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

**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)