A.N. Gorban,
I.V. Karlin
Invariant Manifolds
for Physical and Chemical Kinetics, (15MB)
Lect. Notes
Phys. 660, Springer, Berlin,
Heidelberg, 2005
Contents and pdf files of Chapters
1.1 Ideas and References
1.2 Content and Reading Approaches
2.1 The Boltzmann Equation
2.1.1 The Equation
2.1.2 The Basic Properties of the Boltzmann Equation
2.1.3 Linearized Collision Integral
2.2 Phenomenology and Quasi-Chemical Representation of the Boltzmann Equation
2.3 Kinetic Models
2.4 Methods of Reduced Description
2.4.1 The Hilbert Method
2.4.2 The Chapman–Enskog Method
2.4.3 The Grad Moment Method
2.4.4 Special Approximations
2.4.5 The Method of Invariant Manifold
2.4.6 Quasiequilibrium Approximations
2.5 Discrete Velocity Models
2.6 Direct Simulation
2.7 Lattice Gas and Lattice Boltzmann Models
2.7.1 Discrete Velocity Models for Hydrodynamics
2.7.2 Entropic Lattice Boltzmann Method
2.7.3 Entropic Lattice BGK Method (ELBGK)
2.7.4 Boundary Conditions
2.7.5 Numerical Illustrations of the ELBGK
2.8 Other Kinetic Equations
2.8.1 The Enskog Equation for Hard Spheres
2.8.2 The Vlasov Equation
2.8.3 The Fokker–Planck Equation
2.9 Equations of Chemical Kinetics and Their Reduction
2.9.1 Dissipative Reaction Kinetics
2.9.2 The Problem of Reduced Description in Chemical Kinetics
2.9.3 Partial Equilibrium Approximations
2.9.4 Model Equations
2.9.5 Quasi-Steady State Approximation
2.9.6 Thermodynamic Criteria for the Selection of Important Reactions
2.9.7 Opening
3 Invariance Equation in Differential Form
4 Film Extension of the Dynamics: Slowness as Stability
4.1 Equation for the Film Motion
4.2 Stability of Analytical Solutions
5 Entropy, Quasiequilibrium, and Projectors Field
5.1 Moment Parameterization
5.2 Entropy and Quasiequilibrium
5.3 Thermodynamic Projector without a Priori Parameterization
5.4 Uniqueness of Thermodynamic Projector
5.4.1 Projection of Linear Vector Field
5.4.2 The Uniqueness Theorem
5.4.3 Orthogonality of the Thermodynamic Projector and Entropic Gradient Models
5.4.4 Violation of the Transversality Condition, Singularity of Thermodynamic Projection, and Steps of Relaxation
5.4.5 Thermodynamic Projector, Quasiequilibrium, and Entropy Maximum
5.5 Example: Quasiequilibrium Projector and Defect of Invariance for the Local Maxwellians Manifold of the Boltzmann Equation
5.5.1 Difficulties of Classical Methods of the Boltzmann Equation Theory
5.5.2 Boltzmann Equation
5.5.3 Local Manifolds
5.5.4 Thermodynamic Quasiequilibrium Projector
5.5.5 Defect of Invariance for the LM Manifold
5.6 Example: Quasiequilibrium Closure Hierarchies for the Boltzmann Equation
5.6.1 Triangle Entropy Method
5.6.2 Linear Macroscopic Variables
5.6.3 Transport Equations for Scattering Rates in the Neighbourhood of Local Equilibrium. Second and Mixed Hydrodynamic Chains
5.6.4 Distribution Functions of the Second Quasiequilibrium Approximation for Scattering Rates
5.6.5 Closure of the Second and Mixed Hydrodynamic Chains
5.6.6 Appendix: Formulas of the Second Quasiequilibrium Approximation of the Second and Mixed Hydrodynamic Chains for Maxwell Molecules and Hard Spheres
5.7 Example: Alternative Grad Equations and a “New Determination of Molecular Dimensions” (Revisited)
5.7.1 Nonlinear Functionals Instead of Moments in the Closure Problem
5.7.2 Linearization
5.7.3 Truncating the Chain
5.7.4 Entropy Maximization
5.7.5 A New Determination of Molecular Dimensions (Revisited)
6 Newton Method with Incomplete Linearization
6.1 The Method
6.2 Example: Two-Step Catalytic Reaction
6.3 Example: Non-Perturbative Correction of Local Maxwellian Manifold and Derivation of Nonlinear Hydrodynamics from Boltzmann Equation (1D)
