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About this book

This book is an introduction to a comprehensive and unified dynamic transition theory for dissipative systems and to applications of the theory to a range of problems in the nonlinear sciences. The main objectives of this book are to introduce a general principle of dynamic transitions for dissipative systems, to establish a systematic dynamic transition theory, and to explore the physical implications of applications of the theory to a range of problems in the nonlinear sciences. The basic philosophy of the theory is to search for a complete set of transition states, and the general principle states that dynamic transitions of all dissipative systems can be classified into three categories: continuous, catastrophic and random. The audience for this book includes advanced graduate students and researchers in mathematics and physics as well as in other related fields.

This second edition introduces a unified theory for topological phase transitions, provides a first-principle approach to statistical and quantum physics, and offers a microscopic mechanism of quantum condensates (Bose-Einstein condensation, superfluidity, and superconductivity).

Reviews of first edition:

“The goals of this interesting book are to derive a general principle of dynamic transitions for dissipative systems and to establish a systematic dynamic transition theory for a wide range of problems in the nonlinear sciences. … The intended audience for this book includes students and researchers working on nonlinear problems in physics, meteorology, oceanography, biology, chemistry, and the social sciences.” (Carlo Bianca, Mathematical Reviews, December, 2014)

“This is a clearly written book on numerous types of phase transitions taken in a broad sense when a dynamical dissipative system transforms from one physical state into another. … The book is a very useful literature not only for the professionals in the field of dynamic systems and phase transitions but also for graduate students due to its interdisciplinary coverage and state-of-the-art level.” (Vladimir Čadež, zbMATH, Vol. 1285, 2014)

Table of Contents

Frontmatter

Chapter 1. Introduction to Dynamic Transitions

Abstract
The study of phase transitions is an active field with a long history. This book aims to provide a comprehensive, unified, and balanced account of both dynamic and topological phase transition theories and their applications to statistical systems, quantum systems, classical and geophysical fluid dynamics, biological and chemical systems, and climate dynamics. The dynamic phase transition theory establishes a dynamic transition principle, Principle 1, following the philosophy of searching for a complete set of transition states. We present in this chapter a brief introduction to this dynamic transition theory together with an introduction to first-principle approach to fundamental laws of physics, and to fundamental issues of dynamic phase transitions motivated by problems in the nonlinear sciences.
Tian Ma, Shouhong Wang

Chapter 2. Dynamic Transition Theory

Abstract
This chapter introduces the dynamic transition theory for nonlinear dissipative systems developed recently by the authors. The main focus is to derive a general principle, Principle 1, on dynamic transitions for dissipative systems and to introduce a systematic theory and techniques for studying the types and structure of dynamic transitions.
Tian Ma, Shouhong Wang

Chapter 3. Equilibrium Phase Transitions in Statistical Physics

Abstract
A principal objective in the study of equilibrium phase transitions is to capture the transitions from one equilibrium to another and to study the nature or order of such transitions. The study of equilibrium phase transitions presented in this book involves a combination of modeling, mathematical analysis, and physical predictions. First, the standard model for a thermodynamic system is derived using the recently discovered potential-descending principle (PDP), addressed in Chap. 7. Second, all equilibrium phase transitions are fully characterized by three basic theorems. Third, the dynamical law of fluctuations is derived, and we show that the standard model, together with the dynamic law of fluctuation, offers correct information for critical exponents. Fourth, the theory is applied to gas-liquid transitions, ferromagnetism, binary systems, superconductivity, and liquid heliums, leading to various physical predictions and insights to the underlying physical problems.
Tian Ma, Shouhong Wang

Chapter 4. Fluid Dynamics

Abstract
One important source of dynamic transitions and pattern formation is transition and stability problems in fluid dynamics, including in particular Rayleigh-Bnard convection, the Couette–Taylor problem, the Couette–Poiseuille-Taylor problem, and the parallel shear flow problem. The study of these basic problems leads to a better understanding of turbulent behavior in fluid flows, which in turn often leads to new insights and methods in the solution of other problems in science and engineering.
Tian Ma, Shouhong Wang

Chapter 5. Geophysical Fluid Dynamics and Climate Dynamics

Abstract
Our Earth’s atmosphere and oceans are rotating geophysical fluids that are two important components of the planet’s climate system. The atmosphere and the oceans are extremely rich in their organization and complexity, and many phenomena that they exhibit, involving a broad range of temporal and spatial scales (Charney, 1948), cannot be reproduced in the laboratory. An understanding of the complex scientific issues of geophysical fluid dynamics requires the combined efforts of scientists in many fields. The main objective of this chapter is to initiate a study of dynamic transitions and stability of large-scale atmospheric and oceanic circulations, focusing on a few typical sources of climate variability. Such variability, independently and interactively, may play a significant role in past and future climate change.
Tian Ma, Shouhong Wang

Chapter 6. Dynamical Transitions in Chemistry and Biology

Abstract
This chapter studies dynamic transitions and pattern formations of a few typical nonequilibrium chemical and biological models. The main focus is on Belousov–Zhabotinsky models, the chemotactic model, and a population model.
Tian Ma, Shouhong Wang

Chapter 7. Fundamental Principles of Statistical and Quantum Physics

Abstract
This chapter is aimed for the first-principle approach to statistical physics and quantum physics, in the spirit of the guiding principles of physics: Principles 1.​1.​1 and 1.​1.​2. First, we introduce a new principle, the potential-descending principle (PDP), and show that statistical physics is built upon PDP and the principle of equal probability (PEP). Second, we develop a statistical theory of heat, including in particular the photon number formula of entropy, and the energy level formula of temperature. Also, we demonstrate that the physical carrier of heat is the photons. Third, we introduce a new field theoretical interpretation of quantum wave functions.
Tian Ma, Shouhong Wang

Chapter 8. Quantum Mechanism of Condensates and High Tc Superconductivity

Abstract
The main objective of this chapter is to introduce a new quantum mechanism of condensates and superconductivity based on the field theoretical interpretation of quantum mechanical wave functions presented in Sect. 7.​3.​2, and on recent developments in quantum physics and statistical physics.
Tian Ma, Shouhong Wang

Chapter 9. Topological Phase Transitions

Abstract
This chapter aims to develop a systematic theory of topological phase transitions (TPTs) and explores a few typical examples, including (1) quantum phase transitions (QPTs), (2) galactic spiral structures, (3) electromagnetic eruptions on solar surface, (4) boundary-layer separation of fluid flows, and (5) interior separation of fluid flows.
Tian Ma, Shouhong Wang

Backmatter

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