Time evolution and ergodic properties of harmonic systems

  • Oscar E. LanfordIII
  • Joel L. Lebowitz
I. Statistical Mechanics
Part of the Lecture Notes in Physics book series (LNP, volume 38)


We prove the existence of a time evolution and of a stationary equilibrium measure for the infinite harmonic crystal. The ergodic properties of the system are shown to be related in a simple way to the spectrum of the force matrix; when the spectrum is absolutely continuous, as in the translation invariant crystal, the flow is Bernoulli. The quantum crystal is also discussed.


Canonical Ensemble Point Spectrum Orthogonal Transformation Infinite System Complex Hilbert Space 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. A. Maradudin, E. W. Montroll and G. H. Weiss (with I. P. Ipatova), Theory of Lattice Dynamics in the Harmonic Approximation, Academic Press (1963, (1971)).Google Scholar
  2. [2]
    J. L. Lebowitz in Statistical Mechanics: New Concepts, New Problems, New Applications, S. A. Rice, K. F. Freed and J. C. Light, editors, p. 41, University of Chicago Press (1972).Google Scholar
  3. [3]
    U. M. Titulaer, Physica 70, 257, 70, 276, 70, 456 (1973).Google Scholar
  4. [4]
    J. L. Lebowitz and A. Martin-Löf, Commun. Math. Phys. 25, 272 (1973).Google Scholar
  5. [5]
    V. A. Rohlin, Amer. Math. Soc. Transl. 10, 1 (1962); 49, 171 (1966).Google Scholar
  6. [6]
    T. Hida, Stationary Stochastic Processes, Princeton University Press, 1970.Google Scholar
  7. [7]
    D. Ornstein, Ergodic Theory, Randomness, and Dynamical Systems, Yale University Press, 1974.Google Scholar
  8. [8]
    C.f. A. J. O'Connor and J. L. Lebowitz, J. Math. Phys. 15, 692 (1974), Section 8.Google Scholar
  9. [9]
    P. Mazur and E. W. Montroll, J. Math. Phys. 1, 70 (1960).Google Scholar
  10. [10]
    R. I. Cukier and P. Mazur, Physica 53 (1971) 157Google Scholar
  11. [11]
    C.f. D. Ruelle, Statistical Mechanics, Benjamin (1969).Google Scholar

Copyright information

© Springer-Verlag 1975

Authors and Affiliations

  • Oscar E. LanfordIII
    • 1
  • Joel L. Lebowitz
    • 2
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeley
  2. 2.Belfer Graduate School of ScienceYeshiva University

Personalised recommendations