Chapman-Enskog Method 10.3.2 Israel-Stewart Fourteen-Moment Method 10.3.3 Concluding Remarks 11 RG/E Derivation of Relativistic First-Order Fluid Dynamics 11.1 Introduction 11.2 Preliminaries 11.3 Introduction and Properties of Macroscopic Frame Vector 11.4 Perturbative Solution to Relativistic Boltzmann Equation … 11.4.1 Construction of Approximate Solution Around Arbitrary Time in the Asymptotic Region 11.5 First-Order Fluid Dynamic Equation and Microscopic … 11.6 Properties of First-Order Fluid Dynamic Equation 11.6.1 Uniqueness of Landau-Lifshitz Energy Frame 11.6.2 Generic Stability 12 RG/E Derivation of Relativistic Second-Order Fluid Dynamics 12.1 Introduction 12.2 Preliminaries 12.3 First-Order Solution in the Doublet Scheme 12.4 Second-Order Solution in the Doublet Scheme 12.5 Construction of the Distribution Function Valid in a Global … 12.5.1 RG/E Equation 12.5.2 Reduction of RG/E Equation to a Simpler Form 12.6 Derivation of the Second-Order Fluid Dynamic Equation 12.6.1 Balance Equations and Local Rest Frame of Flow Velocity 12.6.2 Relaxation Equations and Microscopic Representations of Transport Coefficients and Relaxation Times 12.6.3 Derivation of Relaxation Equations 12.7 Properties of Second-Order Fluid Dynamic Equation 12.7.1 Stability 12.7.2 Causality 13 Appendices for Chaps.10, 11, and 12 13.1 Foundation of the Symmetrized Inner Product defined by Eqs. (11.31摥映數爠eflinkeq:ChapAsps4sps2sps00711.3111) and (12.18摥映數爠eflinkeq:relspssecondspsdefspsinnerspsprod12.1812) 13.2 Derivation of Eqs. (10.65摥映數爠eflinkeq:E1spspreliminary10.6510)–(10.67摥映數爠eflinkeq:E3spspreliminary10.6710) 13.3 Detailed Derivation of Explicit Form of µα1 13.4 Computation of Q0F0 in Eq. (12.38摥映數爠eflinkeq:varphi112.3812) 13.5 Proof of Vanishing of Inner Product Between Collision Invariants and B 14 Demonstration of Numerical Calculations of Transport Coefficients and Relaxation Times: Typical Three Models 14.1 Introduction 14.2 Linearized Transport Equations and Solution Method 14.2.1 Reduction of the Integrals in the Linearized Transport Equations in Terms of the Differential Cross Section 14.2.2 Explicit Forms of Kernel Functions 14.2.3 Linearized Transport Equations as Integral Equations 14.2.4 Direct Matrix-Inversion Method Based on Discretization 14.3 Numerical Demonstration: Transport Coefficients and Relaxation … 14.3.1 Accuracy and Efficiency of the Numerical Method: Discretization Errors and Convergence 14.3.2 Numerical Results for Classical, Fermionic, and Bosonic Systems: Comparison of RG and Israel–Stewart Fourteen Moment Method 15 RG/E Derivation of Reactive-Multi-component Relativistic Fluid Dynamics 15.1 Introduction 15.2 Boltzmann Equation in Relativistic Reactive-Multi-component Systems 15.2.1 Collision Invariants and Conservation Laws 15.2.2 Entropy Current 15.3 Reduction of Boltzmann Equation to Reactive-Multi-component Fluid Dynamics 15.3.1 Solving Perturbative Equations 15.3.2 Computation of L-1Q0F(0) 15.3.3 RG Improvement by Envelope Equation 15.3.4 Derivation of Relaxation Equations and Transport Coefficients 15.4 Properties of Derived Fluid Dynamic Equations 15.4.1 Positivity of Transport Coefficients 15.4.2 Onsager's Reciprocal Relation 15.4.3 Positivity of Entropy Production Rate 16 RG/E Derivation of Non-relativistic Second-Order Fluid Dynamics and Application to Fermionic Atomic Gases 16.1 Derivation of Second-Order Fluid Dynamics in Non-relativistic Systems 16.1.1 Non-relativistic Boltzmann Equation 16.1.2 Derivation of Navier–Stokes Equation 16.1.3 Derivation of Second-Order Non-relativistic Fluid Dynamic Equation 16.2 Transport Coefficients and Relaxation Times in Non-relativistic Fluid Dynamics 16.2.1 Analytic Reduction of Transport Coefficients and Relaxation Times for Numerical Studies 16.2.2 Numerical Method 16.2.3 Shear Viscosity and Heat Conductivity 16.2.4 Viscous-Relaxation Time Appendix References Index