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Invariant states of a thermally conducting barrier


For an infinite one-dimensional system representing a thermally conducting barrier and two semi-infinite reservoirs which it separates, we prove the existence of a unique stationary probability distribution, to which essentially any initial distribution converges for large times.

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  1. 1.

    S. Ukai and K. Asano, On the Existence and Stability of Stationary Solutions of the Boltzmann Equation for a Gas Flow Past an Obstacle, inSem. College de France, Res. Notes Math. 53 (to appear).

  2. 2.

    S. Goldstein, J. L. Lebowitz, and K. Ravishankar, Ergodic Properties of a System in Contact with a Heat Bath: A One Dimensional Model,Commun. Math. Phys. 85:419–427 (1982).

  3. 3.

    P. Billingsley,Convergence of Probability Measures (Wiley, New York, 1968).

  4. 4.

    W. Feller,An Introduction to Probability Theory and Its Applications. Vol. I (Wiley, New York, 1968), Vol. II (1971).

  5. 5.

    C. Ryll-Nardzewski, Remarks on Processes of Calls, Proc. of Fourth Berkeley Symposium, Vol. 2, pp. 455–465 (1960).

  6. 6.

    S. R. Foguel,The Ergodic Theory of Markov Processes (Van Nostrand, New York, 1969).

  7. 7.

    S. Goldstein, J. L. Lebowitz, and E. Presutti, Stationary Markov Chains,Colloquia Mathematica Societatis Janos Bolyai 27: Random Fields, J. Fritz, J. L. Lebowitz, and D. Szász, eds. (North-Holland, New York, 1981).

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Research partially supported by NSF grant No. PHY-8003298.

Research partially supported by NSF grant No. PHY-8201708.

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Farmer, J., Goldstein, S. & Speer, E.R. Invariant states of a thermally conducting barrier. J Stat Phys 34, 263–277 (1984).

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Key words

  • Invariant states
  • thermally conducting barrier