Sinitsyn / Dulov / Vedenyapin Kinetic Boltzmann, Vlasov and Related Equations

1. Auflage 2011
ISBN: 978-0-12-387780-2
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 320 Seiten

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

Autoren/Hrsg.

Sinitsyn, Alexander

Dulov, Eugene

Vedenyapin, Victor

Inhaltsverzeichnis

1;Front Cover;1
2;Kinetic Boltzmann, Vlasov and Related Equations;4
3;Copyright;5
4;Table of Contents;6
5;Preface;12
6;About the Authors;16
7;Chapter 1. Principal Concepts of Kinetic Equations;18
7.1;1.1 Introduction;18
7.2;1.2 Kinetic Equations of Boltzmann Kind;18
7.3;1.3 Vlasov's Type Equations;20
7.4;1.4 How did the Concept of Distribution Function Explain Molecular-Kinetic and Gas Laws to Maxwell;21
7.5;1.5 On a Kinetic Approach to the Sixth Hilbert Problem (Axiomatization of Physics);23
7.6;1.6 Conclusions;24
8;Chapter 2. Lagrangian Coordinates;26
8.1;2.1 The Problem of N-Bodies, Continuum of Bodies, and Lagrangian Coordinates in Vlasov Equation;26
8.2;2.2 When the Equations for Continuum of Bodies Become Hamiltonian?;28
8.3;2.3 Oscillatory Potential Example;29
8.4;2.4 Antioscillatory Potential Example;30
8.5;2.5 Hydrodynamical Substitution: Multiflow Hydrodynamics and Euler-Lagrange Description;31
8.6;2.6 Expanding Universe Paradigm;33
8.7;2.7 Conclusions;35
9;Chapter 3. Vlasov-Maxwell and Vlasov-Einstein Equations;36
9.1;3.1 Introduction;36
9.2;3.2 A Shift of Density Along the Trajectories of Dynamical System;36
9.3;3.3 Geodesic Equations and Evolution of Distribution Function on Riemannian Manifold;38
9.4;3.4 How does the Riemannian Space Measure Behave While Being Transformed?;41
9.5;3.5 Derivation of the Vlasov-Maxwell Equation;42
9.6;3.6 Derivation Scheme of Vlasov-Einstein Equation;45
9.7;3.7 Conclusion;45
10;Chapter 4. Energetic Substitution;46
10.1;4.1 System of Vlasov-Poisson Equations for Plasma and Electrons;46
10.2;4.2 Energetic Substitution and Bernoulli Integral;47
10.3;4.3 Boundary-Value Problem for Nonlinear Elliptic Equation;47
10.4;4.4 Verifying the Condition ?' ? 0;48
10.5;4.5 Conclusions;49
11;Chapter 5. Introduction to the Mathematical Theory of Kinetic Equations;52
11.1;5.1 Characteristics of the System;52
11.2;5.2 Vlasov-Maxwell and Vlasov-Poisson Systems;56
11.3;5.3 Weak Solutions of Vlasov-Poisson and Vlasov-Maxwell Systems;57
11.4;5.4 Classical Solutions of VP and VM Systems;59
11.5;5.5 Kinetic Equations Modeling Semiconductors;59
11.6;5.6 Open Problems for Vlasov-Poisson and Vlasov-Maxwell Systems;62
12;Chapter 6. On the Family of the Steady-State Solutions of Vlasov-Maxwell System;66
12.1;6.1 Ansatz of the Distribution Function and Reduction of Stationary Vlasov-Maxwell Equations to Elliptic System;66
12.2;6.2 Boundary Value Problem;71
12.3;6.3 Solutions with Norm;81
13;Chapter 7. Boundary Value Problems for the Vlasov-Maxwell System;86
13.1;7.1 Introduction;86
13.2;7.2 Existence and Properties of the Solutions of the Vlasov-Maxwell and Vlasov-Poisson Systems in the Bounded Domains;88
13.3;7.3 Existence and Properties of Solutions of the VM System in the Bounded Domains;90
13.4;7.4 Collisionless Kinetic Models (Classical and Relativistic Vlasov-Maxwell Systems);91
13.5;7.5 Stationary Solutions of Vlasov-Maxwell System;92
13.6;7.6 Existence of Solutions for the Boundary Value Problem (7.5.28)–(7.5.30);101
13.7;7.7 Existence of Solution for Nonlocal Boundary Value Problem;107
13.8;7.8 Nonstationary Solutions of the Vlasov-Maxwell System;112
13.9;7.9 Linear Stability of the Stationary Solutions of the Vlasov-Maxwell System;122
13.10;7.10 Application Examples with Exact Solutions;129
13.11;7.11 Normalized Solutions for a One-Component Distribution Function;142
