Kinetic Boltzmann, Vlasov and Related Equations (Elsevier Insights)

Kinetic Boltzmann, Vlasov and Related Equations......Page 2
Copyright......Page 3
Preface......Page 4
About the Authors......Page 7
Kinetic Equations of Boltzmann Kind......Page 8
Vlasov's Type Equations......Page 10
How did the Concept of Distribution Function Explain Molecular-Kinetic and Gas Laws to Maxwell......Page 11
On a Kinetic Approach to the Sixth Hilbert Problem (Axiomatization of Physics)......Page 13
Conclusions......Page 14
The Problem of N-Bodies, Continuum of Bodies, and Lagrangian Coordinates in Vlasov Equation......Page 15
When the Equations for Continuum of Bodies Become Hamiltonian?......Page 17
Oscillatory Potential Example......Page 18
Antioscillatory Potential Example......Page 19
Hydrodynamical Substitution: Multiflow Hydrodynamics and Euler-Lagrange Description......Page 20
Expanding Universe Paradigm......Page 22
Conclusions......Page 24
A Shift of Density Along the Trajectories of Dynamical System......Page 25
Method 1. Dirac's δ-Function Method......Page 26
Geodesic Equations and Evolution of Distribution Function on Riemannian Manifold......Page 27
Method 1. Coordinate–Impulse Change of Variables and Hamiltonian Formalism......Page 28
Method 2. Invariant Measure in Coordinate-Velocity Space......Page 29
How does the Riemannian Space Measure Behave While Being Transformed?......Page 30
Derivation of the Vlasov-Maxwell Equation......Page 31
Conclusion......Page 34
System of Vlasov-Poisson Equations for Plasma and Electrons......Page 35
Boundary-Value Problem for Nonlinear Elliptic Equation......Page 36
Verifying the Condition Ψ' ≥ 0
......Page 37
Conclusions......Page 38
Characteristics of the System......Page 40
Vlasov-Maxwell and Vlasov-Poisson Systems......Page 44
Weak Solutions of Vlasov-Poisson and Vlasov-Maxwell Systems......Page 45
Kinetic Equations Modeling Semiconductors......Page 47
Open Problems for Vlasov-Poisson and Vlasov-Maxwell Systems......Page 50
Ansatz of the Distribution Function and Reduction of Stationary Vlasov-Maxwell Equations to Elliptic System......Page 54
Boundary Value Problem......Page 59
Solutions with Norm......Page 69
Introduction......Page 74
Existence and Properties of the Solutions of the Vlasov-Maxwell and Vlasov-Poisson Systems in the Bounded Domains......Page 76
Existence and Properties of Solutions of the VM System in the Bounded Domains......Page 78
Quantum Models: Vigner-Poisson and Schrödinger-Poisson Systems......Page 79
Stationary Solutions of Vlasov-Maxwell System......Page 80
Reduction of the Problem (7.5.1)–(7.5.5) to the System of Nonlinear Elliptic Equations......Page 81
Reduction of System (7.5.22), (7.5.23) to Single Equation......Page 85
Existence of Solutions for the Boundary Value Problem (7.5.28)–(7.5.30)......Page 89
Existence of Solution for Nonlocal Boundary Value Problem......Page 95
Reduction of the Vlasov-Maxwell System to Nonlinear Wave Equation......Page 100
Existence of Nonstationary Solutions of the VLasov-Maxwell System in the Bounded Domain......Page 106
Linear Stability of the Stationary Solutions of the Vlasov-Maxwell System......Page 110
Application Examples with Exact Solutions......Page 117
Normalized Solutions for a One-Component Distribution Function......Page 130
Introduction......Page 137
Bifurcation of Solutions of Nonlinear Equations in Banach Spaces......Page 140
Conclusions......Page 146
Statement of Boundary Value Problem and the Problem on Point of Bifurcation of System (8.4.7), (8.4.13)......Page 147
Resolving Branching Equation......Page 157
The Existence Theorem for Bifurcation Points and the Construction of Asymptotic Solutions......Page 160
Collision Integral......Page 168
Conservation Laws and H- Theorem......Page 169
Conservation Laws......Page 170
Beams......Page 171
Boundary-value problem......Page 172
Boltzmann Equation for Mixtures......Page 173
Chapman-Enskog Method......Page 175
Quantum Kinetic Equations (Uehling-Uhlenbeck Equations)......Page 176
Peculiarity of Hydrodynamic Equations, Obtained from Kinetic Equations......Page 177
Linear Boltzmann Equation and Markovian Processes......Page 178
Boltzmann Extremals......Page 180
Boltzmann Extremals and the Liouville Equation......Page 181
Boltzmann Extremals and the Ergodic Theorem......Page 182
Calerman Model......Page 186
Godunov-Sultangazin Model......Page 187
Carleman Model......Page 188
The Class of Decreasing Functionals for Discrete Models: Uniqueness Theorem of the Boltzmann H- Function......Page 189
Relaxation Problem......Page 190
Chemical Kinetics Equations and H- Theorem: Conditions of Chemical Equilibrium......Page 191
Conclusion......Page 195
Linear Operators Commuting with Rotation Group......Page 197
Bilinear Operators Commuting with Rotation Group......Page 198
Computations of Eigenvalues......Page 199
Fourier Transformation of Collision Integral......Page 200
Momentum System and Maxwellian Gas Relaxation to Equilibrium. Bobylev Symmetry......Page 202
Equations of the Form ∂u/∂t = F(u)......Page 204
Waves Interaction Series......Page 205
Investigating Resonance Relations......Page 206
Convergence of the Series of n Interacting Traveling Waves......Page 207
Final Remarks......Page 210
Discrete Models with Impulses on the Lattice......Page 212
Invariants......Page 214
Inductive Process......Page 215
On Solution of Diophantine Equations of Conservation Laws and Classification of Collisions......Page 216
Models in One-Dimensional Case......Page 217
The Models in Two-Dimensional Cases......Page 218
Photo-, Electro-, Magneto-, and Thermophoresis and Reactive Forces......Page 220
Model and a System of Equations......Page 222
System of Equations in Maxwell Equilibrium for Small Velocities of a Particle......Page 224
Applications......Page 226
Conclusions......Page 227
Conservation Laws for Polynomial Hamiltonians......Page 228
Conservation Laws for Kinetic Equations......Page 230
Stokes Scattering......Page 234
Raman Scattering Considering Anti-Stokes Component......Page 236
The Systems of Special Polynomials in the Problems of Quantum Optics......Page 237
Representation of General Commutation Relations......Page 238
Tower of Mathematical Physics......Page 239
Conclusions......Page 240
Description of Vacuum Diode......Page 242
Description of the Mathematical Model......Page 244
Solution Trajectory, Upper and Lower Solutions......Page 248
Existence of Solutions for System (14.3.18)–(14.3.22)......Page 255
Analysis of the Known Upper and Lower Solutions......Page 257
Conclusions......Page 260
Introduction......Page 261
Problem Statement......Page 265
An Approach of M.E. Tuckerman Group on the Classical Mechanics of Non-Hamiltonian Systems
......Page 267
Small–Time Parameter Method......Page 269
Hermite Polynomial Decomposition......Page 272
Advanced Convergence Results......Page 275
Partial Conclusions......Page 278
Eigen Expansion of Generalized Liouville Operator......Page 279
Hermitian Function Expansion......Page 282
Another Application Example for Hermite Polynomial Decomposition......Page 285
Glossary of Terms and Symbols......Page 287
Bibliography......Page 288