Journal of Statistical Physics

, Volume 27, Issue 1, pp 65–74 | Cite as

Heat flow in an exactly solvable model

  • C. Kipnis
  • C. Marchioro
  • E. Presutti


A chain of one-dimensional oscillators is considered. They are mechanically uncoupled and interact via a stochastic process which redistributes the energy between nearest neighbors. The total energy is kept constant except for the interactions of the extremal oscillators with reservoirs at different temperatures. The stationary measures are obtained when the chain is finite; the thermodynamic limit is then considered, approach to the Gibbs distribution is proven, and a linear temperature profile is obtained.

Key words

Heat conduction Fourier's law stochastic dynamics for infinitely many particles additive processes 


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  1. 1.
    Z. Rieder, J. L. Lebowitz, and E. Lieb,J. Math. Phys. 8:1073 (1967); H. Spohn and J. L. Lebowitz,Commun. Math. Phys. 54:97 (1977).Google Scholar
  2. 2.
    M. Bolsterli, M. Rieh, and W. M. Visscher, Simulation of non-harmonic interactions in a crystal by self-consistent reservoirs,Phys. Rev. A 1:1086–1088 (1970); M. Rich and W. M. Visscher, Disordered harmonic chain with self consistent reservoirs,Phys. Rev. B 11:2164–2170 (1975).Google Scholar
  3. 3.
    A. Galves, C. Kipnis, C. Marchioro, and E. Presutti, Non equilibrium measures which exhibit a temperature gradient: study of a model,Commun. Math. Phys. 81:127–147 (1981).Google Scholar
  4. 4.
    V. B. Minassian, Some results concerning local equilibrium Gibbs distributions,Sov. Math. Dokl. 19:1238–1242 (1978).Google Scholar
  5. 5.
    T. M. Liggett, The stochastic evolution of infinite systems of interacting particles,Lecture Notes in Mathematics, No. 598, pp. 188–248 (Springer, Berlin, 1976).Google Scholar
  6. 6.
    E. B. Davies, A model of heat conduction,J. Stat. Phys. 18:161–170 (1978).Google Scholar

Copyright information

© Plenum Publishing Corporation 1982

Authors and Affiliations

  • C. Kipnis
    • 1
  • C. Marchioro
    • 2
  • E. Presutti
    • 3
    • 4
  1. 1.Centre de Mathématique de l'Ecole PolytechniquePalaiseau CedexFrance
  2. 2.Dipartimento di MatematicaLibera Università di TrentoPovoItaly
  3. 3.Istituto MatematicoUniversità di RomaRomaItaly
  4. 4.Institut des Hautes Etudes ScientifiquesBures-sur-YvetteFrance

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