14;Chapter 8. Bifurcation of Stationary Solutions of the Vlasov-Maxwell System;150
14.1;8.1 Introduction;150
14.2;8.2 Bifurcation of Solutions of Nonlinear Equations in Banach Spaces;153
14.3;8.3 Conclusions;159
14.4;8.4 Statement of Boundary Value Problem and the Problem on Point of Bifurcation of System (8.4.7), (8.4.13);160
14.5;8.5 Resolving Branching Equation;170
14.6;8.6 The Existence Theorem for Bifurcation Points and the Construction of Asymptotic Solutions;173
15;Chapter 9. Boltzmann Equation;182
15.1;9.1 Collision Integral;182
15.2;9.2 Conservation Laws and H- Theorem;183
15.3;9.3 Boltzmann Equation for Mixtures;187
15.4;9.4 Quantum Kinetic Equations (Uehling-Uhlenbeck Equations);190
15.5;9.5 Peculiarity of Hydrodynamic Equations, Obtained from Kinetic Equations;191
15.6;9.6 Linear Boltzmann Equation and Markovian Processes;192
15.7;9.7 Time Averages and Boltzmann Extremals;194
16;Chapter 10. Discrete Models of Boltzmann Equation;200
16.1;10.1 General Discrete Models of Boltzmann Equation;200
16.2;10.2 Calerman, Godunov-Sultangazin, and Broadwell Models;200
16.3;10.3 H- Theorem and Conservation Laws;202
16.4;10.4 The Class of Decreasing Functionals for Discrete Models: Uniqueness Theorem of the Boltzmann H- Function;203
16.5;10.5 Relaxation Problem;204
16.6;10.6 Chemical Kinetics Equations and H- Theorem: Conditions of Chemical Equilibrium;205
17;Chapter 11. Method of Spherical Harmonics and Relaxation of Maxwellian Gas;212
17.1;11.1 Linear Operators Commuting with Rotation Group;212
17.2;11.2 Bilinear Operators Commuting with Rotation Group;213
17.3;11.3 Momentum System and Maxwellian Gas Relaxation to Equilibrium. Bobylev Symmetry;217
17.4;11.4 Exponential Series and Superposition of Travelling Waves;219
18;Chapter 12. Discrete Boltzmann Equation Models for Mixtures;228
18.1;12.1 Discrete Models with Impulses on the Lattice;228
18.2;12.2 Invariants;230
18.3;12.3 Inductive Process;231
18.4;12.4 On Solution of Diophantine Equations of Conservation Laws and Classification of Collisions;232
18.5;12.5 Boltzmann Equation for the Mixture in One-Dimensional Case;233
18.6;12.6 Models in One-Dimensional Case;233
18.7;12.7 The Models in Two-Dimensional Cases;234
18.8;12.8 Conclusions;236
18.9;12.9 Photo-, Electro-, Magneto-, and Thermophoresis and Reactive Forces;236
19;Chapter 13. Quantum Hamiltonians and Kinetic Equations;244
19.1;13.1 Conservation Laws for Polynomial Hamiltonians;244
19.2;13.2 Conservation Laws for Kinetic Equations;246
19.3;13.3 The Asymptotics of Spectrum for Hamiltonians of Raman Scattering;250
19.4;13.4 The Systems of Special Polynomials in the Problems of Quantum Optics;253
19.5;13.5 Representation of General Commutation Relations;254
19.6;13.6 Tower of Mathematical Physics;255
19.7;13.7 Conclusions;256
20;Chapter 14. Modeling of the Limit Problem for the Magnetically Noninsulated Diode;258
20.1;14.1 Introduction;258
20.2;14.2 Description of Vacuum Diode;258
20.3;14.3 Description of the Mathematical Model;260
20.4;14.4 Solution Trajectory, Upper and Lower Solutions;264
20.5;14.5 Existence of Solutions for System (14.3.18)–(14.3.22);271
20.6;14.6 Analysis of the Known Upper and Lower Solutions;273
20.7;14.7 Conclusions;276
21;Chapter 15. Generalized Liouville Equation and Approximate Orthogonal Decomposition Methods;278
21.1;15.1 Introduction;278
21.2;15.2 Problem Statement;282
21.3;15.3 The Overview of Preceeding Results;284
21.4;15.4 Eigen Expansion of Generalized Liouville Operator;296
21.5;15.5 Hermitian Function Expansion;299
21.6;15.6 Another Application Example for Hermite Polynomial Decomposition;302
22;Glossary of Terms and Symbols;304
23;Bibliography;306