Communications in Mathematical Physics

, Volume 68, Issue 1, pp 69–82 | Cite as

Transmission coefficient and heat conduction of a harmonic chain with random masses: Asymptotic estimates on products of random matrices

  • Theo Verheggen


We find upper and lower bounds for the transmission coefficient of a chain of random masses. Using these bounds we show that the heat conduction in such a chain does not obey Fourier's law: For different temperatures at the ends of a chain containingN particles the energy flux falls off likeN−1/2 rather thanN−1.


Neural Network Fourier Statistical Physic Complex System Lower Bound 
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  1. 1.
    Casher, A., Lebowitz, J.L.: J. Math. Phys.12, 1701 (1971)Google Scholar
  2. 2.
    Rubin, R.J., Greer, W.L.: J. Math. Phys.12, 1686 (1971)Google Scholar
  3. 3.
    O'Connor, A.J., Lebowitz, J.L.: J. Math. Phys.15, 692 (1974)Google Scholar
  4. 4.
    Pastur, L.A., Feldman, E.P.: Sov. Phys. JETP40, 241 (1975)Google Scholar
  5. 5.
    O'Connor, A.J.: Commun. Math. Phys.45, 63 (1975)Google Scholar
  6. 6.
    Papanicolau, G.C.: On the instability of the random harmonic oscillator. Preprint (1978)Google Scholar
  7. 7.
    Spohn, H., Lebowitz, J.L.: Commun. Math. Phys.54, 97 (1977)Google Scholar
  8. 8.
    Matsuda, M., Ishii, K.: Suppl. Prog. Theor. Phys.45, 56 (1970)Google Scholar
  9. 9.
    Payton, D.N., Rich, M., Visscher, W.M.: Phys. Rev.160, 706 (1967)Google Scholar
  10. 10.
    Keller, J.B., Papanicolau, G.C., Weilenmann, J.: Commun. Pure Appl. Math.31, 583 (1978)Google Scholar
  11. 11.
    Papanocilau, G.C.: Unpublished notes (1976)Google Scholar
  12. 12.
    Yoshioka, Y.: Proc. Jpn. Acad.49, 665 (1973)Google Scholar
  13. 13.
    Furstenberg, H.: Trans. A.M.S.108, 377 (1963)Google Scholar
  14. 14.
    Doob, J.L.: Stochastic processes. New York: Wiley 1953Google Scholar
  15. 15.
    Gikhman, I.I., Skorokhod, A.V.: Introduction to the theory of random processes. Philadelphia, London, Toronto: Saunders 1969Google Scholar
  16. 16.
    Krein, M.G., Rutman, M.A.: Usp. Mat. Nauk.23, 3 (1948) (A.M.S Transl. no. 26 (1950))Google Scholar
  17. 17.
    Visscher, W.M.: Prog. Theor. Phys.46, 729 (1971)Google Scholar
  18. 18.
    Visscher, W.M.: Methods Comput. Phys.15, 371 (1976)Google Scholar
  19. 19.
    Goldshtein, I.Ya, Molchanov, S.A., Pastur, L.A.: Funct. Anal. Appl.11, 1 (1977)Google Scholar
  20. 20.
    Ruelle, D.: Ann. N.Y. Acad. Sci. Conf. on Bifurcation (1978). In pressGoogle Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • Theo Verheggen
    • 1
  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA

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