Keywords:
Condensed matter.
;
Electronic books.
Description / Table of Contents:
This book introduces a general principle of dynamic transitions for dissipative systems, establishes a systematic dynamic transition theory, and explores the physical implications of applications of the theory to a range of problems in the nonlinear sciences.
Type of Medium:
Online Resource
Pages:
1 online resource (575 pages)
Edition:
1st ed.
ISBN:
9781461489634
URL:
https://ebookcentral.proquest.com/lib/geomar/detail.action?docID=1592929
DDC:
530.414
Language:
English
Note:
Intro -- Preface -- Contents -- Introduction -- 1 Introduction to Dynamic Transitions -- 1.1 First Principles and Dynamic Models -- 1.1.1 Physical Laws and Mathematical Models -- 1.1.2 Rayleigh-Bénard Convection -- 1.1.3 Mathematical Formulation of Physical Problems -- 1.2 Introduction to Dynamic Transition Theory -- 1.2.1 Motivation and Key Philosophy -- 1.2.2 Principle of Exchange of Stability -- 1.2.3 Equation of Critical Parameters -- 1.2.4 Classifications of Dynamic Transitions -- 1.2.5 Structure and Characterization of Dynamic Transitions -- 1.2.6 General Features of Dynamic Transitions -- 1.3 Examples of Typical Phase Transition Problems -- 1.3.1 Rayleigh-Bénard Convection -- 1.3.2 El Niño Southern Oscillation -- 1.3.3 Dynamic Transition Versus Transition in Physical Space -- 1.3.4 Andrews Critical Point and Third-Order Gas-Liquid Transition -- 1.3.5 Binary Systems -- 2 Dynamic Transition Theory -- 2.1 General Dynamic Transition Theory -- 2.1.1 Classification of Dynamic Transitions -- 2.1.2 Characterization of Transition Types -- 2.1.3 Local Topological Structure of Transitions -- 2.2 Continuous Transition -- 2.2.1 Finite-Dimensional Systems -- 2.2.2 S1-Attractor Bifurcation -- 2.2.3 Sm-Attractor Bifurcation -- 2.2.4 Structural Stability of Dynamic Transitions -- 2.2.5 Infinite-Dimensional Systems -- 2.3 Transition from Simple Eigenvalues -- 2.3.1 Real Simple Eigenvalues -- 2.3.2 Transitions from Complex Simple Eigenvalues -- 2.3.3 Computation of b -- 2.4 Transition from Eigenvalues with Multiplicity Two -- 2.4.1 Index Formula for Second-Order Nondegenerate Singularities -- 2.4.2 Bifurcation at Second-Order Singular Points -- 2.4.3 The Case ind (F,0)=-2 -- 2.4.4 The Case ind (F,0)=2 -- 2.4.5 The Case ind (F,0)=0 -- 2.4.6 Indices of kth-order Nondegenerate Singularities -- 2.4.7 Structure of kth-Order Nondegenerate Singularities.
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2.4.8 Transition from kth-Order Nondegenerate Singularities -- 2.4.9 Bifurcation to Periodic Orbits -- 2.4.10 Application to Parabolic Systems -- 2.5 Singular Separation -- 2.5.1 General Principle -- 2.5.2 Saddle-Node Bifurcation -- 2.5.3 Singular Separation of Periodic Orbits -- 2.6 Perturbed Systems -- 2.6.1 General Eigenvalues -- 2.6.2 Simple Eigenvalues -- 2.6.3 Complex Eigenvalues -- 2.7 Notes -- 3 Equilibrium Phase Transition in Statistical Physics -- 3.1 Dynamic Models for Equilibrium Phase Transitions -- 3.1.1 Thermodynamic Potentials -- 3.1.2 Time-Dependent Equations -- 3.2 Classification of Equilibrium Phase Transitions -- 3.3 Third-Order Gas-Liquid Phase Transition -- 3.3.1 Introduction -- 3.3.2 Time-Dependent Models for PVT Systems -- 3.3.3 Phase Transition Dynamics for PVT Systems -- 3.3.4 Physical Conclusions -- 3.4 Ferromagnetism -- 3.4.1 Classical Theory of Ferromagnetism -- 3.4.2 Dynamic Transitions in Ferromagnetism -- 3.4.3 Physical Implications -- 3.4.4 Asymmetry of Fluctuations -- 3.5 Phase Separation in Binary Systems -- 3.5.1 Modeling -- 3.5.2 Phase Transition in General Domains -- 3.5.3 Phase Transition in Rectangular Domains -- 3.5.4 Spatial Geometry, Transitions, and Pattern Formation -- 3.5.5 Phase Diagrams and Physical Conclusions -- 3.6 Superconductivity -- 3.6.1 Ginzburg-Landau Model -- 3.6.2 TGDL as a Gradient-Type System -- 3.6.3 Phase Transition Theorems -- 3.6.4 Model Coupled with Entropy -- 3.6.5 Physical Conclusions -- 3.7 Liquid Helium-4 -- 3.7.1 Dynamic Model for Liquid Helium-4 -- 3.7.2 Dynamic Phase Transition for Liquid 4He -- 3.8 Superfluidity of Helium-3 -- 3.8.1 Dynamic Model for Liquid 3He with Zero Applied Field -- 3.8.2 Critical Parameter Curves and PT-Phase Diagram -- 3.8.3 Classification of Superfluid Transitions -- 3.8.4 Liquid 3He with Nonzero Applied Field -- 3.8.5 Physical Remarks.
