This 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.

Teiji Kunihiro is Professor Emeritus at Kyoto University in Japan and specializes in research in nuclear and hadron physics theory and mathematical physics. He received his Doctor of Science in Physics from Kyoto University in 1981. After serving as Associate Professor and Professor at Ryukoku University, he was appointed as Professor at the Yukawa Institute for Theoretical Physics, Kyoto University in 2000, and was Vice Director of the institute from 2006 to 2007, before moving to the Department of Physics in 2008.

Yuta Kikuchi is a Goldhaber Fellow at Brookhaven National Laboratory in the USA at the completion of the present book, and appointed to a scientist at Cambridge Quantum Computing starting in 2022. He received his Doctor of Science in Physics from Kyoto University in 2018. He was awarded the Research Fellowship for Young Scientists by Japan Society for the Promotion of Science (JSPS) in 2015, and the Goldhaber distinguished fellowship by Brookhaven National Laboratory in 2020. He currently focuses on designing quantum algorithms for near-term quantum computing.

Kyosuke Tsumura is Primary Research Scientist at the Analysis Technology Center, Fujifilm Corporation in Japan. He joined the Analysis Technology Center as a Researcher in 2006 and was promoted to current position. He received his Doctor of Science in Physics from Kyoto University in 2013. He is Leader of a project developing a novel computational method for efficient drug design.

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

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