Kinetic Boltzmann, Vlasov and Related Equations

Boltzmann and Vlasov equations played a great role in the past and still play an important role in modern natural sciences, technique and even philosophy of science. Classical Boltzmann equation derived in 1872 became a cornerstone for the molecular-kinetic theory, the second law of thermodynamics (increasing entropy) and derivation of the basic hydrodynamic equations. After modifications, the fields and numbers of its applications have increased to include diluted gas, radiation, neutral particles transportation, atmosphere optics and nuclear reactor modelling. Vlasov equation was obtained in 1938 and serves as a basis of plasma physics and describes large-scale processes and galaxies in astronomy, star wind theory.

This book provides a comprehensive review of both equations and presents both classical and modern applications. In addition, it discusses several open problems of great importance.

• Reviews the whole field from the beginning to today
• Includes practical applications
• Provides classical and modern (semi-analytical) solutions

Mathematicians and postgraduates in maths, physics, chemistry and astronomy

Preface

About the Authors

1. Principal Concepts of Kinetic Equations

1.1. Introduction

1.2. Kinetic Equations of Boltzmann Kind

1.3. Vlasov's Type Equations

1.4. How did the Concept of Distribution Function Explain Molecular-Kinetic and Gas Laws to Maxwell

1.5. On a Kinetic Approach to the Sixth Hilbert Problem (Axiomatization of Physics)

1.6. Conclusions

2. Lagrangian Coordinates

2.1. The Problem of N-Bodies, Continuum of Bodies, and Lagrangian Coordinates in Vlasov Equation

2.2. When the Equations for Continuum of Bodies Become Hamiltonian?

2.3. Oscillatory Potential Example

2.4. Antioscillatory Potential Example

2.5. Hydrodynamical Substitution: Multiflow Hydrodynamics and Euler-Lagrange Description

2.6. Expanding Universe Paradigm

2.7. Conclusions

3. Vlasov-Maxwell and Vlasov-Einstein Equations

3.1. Introduction

3.2. A Shift of Density Along the Trajectories of Dynamical System

3.3. Geodesic Equations and Evolution of Distribution Function on Riemannian Manifold

3.4. How does the Riemannian Space Measure Behave While Being Transformed?

3.5. Derivation of the Vlasov-Maxwell Equation

3.6. Derivation Scheme of Vlasov-Einstein Equation

3.7. Conclusion

4. Energetic Substitution

4.1. System of Vlasov-Poisson Equations for Plasma and Electrons

4.2. Energetic Substitution and Bernoulli Integral

4.3. Boundary-Value Problem for Nonlinear Elliptic Equation

4.4. Verifying the Condition Ψ′ ≥ 0

4.5. Conclusions

5. Introduction to the Mathematical Theory of Kinetic Equations

5.1. Characteristics of the System

5.2. Vlasov-Maxwell and Vlasov-Poisson Systems

5.3. Weak Solutions of Vlasov-Poisson and Vlasov-Maxwell Systems

5.4. Classical Solutions of VP and VM Systems

5.5. Kinetic Equations Modeling Semiconductors

5.6. Open Problems for Vlasov-Poisson and Vlasov-Maxwell Systems

6. On the Family of the Steady-State Solutions of Vlasov-Maxwell System

6.1. Ansatz of the Distribution Function and Reduction of Stationary Vlasov-Maxwell Equations to Elliptic System

6.2. Boundary Value Problem

6.3. Solutions with Norm

7. Boundary Value Problems for the Vlasov-Maxwell System

7.1. Introduction

7.2. Existence and Properties of the Solutions of the Vlasov-Maxwell and Vlasov-Poisson Systems in the Bounded Domains

7.3. Existence and Properties of Solutions of the VM System in the Bounded Domains

7.4. Collisionless Kinetic Models (Classical and Relativistic Vlasov-Maxwell Systems)

