**A.N. Gorban,
I.V. Karlin
Invariant Manifolds
for Physical and Chemical Kinetics, (15MB)
**

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**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 1*D*10*M *Grad Equations

8.3.2 The 3*D*10*M *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 1*D*13*M *Grad System

8.4.5 Invariance Equation
for the 3*D*13*M *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)