Description
Book SynopsisThis 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.
Table of ContentsPART 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