6.3.1 Positivity and Normalization
6.3.2 Galilean Invariance of Invariance Equation
6.3.3 Equation of the First Iteration
6.3.4 Parametrix Expansion
6.3.5 Finite-Dimensional Approximations to Integral Equations
6.3.6 Hydrodynamic Equations
6.3.7 Nonlocality
6.3.8 Acoustic Spectra
6.3.9 Nonlinearity
6.4 Example: Non-Perturbative Derivation of Linear Hydrodynamics from the Boltzmann Equation (3D)
6.5 Example: Dynamic Correction to Moment Approximations
6.5.1 Dynamic Correction or Extension of the List of Variables?
6.5.2 Invariance Equation for Thirteen-Moment Parameterization
6.5.3 Solution of the Invariance Equation
6.5.4 Corrected Thirteen-Moment Equations
6.5.5 Discussion: Transport Coefficients, Destroying the Hyperbolicity, etc.
7 Quasi-Chemical Representation
7.1 Decomposition of Motions, Non-Uniqueness of Selection of Fast Motions, Self-Adjoint Linearization, Onsager Filter, and Quasi-Chemical Representation
7.2 Example: Quasi-Chemical Representation and Self-Adjoint Linearization of the Boltzmann Collision Operator
8 Hydrodynamics From Grad’s Equations: What Can We Learn From Exact Solutions?
8.1 The “Ultra-Violet Catastrophe” of the Chapman-Enskog Expansion
8.2 The Chapman–Enskog Method for Linearized Grad’s Equations
8.3 Exact Summation of the Chapman–Enskog Expansion
8.3.1 The 1D10M Grad Equations
8.3.2 The 3D10M Grad Equations
8.4 The Dynamic Invariance Principle
8.4.1 Partial Summation of the Chapman–Enskog Expansion
8.4.2 The Dynamic Invariance
8.4.3 The Newton Method
8.4.4 Invariance Equation for the 1D13M Grad System
8.4.5 Invariance Equation for the 3D13M Grad System
8.4.6 Gradient Expansions in Kinetic Theory of Phonons
8.4.7 Nonlinear Grad Equations
8.5 The Main Lesson
9.1 “Large Stepping” for the Equation of the Film Motion
9.2 Example: Relaxation Method for the Fokker-Planck Equation
9.2.1 Quasi-Equilibrium Approximations for the Fokker-Planck Equation
9.2.2 The Invariance Equation for the Fokker-Planck Equation
9.2.3 Diagonal Approximation
9.3 Example: Relaxational Trajectories: Global Approximations
9.3.1 Initial Layer and Large Stepping
9.3.2 Extremal Properties of the Limiting State
9.3.3 Approximate Trajectories
9.3.4 Relaxation of the Boltzmann Gas
9.3.5 Estimations
9.3.6 Discussion
10.1 Invariant Grids
10.2 Grid Construction Strategy
10.2.1 Growing Lump
10.2.2 Invariant Flag
10.2.3 Boundaries Check and the Entropy
10.3 Instability of Fine Grids
10.4 Which Space is Most Appropriate for the Grid Construction?
10.5 Carleman’s Formula in the Analytical Invariant Manifolds Approximations. First Benefit of Analyticity: Superresolution
10.6 Example: Two-Step Catalytic Reaction
10.7 Example: Model Hydrogen Burning Reaction
10.8 Invariant Grid as a Tool for Data Visualization
11 Method of Natural Projector
11.1 Ehrenfests’ Coarse-Graining Extended to a Formalism of Nonequilibrium Thermodynamics
11.2 Example: From Reversible Dynamics to Navier–Stokes and Post-Navier–Stokes Hydrodynamics by Natural Projector
11.2.1 General Construction
11.2.2 Enhancement of Quasiequilibrium Approximations for Entropy-Conserving Dynamics
11.2.3 Entropy Production
11.2.4 Relation to the Work of Lewis
11.2.5 Equations of Hydrodynamics
11.2.6 Derivation of the Navier–Stokes Equations
11.2.7 Post-Navier–Stokes Equations
11.3 Example: Natural Projector for the Mc Kean Model
11.3.1 General Scheme
11.3.2 Natural Projector for Linear Systems
11.3.3 Explicit Example of the Fluctuation–Dissipation Formula
11.3.4 Comparison with the Chapman–Enskog Method and Solution of the Invariance Equation
12 Geometry of Irreversibility: The Film of Nonequilibrium States
12.1 The Thesis About Macroscopically Definable Ensembles and the Hypothesis About Primitive Macroscopically Definable Ensembles
12.2 The Problem of Irreversibility
12.2.1 The Phenomenon of the Macroscopic Irreversibility
12.2.2 Phase Volume and Dynamics of Ensembles
