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

  1. Principal Concepts of Kinetic Equations
  2. Lagrangian Coordinates
  3. Vlasov-Maxwell and Vlasov-Einstein Equations
  4. Energetic Substitution
  5. Introduction in Mathematical Theory of Kinetic Equations
  6. On the Family of the Steady-State Solutions of Vlasov-Maxwell System
  7. Boundary Value Problems for the Vlasov-Maxwell System
  8. Bifurcation of Stationary Solutions of the Vlasov-Maxwell System
  9. Boltzmann Equation
  10. Discrete Models of Boltzmann Equation
  11. Method of Spherical Harmonics and Relaxation of Maxwellian Gas
  12. Discrete Boltzmann equation Models for Mixtures
  13. Quantum Hamiltonians and Kinetic Equations
  14. Modelling of the Limit Problem for the Magnetically Noninsulated Diode
  15. Generalized Liouville Equation and Approximate Orthogonal Decomposition Methods



No. of pages:
© 2011
Print ISBN:
Electronic ISBN:

About the authors

Alexander Sinitsyn

Affiliations and Expertise

Universidad Nacional de Colombia, Bogota, Colombia & Institute for System Dynamics and Control Theory, Russia

Eugene Dulov

Affiliations and Expertise

Universidad Nacional de Colombia, Bogotá, Colombia

Victor Vedenyapin

Affiliations and Expertise

Keldysh Institute of Applied Mathematics, Russia


"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