Geometrical Formulation of Renormalization-Group Method as an Asymptotic Analysis

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          Lindstedt's Method

3.3          Krylov-Bogoliubov-Mitropolsky's 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 Kuramoto's 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

10       Dynamical Reduction of A Generic Non-linear Evolution Equation with Semi-simple Linear Operator11       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

17.1 

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