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3.9 Mixture of He-3 and He-4 -- 3.9.1 Model for Liquid Mixture of 3He and 4He -- 3.9.2 Critical Parameter Curves -- 3.9.3 Transition Theorems -- 3.9.4 Physical Conclusions -- 4 Fluid Dynamics -- 4.1 Rayleigh-Bénard Convection -- 4.1.1 Bénard Problem -- 4.1.2 Boussinesq Equations -- 4.1.3 Dynamic Transition Theorems -- 4.1.4 Topological Structure and Pattern Formation -- 4.1.5 Asymptotic Structure of Solutions for the BénardProblem -- 4.1.6 Structure of Bifurcated Attractors -- 4.1.7 Physical Remarks -- 4.2 Taylor-Couette Flow -- 4.2.1 Taylor Problem -- 4.2.2 Governing Equations -- 4.2.3 Narrow-Gap Case with Axisymmetric Perturbations -- 4.2.4 Asymptotic Structure of Solutions and Taylor Vortices -- 4.2.5 Taylor Problem with z-Periodic Boundary Condition -- 4.2.6 Other Boundary Conditions -- 4.2.7 Three-Dimensional Perturbation for the Narrow-GapCase -- 4.2.8 Physical Remarks -- 4.3 Boundary-Layer and Interior Separations in the Taylor-Couette-Poiseuille Flow -- 4.3.1 Model for the Taylor-Couette-Poiseuille Problem -- 4.3.2 Phase Transition of the TCP Problem -- 4.3.3 Boundary-Layer Separation from the Couette-Poiseuille Flow -- 4.3.4 Interior Separation from the Couette-Poiseuille Flow -- 4.3.5 Nature of Boundary-Layer and Interior Separations -- 4.4 Rotating Convection Problem -- 4.4.1 Rotating Boussinesq Equations -- 4.4.2 Eigenvalue Problem -- 4.4.3 Principle of Exchange of Stabilities -- 4.4.4 Transition from First Real Eigenvalues -- 4.4.5 Transition from First Complex Eigenvalues -- 4.4.6 Physical Remarks -- 4.5 Convection Scale Theory -- 5 Geophysical Fluid Dynamics and Climate Dynamics -- 5.1 Modeling and General Characteristics of Geophysical Flows -- 5.2 El Niño-Southern Oscillation -- 5.2.1 Walker Circulation and ENSO -- 5.2.2 Equatorial Circulation Equations -- 5.2.3 Walker Circulation Under Idealized Conditions.
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5.2.4 Walker Circulation Under Natural Conditions -- 5.2.5 ENSO: Metastable Oscillation Theory -- 5.3 Thermohaline Ocean Circulation -- 5.3.1 Boussinesq Equations -- 5.3.2 Linear Analysis -- 5.3.3 Nonlinear Dynamic Transitions -- 5.3.4 Convection Scales and Dynamic Transition -- 5.4 Arctic Ocean Circulations -- 5.4.1 Model -- 5.4.2 Linear Theory -- 5.4.3 Transition Theorems -- 5.4.4 Revised Transition Theory -- 5.4.5 Physical Conclusions -- 5.5 Large-Scale Meridional Atmospheric Circulation -- 5.5.1 Polar, Ferrel, and Hadley Cells -- 5.5.2 -Plane Assumption -- 5.5.3 Meridional Circulation Under Idealized Conditions -- 5.5.4 Physical Implications -- 6 Dynamical Transitions in Chemistry and Biology -- 6.1 Modeling -- 6.1.1 Dynamical Equations of Chemical Reactions -- 6.1.2 Population Models of Biological Species -- 6.2 Belousov-Zhabotinsky Chemical Reactions: Oregonator -- 6.2.1 The Field-Korös-Noyes Equations -- 6.2.2 Transition Under the Dirichlet Boundary Condition -- 6.2.3 Transitions Under the Neumann Boundary Condition -- 6.2.4 Phase Transition in the Realistic Oregonator -- 6.3 Belousov-Zhabotinsky Reactions: Brusselator -- 6.3.1 Prigogine-Lefever Model -- 6.3.2 Linearized Problem -- 6.3.3 Transition from Real Eigenvalues -- 6.3.4 Transition from Complex Eigenvalues -- 6.4 Bacterial Chemotaxis -- 6.4.1 Keller-Segel Models -- 6.4.2 Dynamic Transitions for a Rich Stimulant System -- 6.4.3 Transition of Three-Component Systems -- 6.4.4 Biological Conclusions -- 6.5 Biological Species -- 6.5.1 Modeling -- 6.5.2 Predator-Prey Systems -- 6.5.3 Three-Species Systems -- Appendix A -- A.1 Formulas for Center Manifold Functions -- A.2 Dynamics of Gradient-Type Systems -- References -- Index.
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