Skip to main content
SpringerLink
Log in

The Boltzmann equation and its properties

  • Conference paper
  • First Online:
Kinetic Theories and the Boltzmann Equation

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1048))

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

Access this chapter

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 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 46.00
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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References and Footnotes

  1. We adopt the notation and techniques described by James J. Duderstadt and William K. Martin ‘Transport Theory" (John Wiley, New York, 1979). Chapter 3. This textbook contains a number of references to the earlier literature.

    MATH  Google Scholar 

  2. The treatment follows that of Kerson Huang "Statistical Mechanics" (John Wiley, New York, 1963) Chapter 4.

    Google Scholar 

  3. Joel L. Lebowitz and Peter G. Bergmann, Ann. Phys. (N.Y.) 1, 1(1957).

    Article  ADS  MathSciNet  Google Scholar 

  4. H. Grad in Handbuch der Physik, Vol. XII. "Thermodynamics of Gases" (Springer-Verlag, Berlin, 1968).

    Google Scholar 

  5. Sometimes, in order to compress notation, we shall use the symbol (\(\vec w\)) to indicate the molecule whose velocity is \(\vec w\).

    Google Scholar 

  6. See, for example, P. F. Zweifel "Reactor Physics" (McGraw-Hill, New York, 1973) Appendix E.

    Google Scholar 

  7. H. Goldstein. "Classical Mechanics" (Addison-Wesley, Cambridge, Mass., 1953) page 82.

    Google Scholar 

  8. Michael Reed and Barry Simon "Methods of Modern Mathematical Physics-II. Fourier Analysis, Self-Adjointness (Academic Press, New York, 1975) Sec. X. 13.

    MATH  Google Scholar 

  9. Walter Rudin. "Real and Complex Analysis" (McGraw-Hill, New York, 1966) page 21.

    MATH  Google Scholar 

  10. T. Carleman. "Problemès mathématiques dans la théorie cinétique des gaz," (Almqvist and Wiksells, Uppsala. 1957).

    MATH  Google Scholar 

  11. E. Wild, Proc. Camb. Phil. Soc. 47, 602 (1951).

    Article  ADS  MathSciNet  Google Scholar 

  12. D. Morgenstern, Proc. Nat. Acad. Sci. U.S.A. 40, 719 (1954).

    Article  ADS  MathSciNet  Google Scholar 

  13. Lief Arkeryd Arch. Rat. Mech. and Anal. 95, 1 (1972)

    ADS  MathSciNet  Google Scholar 

  14. 17 (1972). c.f. also

    Google Scholar 

  15. "Intermolecular Forces of Infinite Range and the Boltzmann Equation," Chalmers University of Technology (Sweden) preprint (1970).

    Google Scholar 

  16. A. Ja. Povzner, Mat. Sbornik 58, 65 (1962).

    MathSciNet  Google Scholar 

  17. G. Di Blasio, Boll. U.M.I. 8, 127 (1973); Comm. Math. Phys. 38, 331 (1974).

    MathSciNet  Google Scholar 

  18. Shmuel Kaniel and Marvin Shinbrot, Comm. Math. Phys. 58, 65 (1978).

    Article  ADS  MathSciNet  Google Scholar 

  19. Alexander Glikson, Arch. Rat. Mech. Anal. 45, 35 (1972); Bull. Australian Math. Soc. 16, 321 (1977).

    Article  MathSciNet  Google Scholar 

  20. D. Morgenstern, J. Rat. Anal. 4, 533 (1955).

    MathSciNet  Google Scholar 

  21. Barry Simon "The P(φ)2 Euclidean (Quantum) Field Theory" (Princeton Univ. Press, Princeton, N.J., 1974).

    Google Scholar 

  22. Michael Reed and Barry Simon "Methods of Modern Mathematical Physics—I; Functional Analysis" (Academic Press, New York, 1972) p. 151.

    MATH  Google Scholar 

  23. The proof of Lemma 2 requires that R(t1)R(t2) = R(t2)R(t1), otherwise, the "time-ordered" exponential must be used. cf. Chapter VI for a situation in which this is necessary.

    Google Scholar 

  24. H. Grad in "Applications of Nonlinear Partial Differential Equations in Mathematical Physics" (Amer. Math. Soc., Providence, R.I., 1965) p. 154.

    Google Scholar 

  25. cf. Ref. 5, p. 237.

    Google Scholar 

  26. cf. Ref. 24, Appendix.

    Google Scholar 

  27. cf. Ref. 2, p. 131.

    Google Scholar 

  28. H. Grad, Comm. Pure Appl. Math. XVIII, 315 (1965).

    Google Scholar 

  29. H. Grad, Phys. Fluids 6, 147 (1963).

    Article  ADS  MathSciNet  Google Scholar 

  30. Y. Shizuta and K. Asano, Proc. Japan Acad. 53, 3 (1977).

    Article  MathSciNet  Google Scholar 

  31. T. Kato "Perturbation Theory for Linear Operators" (Springer-Verlag, New York, 1966) Chapter IX.

    Book  MATH  Google Scholar 

  32. T. Nishida and K. Imai, Publ. RIMS Kyoto Univ. 229 (1976).

    Google Scholar 

  33. J. P. Guraud, Coll. Int. C.N.R.S. No. 236.

    Google Scholar 

  34. Y. Shizuta, "On the Classical Solutions of the Boltzmann Equation." Preprint.

    Google Scholar 

  35. S. Ukai, Proc. Japan Acad. 50, 179 (1974); C. R. Acad. Sci. Paris 282, A-317 (1976).

    Article  MathSciNet  Google Scholar 

  36. C. Cercignani, W. Greenberg and P. F. Zweifel, J. Stat. Phys. 20, 449 (1979).

    Article  ADS  MathSciNet  Google Scholar 

  37. Herbert Spohn, J. Stat. Phys. 20, 463 (1979).

    Article  ADS  MathSciNet  Google Scholar 

  38. W. Greenberg, J. Voigt and P. F. Zweifel, J. Stat. Phys. 21, 649 (1979).

    Article  ADS  MathSciNet  Google Scholar 

  39. cf. Sec. 12, Lemma 2. Morgenstern's operator R(t;f) is a multiplicative operator, and thus the evolution is given simply by \(\int\limits_0^t {R(t;f(\tau ))d\tau }\) In our case, T(n;t1,t2) is generated by A + v (n) and is hence a non-diagonal matrix, which is why we find it necessary to introduce "time-ordering." Somehow, Morgenstern has been able to circumvent this difficulty by working along the free trajectories.

    Google Scholar 

  40. See Ref. 31, pp. 487 ff.

    Google Scholar 

  41. J. P. Ginzburg, Am. Math. Soc. Translations, Series 2, 96, 189 (1970).

    Google Scholar 

  42. Ref. 21, Sec. VIII.8.

    Google Scholar 

  43. J. Voigt, T.T.S.P. 8, 17 (1979).

    MathSciNet  Google Scholar 

  44. M. Moreau, J. Math. hys. 19, 2494 (1978).

    ADS  MathSciNet  Google Scholar 

  45. J. Voigt "H-theorem for Boltzmann Type Equations." Preprint, Laboratory for Transport Theory and Mathematical Physics, Va. Polytech. Inst. (1979).

    Google Scholar 

  46. H. L. Royden "Real Analysis" (MacMillan, London, 1968) p. 110.

    Google Scholar 

Download references

Authors
  1. Paul F. Zweifel
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Carlo Cercignani

Rights and permissions

Reprints and permissions

Copyright information

© 1984 Springer-Verlag

About this paper

Cite this paper

Zweifel, P.F. (1984). The Boltzmann equation and its properties. In: Cercignani, C. (eds) Kinetic Theories and the Boltzmann Equation. Lecture Notes in Mathematics, vol 1048. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0071879

Download citation

  • DOI: https://doi.org/10.1007/BFb0071879

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12899-1

  • Online ISBN: 978-3-540-38777-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation