Geometrical Formulation of Renormalization-Group Method as an Asymptotic Analysis: With Applications to Derivation of Causal Fluid Dynamics

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.

Table of Contents:
Notion of Effective Theories in Physical Sciences.- Divergence and Secular Term in the Perturbation Series of Ordinary Differential Equations.- Traditional Resummation Methods.- Elementary Introduction of the RG method in Terms of the Notion of Envelopes.- Ei-Fujii-Kunihiro Formulation and Relation to Kuramoto’s reduction scheme.- Relation to the RG Theory in Quantum Field Theory.- Resummation of the Perturbation Series in Quantum Methods.- Illustrative Examples.- Slow Dynamics Around Critical Point in Bifurcation Phenomena.- Dynamical Reduction of A Generic Non-linear Evolution Equation with Semi-simple Linear Operator.- A Generic Case when the Linear Operator Has a Jordan-cell Structure.- Dynamical Reduction of Difference Equations.- Slow Dynamics in Some Partial Differential Equations.- Some Mathematical Formulae.- Dynamical Reduction of Kinetic Equations.- Relativistic First-Order Fluid Dynamic Equation.- Doublet Scheme and its Applications.- Relativistic Causal Fluid dynamic Equation.- Numerical Analysis of Transport Coefficients and Relaxation Times.- Reactive Multi-component Systems.- Non-relativistic Case and Application to Cold Atoms.- Summary and Future Prospects.