Geometrical Formulation of Renormalization-Group Method as an Asymptotic Analysis: With Applications to Derivation of Causal Fluid Dynamics, Kunihiro Teiji, Kikuchi Yuta, Tsumura Kyosuke

PART I Introduction to Renormalization Group (RG) Method

1 Introduction: Notion of Effective Theories in Physical Sciences

2 Divergence and Secular Term in the Perturbation Series of Ordinary Differential Equations

3 Traditional Resummation Methods

3.1 Reductive Perturbation Theory

3.2 Lindstedts Method

3.3 Krylov-Bogoliubov-Mitropolskys Method for Nonlinear Oscillators

4 Elementary Introduction of the RG method in Terms of the Notion of Envelopes

4.1 Notion of Envelopes of Family of Curves Adapted for a Geometrical Formulation of the RG Method

4.2 Elementary Examples: Damped Oscillator and Boundary-Layer Problem

5 General Formulation and Foundation of the RG Method: Ei-Fujii-Kunihiro

Formulation and Relation to Kuramotos reduction scheme

6 Relation to the RG Theory in Quantum Field Theory

7 Resummation of the Perturbation Series in Quantum Methods

PART II Extraction of Slow Dynamics Described by Differential and Difference Equations

8 Illustrative Examples

8.1 Rayleigh/Van der Pol equation and jumping phenomena

8.2 Lotka-Volterra Equation

8.3 Lorents Model

9 Slow Dynamics Around Critical Point in Bifurcation Phenomena

11 A Generic Case when the Linear Operator Has a Jordan-cell Structure

12 Dynamical Reduction of Difference Equations (Maps)

13 Slow Dynamics in Some Partial Differential Equations

13.1 Dissipative One-Dimensional Hyperbolic Equation

13.2 Swift-Hohenberg Equation

13.3 Damped Kuramoto-Shivashinsky Equation

13.4 Diffusion in Porus Medium --- Barrenblatt Equation

14 Appendix: Some Mathematical Formulae

PART III Application to Extracting Slow Dynamics of Non-equilibrium Phenomena

15 Dynamical Reduction of Kinetic Equations

15.1 Derivation of Boltzmann Equation from Liouville Equation

15.2 Derivation of the Fokker-Planck (FP) Equation from Langevin Equation

15.3 Adiabatic Elimination of Fast Variables in FP Equation: Derivation of Generalized Kramers Equations

16 Relativistic First-Order Fluid Dynamic Equation

17 Doublet Scheme and its Applications