Journal of Statistical Physics

, Volume 30, Issue 1, pp 123–155 | Cite as

Convergence to stationary states for infinite harmonic systems

  • Carlo Boldrighini
  • Alessandro Pellegrinotti
  • Livio Triolo


We study the evolution of the states for one-dimensional infinite harmonic systems, interacting through a translation invariant force of rapid decrease. We prove that for a large class of initial states convergence to a Gaussian limiting state, as time goes to infinity, is equivalent to convergence of the covariance. The main assumption on the initial states is a kind of weak dependence between distant regions (mixing condition). We prove also convergence of the covariance under some general assumptions. We show furthermore that there are two countable families of intensive constants of the motion, which are “inherited” from the corresponding finite systems. The translation invariant limiting states are in one-to-one correspondence with the admissible values of these constants of the motion. Moreover, under some additional regularity assumption, such states are shown to be Gibbs states, obtained by a “Boltzmann-Gibbs” prescription.

Key words

Harmonic oscillators convergence to equilibrium first integrals 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. L. Dobrushin and Yu. M. Sukhov, On the Problem of the Mathematical Foundation of the Gibbs Postulate in Classical Statistical Mechanics, inMathematical Problems in Theoretical Physics-Proceedings, Rome 1977. Lecture Notes in Physics No. 80 (Springer-Verlag, Berlin, 1978).Google Scholar
  2. 2.
    R. L. Dobrushin and J. Fritz,Commun. Math. Phys. 55:275–292 (1977); andCommun. Math. Phys. 57:67-81 (1977).Google Scholar
  3. 3.
    O. E. Lanford III, J. L. Lebowitz, and E. H. Lieb,J. Stat. Phys 16:453–461 (1977); C. Marchioro, A. Pellegrinotti, and M. Pulvirenti,J. Math. Phys. 22:1740-1745 (1981).Google Scholar
  4. 4.
    Ya. G. Sinai and K. L. Volkoviyski,Fund. Anal. Its Appl. 5:185–187 (1971).Google Scholar
  5. 5.
    Ya. G. Sinai,Funct. Anal. Its Appl. 6:35–43 (1972).Google Scholar
  6. 6.
    C. Boldrighini, R. L. Dobrushin, and Yu. M. Sukhov, Time Asymptotics for Some Degenerate Models of the Evolution of Infinite Particle Systems, Universita di Camerino (1980).Google Scholar
  7. 7.
    A. A. Maradudin, E. W. Montroll, and G. H. Weiss,Theory of Lattice Dynamics in the Harmonic Approximation (Academic Press, New York, 1971).Google Scholar
  8. 8.
    G. Klein and I. Prigogine,Physica 19:1053–1071 (1953).Google Scholar
  9. 9.
    O. E. Lanford III and J. L. Lebowitz, Time Evolution and Ergodic Properties of Harmonic Systems, inDynamical Systems, Theory and Applications, Lecture Notes in Physics No. 38 (Springer-Verlag, Berlin, 1975).Google Scholar
  10. 10.
    J. L. Van Hemmen,Phys. Rep. 65:43–149 (1980).Google Scholar
  11. 11.
    H. Spohn and J. Lebowitz,Commun. Math. Phys. 54:97–120 (1977).Google Scholar
  12. 12.
    O. E. Lanford III and D. Robinson,Commun. Math. Phys. 24:193–210 (1971).Google Scholar
  13. 13.
    I. A. Ibragimov and Yu. V. Linnik,Independent and Stationary Sequences of Random Variables (Noordhoff, Groningen, 1971).Google Scholar
  14. 14.
    Y. Katznelson,An Introduction to Harmonic Analysis (Dover, New York, 1976).Google Scholar
  15. 15.
    W. S. Stout,Almost Sure Convergence (Academic Press, New York, 1974).Google Scholar
  16. 16.
    N. Wiener,The Fourier Integral and Certain of its Applications (Dover, New York, 1958).Google Scholar
  17. 17.
    R. L. Dobrushin, Gaussian Random Fields — Gibbsian Point of View, inMulticomponent Random Systems, R. L. Dobrushin and Ya. G. Sinai, eds.,Advances in Probability and Related Topics, Vol. 6 (Marcel Dekker Inc., New York, 1980).Google Scholar
  18. 18.
    M. V. Fedoriuk,The Saddle Point Method (In russian) (Nauka, Moscow, 1977).Google Scholar
  19. 19.
    T. Kawata,Fourier Analysis in Probability Theory (Academic Press, New York, 1972).Google Scholar

Copyright information

© Plenum Publishing Corporation 1983

Authors and Affiliations

  • Carlo Boldrighini
    • 1
  • Alessandro Pellegrinotti
    • 1
  • Livio Triolo
    • 2
  1. 1.Instituto Matematico, Universitá di CamerinoG.N.F.M., C.N.R.CamerinoItaly
  2. 2.Instituto di Matematica Applicata, Universita di RomaG.N.F.M., C.N.R.RomaItaly

Personalised recommendations