Journal of Statistical Physics

, Volume 50, Issue 3–4, pp 657–687 | Cite as

Statistical mechanics of the nonlinear Schrödinger equation

  • Joel L. Lebowitz
  • Harvey A. Rose
  • Eugene R. Speer
Articles

Abstract

We investigate the statistical mechanics of a complex fieldø whose dynamics is governed by the nonlinear Schrödinger equation. Such fields describe, in suitable idealizations, Langmuir waves in a plasma, a propagating laser field in a nonlinear medium, and other phenomena. Their Hamiltonian
$$H(\phi ) = \int_\Omega {[\frac{1}{2}|\nabla \phi |^2 - (1/p) |\phi |^p ] dx}$$
is unbounded below and the system will, under certain conditions, develop (self-focusing) singularities in a finite time. We show that, whenΩ is the circle and theL2 norm of the field (which is conserved by the dynamics) is bounded byN, the Gibbs measureυ obtained is absolutely continuous with respect to Wiener measure and normalizable if and only ifp andN are such that classical solutions exist for all time—no collapse of the solitons. This measure is essentially the same as that of a one-dimensional version of the more realisitc Zakharov model of coupled Langmuir and ion acoustic waves in a plasma. We also obtain some properties of the Gibbs state, by both analytic and numerical methods, asN and the temperature are varied.

Key words

Nonlinear Schrödinger equation statistical mechanics unbounded Hamiltonians singularities Gibbs measures 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J.-P. Eckmann and D. Ruelle,Rev. Mod. Phys. 57:617–656 (1985).Google Scholar
  2. 2.
    V. E. Zakharov,Sov. Phys. JETP 35:908–914 (1972).Google Scholar
  3. 3.
    M. V. Goldman,Rev. Mod. Phys. 56:709–735 (1984).Google Scholar
  4. 4.
    H. A. Rose, D. F. DuBois, and B. Bezzerides,Phys. Rev. Lett. 58:2547–2550 (1985).Google Scholar
  5. 5.
    D. W. McLaughlin, G. C. Papanicolaou, C. Sulem, and P. L. Sulem,Phys. Rev. A 34:1200–1210 (1986).Google Scholar
  6. 6.
    D. Ruelle,Statistical Mechanics (Benjamin, Reading, Massachusetts, 1974).Google Scholar
  7. 7.
    J. Glimm and A. Jaffe,Quantum Physics (Springer-Verlag, New York, 1981).Google Scholar
  8. 8.
    J. Ginibre and G. Velo,Ann. Inst. Henri Poincaré 28:287–316 (1978).Google Scholar
  9. 9.
    V. E. Zakharov and A. B. Shabat,Sov. Phys. JETP 34:62–69 (1972).Google Scholar
  10. 10.
    D. Russell, D. F. DuBois, and H. A. Rose,Phys. Rev. Lett. 56:838–841 (1986).Google Scholar
  11. 11.
    M. Weinstein,Commun. Math. Phys. 87:567–576 (1983).Google Scholar
  12. 12.
    S. H. Schochet and M. I. Weinstein,Commun. Math. Phys. 106:569–580 (1986).Google Scholar
  13. 13.
    H. Tasso,Phys. Lett. A 120:464–465 (1987).Google Scholar
  14. 14.
    G. Pelletier,J. Plasma Phys. 24:287–297 (1980).Google Scholar
  15. 15.
    G. Pelletier,J. Plasma Phys. 24:421–443 (1980).Google Scholar
  16. 16.
    G. Z. Sun, D. R. Nicholson, and H. A. Rose,Phys. Fluids 28:2395–2405 (1985).Google Scholar
  17. 17.
    R. H. Kraichnan and D. Montgomery,Rep. Prog. Phys. 43:547–619 (1979).Google Scholar
  18. 18.
    L. Nirenberg,Commun. Pure Appl. Math. 8:648–674 (1955).Google Scholar
  19. 19.
    R. T. Glassey,J. Math. Phys. 18:1794–1797 (1977).Google Scholar
  20. 20.
    O. Kavian, A Remark on the blowing-up of solutions to the Cauchy problem for the non-linear Schrödinger equation, preprint, Lab. Ana. Nu., Universite P. & M. Curie.Google Scholar
  21. 21.
    B. Simon,Functional Integration and Quantum Physics (Academic Press, New York, 1979).Google Scholar
  22. 22.
    W. Feller,An Introduction to Probability Theory and Its Applications, Vol. II (Wiley, New York, 1971).Google Scholar
  23. 23.
    D. Dürr and A. Bach,Commun. Math. Phys. 60:153–170 (1978).Google Scholar
  24. 24.
    G. H. Hardy and J. E. Littlewood,Acta Mathematica 37:193–238 (1914).Google Scholar
  25. 25.
    E. Hille and R. S. Phillips,Functional Analysis and Semi-Groups (American Mathematical Society, Providence, Rhode Island, 1957).Google Scholar
  26. 26.
    A. Zygmund,Trigonometrical Series (Warsaw, 1935).Google Scholar
  27. 27.
    B. Simon,The P(φ) Euclidean (Quantum) Field Theory (Princeton University Press, Princeton, New Jersey, 1974).Google Scholar
  28. 28.
    G. Papanicolaou, private communication.Google Scholar

Copyright information

© Plenum Publishing Corporation 1988

Authors and Affiliations

  • Joel L. Lebowitz
    • 1
  • Harvey A. Rose
    • 2
  • Eugene R. Speer
    • 3
  1. 1.Departments of Mathematics and PhysicsRutgers UniversityNew Brunswick
  2. 2.Los Alamos National LaboratoryLos Alamos
  3. 3.Department of MathematicsRutgers UniversityNew Brunswick

Personalised recommendations