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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
Articles

Abstract

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 

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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

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