Geometrical Formulation of Renormalization-Group Method as an Asymptotic Analysis: With Applications to Derivation of Causal Fluid Dynamics (Fundamental Theories of Physics, 206)

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