Skip to main content
SpringerLink
Log in
Book cover

Kinetic Theory and Gas Dynamics pp 183–211Cite as

Kinetic Equations from Hamiltonian Dynamics: The Markovian Approximations

  • Chapter

Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 293))

Abstract

The goal of non-equilibrium statistical mechanics is to explain the macroscopic behavior of matter from the dynamics of its microscopic constituents, i.e. atoms or molecules. Because of the large number of particles involved, this is a rather ambitious program. Therefore, as an intermediate step, one tries to write down an approximately valid dynamics as governed by a kinetic equation. Examples are the Boltzmann equation for a dilute gas, the hydrodynamic equations for an “aged” fluid, etc., There is a lot known about the interrelationship between the microscopic and kinetic description of a physical system and it would be rather hopeless to press all these results into one lecture. Therefore I would like to do three things.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. O. E. Lanford, Time evolution of large classical systems. In: Dynamical Systems, Theory and Applications, ed. J. Moser, Lecture Notes in Physics 38, p. 1. Springer, Berlin (1975)

    Chapter  Google Scholar 

  2. O. E. Lanford, On the derivation of the Boltzmann equation. Soc. Math. de France, Astérisque 40, 117 (1976)

    MathSciNet  Google Scholar 

  3. F. King, BBGKY hierarchy for positive potentials. Ph. D. Thesis, Dep. of Math., Univ. of Cal. at Berkeley (1975)

    Google Scholar 

  4. H. van Beijeren, O. E. Lanford, J. L. Lebowitz, H. Spohn, Equilibrium time correlation functions in the low density limit, preprint (1979)

    Google Scholar 

  5. H. Spohn, Kinetic equations from Hamiltonian dynamics: The Markovian limit. Lecture Notes, University of Leuven, 1979

    Google Scholar 

  6. G. Papanicalaou, private communication

    Google Scholar 

  7. H. Neunzert, Neuere qualitative und numerische Methoden in.er Plasmaphysik, Vorlesungsmanuskript Paderborn, 1975

    Google Scholar 

  8. W. Braun, K. Hepp, Comm. Math. Phys. 56, 101 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  9. G. Gallavotti, Lectures on the billiard. In: Dynamical Systems, Theory and Application, ed. J. Moser. Lecture Notes in Physics 38, p. 236. Springer, Berlin (1975)

    Chapter  Google Scholar 

  10. G. Papanicalaou, Bull. AMS 81, 330 (1975)

    Article  Google Scholar 

  11. E. H. Hauge, What can one learn from Lorentz models? In; Transport Phenomena, ed. L. Garrido, Lecture Notes in Physics 31, p. 338. Springer, Berlin (1974)

    Google Scholar 

  12. B. J. Alder, W. E. Alley, J. Stat. Phys. 19, 341 (1978)

    Article  Google Scholar 

  13. J. C. Lewis, J. A. Tjon, Physics Letters 66A, 349 (1978)

    Article  Google Scholar 

  14. R. K. Alexander, The infinite hard sphere system. Ph. D. Thesis, Dep. of Math. Univ. of Cal. at Berkeley (1975)

    Google Scholar 

  15. C. Cercignani, Transport Theory and Statistical Physics 2, 211 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  16. O. E. Lanford, Notes of the lectures at the troisième cycle at Lausanne, 1978, unpublished

    Google Scholar 

  17. A. A. Abrikosov, I. M. Khalatnikov, Soviet Phys. JETP 34, 135 (1958)

    MathSciNet  Google Scholar 

  18. M. Bixon, R. Zwanzig, Phys. Rev. 187, 267 (1969)

    Article  MathSciNet  Google Scholar 

  19. R. F. Fox, G. E. Uhlenbeck, Phys. Fluids 13, 1893, 2881 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  20. T. Kirkpatrick, E. G. D. Cohen, J. R. Dorfman, Phys. Rev. Lett. 42, 862 (1979)

    Article  Google Scholar 

  21. N. G. van Kampen, Phys. Lett. A13, 458 (1976)

    Google Scholar 

  22. M. Kac, J. Logan, Fluctuations. Studies in Statistical Mechanics, to appear

    Google Scholar 

  23. M. Kac, Fluctuations near and far from equilibrium. In: Statistical Mechanics and Statistical Methods in Theory and Application. Ed. U. Landman. Plenum Press, New York (1977)

    Google Scholar 

  24. M. Klaus, Helv. Phys. Acta 48, 99 (1975)

    MathSciNet  Google Scholar 

  25. R. Ellis, M. Pinsky. J. de Mathématique Pure et Appliqué 54, 125 (1975)

    MATH  MathSciNet  Google Scholar 

  26. N. G. van Kampen, The expansion of the master equation. In: Advances in Chemical Physics 34, p. 245, ed. I. Prigogine. John Wiley, New York (1976)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universität München, München, Germany

    H. Spohn

Authors
  1. H. Spohn
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Politecnico di Milano, Italy

    C. Cercignani

Rights and permissions

Reprints and permissions

Copyright information

© 1988 Springer-Verlag Wien

About this chapter

Cite this chapter

Spohn, H. (1988). Kinetic Equations from Hamiltonian Dynamics: The Markovian Approximations. In: Cercignani, C. (eds) Kinetic Theory and Gas Dynamics. International Centre for Mechanical Sciences, vol 293. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2762-9_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-7091-2762-9_6

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-82090-2

  • Online ISBN: 978-3-7091-2762-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation