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


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 


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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

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