Kinetic Boltzmann, Vlasov and Related Equations

1st Edition

Print ISBN: 9780323165303
eBook ISBN: 9780123877802
Imprint: Elsevier
Published Date: 17th June 2011
Page Count: 320
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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.

Key Features

  • 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

Table of Contents


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. K


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"The reviewed collective monograph presents not only the basics and common facts, but also recent results in the theory of kinetic equations and their many applications."--Zentralblatt MATH 1230-1