7.5. Stationary Solutions of Vlasov-Maxwell System

7.6. Existence of Solutions for the Boundary Value Problem (7.5.28)–(7.5.30)

7.7. Existence of Solution for Nonlocal Boundary Value Problem

7.8. Nonstationary Solutions of the Vlasov-Maxwell System

7.9. Linear Stability of the Stationary Solutions of the Vlasov-Maxwell System

7.10. Application Examples with Exact Solutions

7.11. Normalized Solutions for a One-Component Distribution Function

8. Bifurcation of Stationary Solutions of the Vlasov-Maxwell System

8.1. Introduction

8.2. Bifurcation of Solutions of Nonlinear Equations in Banach Spaces

8.3. Conclusions

8.4. Statement of Boundary Value Problem and the Problem on Point of Bifurcation of System (8.4.7), (8.4.13)

8.5. Resolving Branching Equation

8.6. The Existence Theorem for Bifurcation Points and the Construction of Asymptotic Solutions

9. Boltzmann Equation

9.1. Collision Integral

9.2. Conservation Laws and H-Theorem

9.3. Boltzmann Equation for Mixtures

9.4. Quantum Kinetic Equations (Uehling-Uhlenbeck Equations)

9.5. Peculiarity of Hydrodynamic Equations, Obtained from Kinetic Equations

9.6. Linear Boltzmann Equation and Markovian Processes

9.7. Time Averages and Boltzmann Extremals

10. Discrete Models of Boltzmann Equation

10.1. General Discrete Models of Boltzmann Equation

10.2. Calerman, Godunov-Sultangazin, and Broadwell Models

10.3. H- Theorem and Conservation Laws

10.4. The Class of Decreasing Functionals for Discrete Models: Uniqueness Theorem of the Boltzmann H- Function

10.5. Relaxation Problem

10.6. Chemical Kinetics Equations and H- Theorem: Conditions of Chemical Equilibrium

11. Method of Spherical Harmonics and Relaxation of Maxwellian Gas

11.1. Linear Operators Commuting with Rotation Group

11.2. Bilinear Operators Commuting with Rotation Group

11.3. Momentum System and Maxwellian Gas Relaxation to Equilibrium. Bobylev Symmetry

11.4. Exponential Series and Superposition of Travelling Waves

12. Discrete Boltzmann Equation Models for Mixtures

12.1. Discrete Models with Impulses on the Lattice

12.2. Invariants

12.3. Inductive Process

12.4. On Solution of Diophantine Equations of Conservation Laws and Classification of Collisions

12.5. Boltzmann Equation for the Mixture in One-Dimensional Case

12.6. Models in One-Dimensional Case

12.7. The Models in Two-Dimensional Cases

12.8. Conclusions

12.9. Photo-, Electro-, Magneto-, and Thermophoresis and Reactive Forces

13. Quantum Hamiltonians and Kinetic Equations

13.1. Conservation Laws for Polynomial Hamiltonians

13.2. Conservation Laws for Kinetic Equations

13.3. The Asymptotics of Spectrum for Hamiltonians of Raman Scattering

13.4. The Systems of Special Polynomials in the Problems of Quantum Optics

13.5. Representation of General Commutation Relations

13.6. Tower of Mathematical Physics

13.7. Conclusions

14. Modeling of the Limit Problem for the Magnetically Noninsulated Diode

14.1. Introduction

14.2. Description of Vacuum Diode

14.3. Description of the Mathematical Model

14.4. Solution Trajectory, Upper and Lower Solutions

14.5. Existence of Solutions for System (14.3.18)–(14.3.22)

14.6. Analysis of the Known Upper and Lower Solutions

14.7. Conclusions

15. Generalized Liouville Equation and Approximate Orthogonal Decomposition Methods

15.1. Introduction

15.2. Problem Statement

15.3. The Overview of Preceeding Results

15.4. Eigen Expansion of Generalized Liouville Operator

15.5. Hermitian Function Expansion

15.6. Another Application Example for Hermite Polynomial Decomposition

Glossary of Terms and Symbols

Bibliography