12.2.3 Macroscopically Definable Ensembles and Quasiequilibria
12.2.4 Irreversibility and Initial Conditions
12.2.5 Weak and Strong Tendency to Equilibrium, Shaking and Short Memory
12.2.6 Subjective Time and Irreversibility
12.3 Geometrization of Irreversibility
12.3.1 Quasiequilibrium Manifold
12.3.2 Quasiequilibrium Approximation
12.4 Natural Projector and Models of Nonequilibrium Dynamics
12.4.1 Natural Projector
12.4.2 One-Dimensional Model of Nonequilibrium States
12.4.3 Curvature and Entropy Production: Entropic Circle and First Kinetic Equations
12.5 The Film of Non-Equilibrium States
12.5.1 Equations for the Film
12.5.2 Thermodynamic Projector on the Film
12.5.3 Fixed Points of the Film Equation
12.5.4 The Failure of the Simplest Galerkin-Type Approximations for Conservative Systems
12.5.5 Second Order Kepler Models of the Film
12.5.6 The Finite Models: Termination at the Horizon Points
12.5.7 The Transversal Restart Lemma
12.5.8 The Time Replacement, and the Invariance of the Projector
12.5.9 Correction to the Infinite Models
12.5.10 The Film, and the Macroscopic Equations
12.5.11 New in the Separation of the Relaxation Times
12.6 The Main Results
13 Slow Invariant Manifolds for Open Systems
13.1 Slow Invariant Manifold for a Closed System Has Been Found. What Next?
13.2 Slow Dynamics in Open Systems. Zero-Order Approximation and the Thermodynamic Projector
13.3 Slow Dynamics in Open Systems. First-Order Approximation
13.4 Beyond the First-Order Approximation: Higher-Order Dynamic Corrections, Stability Loss and Invariant Manifold Explosion
13.5 Example: The Universal Limit in Dynamics of Dilute Polymeric Solutions
13.5.1 The Problem of Reduced Description in Polymer Dynamics
13.5.2 The Method of Invariant Manifold for Weakly Driven Systems
13.5.3 Linear Zero-Order Equations
13.5.4 Auxiliary Formulas. 1. Approximations to Eigenfunctions of the Fokker–Planck Operator
13.5.5 Auxiliary Formulas. 2. Integral Relations
13.5.6 Microscopic Derivation of Constitutive Equations
13.5.7 Tests on the FENE Dumbbell Model
13.5.8 The Main Results of this Example are as Follows
13.6 Example: Explosion of Invariant Manifold, Limits of Macroscopic Description for Polymer Molecules, Molecular Individualism, and Multimodal Distributions
13.6.1 Dumbbell Models and the Problem of the Classical Gaussian Solution Stability
13.6.2 Dynamics of the Moments and Explosion of the Gaussian Manifold
13.6.3 Two-Peak Approximation for Polymer Stretching in Flow and Explosion of the Gaussian Manifold
13.6.4 Polymodal Polyhedron and Molecular Individualism
14 Dimension of Attractors Estimation
14.1 Lyapunov Norms, Finite-Dimensional Asymptotics and Volume Contraction
14.2 Examples: Lyapunov Norms for Reaction Kinetics
14.3 Examples: Infinite-Dimensional Systems With Finite-Dimensional Attractors
14.4 Systems with Inheritance: Dynamics of Distributions with Conservation of Support, Natural Selection and Finite-Dimensional Asymptotics
14.4.1 Introduction: Unusual Conservation Law
14.4.2 Optimality Principle for Limit Diversity
14.4.3 How Many Points Does the Limit Distribution Support Hold?
14.4.4 Selection Efficiency
14.4.5 Gromov’s Interpretation of Selection Theorems
14.4.6 Drift Equations
14.4.7 Three Main Types of Stability
14.4.8 Main Results About Systems with Inheritance
14.5 Example: Cell Division Self-Synchronization
15 Accuracy Estimation and Post-Processing in Invariant Manifolds Construction
15.1 Formulas for Dynamic and Static Post-Processing
15.2 Example: Defect of Invariance Estimation and Switching from the Microscopic Simulations to Macroscopic Equations
15.2.1 Invariance Principle and Micro-Macro Computations
15.2.2 Application to Dynamics of Dilute Polymer Solution
Mathematical Notation and Some Terminology
Sauro Succi, Review in Bull. London Math. Soc. 38 (2006)
Eugene Kryachko, Review in Zentralblatt Math. (2006)