Kunihiro, Teiji, Kikuchi, Yuta, Tsumura, Kyosuke 　著

内容

目次

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 Operator 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 17.1 General Formulation 17.2 Lorentz Model Revisited 18 Relativistic Causal Fluid dynamic Equation 19 Numerical Analysis of Transport Coefficients and Relaxation Times 20 Reactive-Multi-component Systems 21 Non-relativistic Case and Application to Cold Atoms PART IV Summary and Future Prospect 22 Summary and Future Prospects