Journal of Statistical Physics

, Volume 80, Issue 1–2, pp 125–146

Effective conductivity and skew Brownian motion

  • Reinhard Lang
Articles

Abstract

We consider heat conduction in a periodic body which is composed of finitely many different components. The effective conductivity is represented in terms of skew Brownian motion. The representation formula is a fluctuation-dissipation relation. The dissipation term in this formula is related to the transmission of heat through the surface separating the different components of the body; it is described by the skew reflections of Brownian motion at these surfaces. The problems caused by the discontinuity of the microscopic conductivity are handled in the framework of Dirichlet forms.

Key Words

Effective conductivity Green-Kubo formula discontinuous random media skew Brownian motion reversible diffusions Dirichlet forms 

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References

  1. 1.
    S. V. Anulova, Diffusion processes with singular characteristics, inStochastic Differential Systems. Filtering and Control, B. Grigelionis, ed. (Springer, Berlin, 1980), pp. 264–269.Google Scholar
  2. 2.
    R. F. Bass and P. Hsu, The semimartingale structure of reflecting Brownian motion,Proc. Am. Math. Soc. 108:1007–1010 (1990).Google Scholar
  3. 3.
    R. F. Bass and P. Hsu, Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains,Ann. Prob. 19:486–508 (1991).Google Scholar
  4. 4.
    J. R. Baxter and G. A. Brosamler, Energy and the law of the iterated logarithm,Math. Scand. 38:115–136 (1976).Google Scholar
  5. 5.
    A. De Masi, P. A. Ferrari, S. Goldstein, and D. W. Wick, An invariance principle for reversible Markov processes: Applications to random motions in random environments,J. Stat. Phys. 55:787–855 (1989)Google Scholar
  6. 6.
    M. Fukushima,Dirichlet Forms and Markov Processes (North-Holland, Amsterdam, 1980).Google Scholar
  7. 7.
    K. Golden and G. Papanicolaou, Bounds for effective parameters of heterogeneous media by analytic continuation,Comm. Math. Phys. 90:473–491 (1983).Google Scholar
  8. 8.
    P. Hsu, On excursions of reflecting Brownian motion,Trans. Am. Math. Soc. 296:239–264 (1986).Google Scholar
  9. 9.
    K. Itô and H. P. McKean, Brownian motions on a half line,Illinois J. Math. 7:181–231 (1963).Google Scholar
  10. 10.
    R. Lang, Une représentation probabiliste de la conductivité effective,C. R. Acad. Sci. Paris 317 (Ser. I):967–969 (1993).Google Scholar
  11. 11.
    K. Lichtenecker, Die Dielektrizitätskonstante natürlicher und künstlicher Mischkörper,Phys. Z. 27:115–158 (1926).Google Scholar
  12. 12.
    H. Osada, Homogenization of diffusion processes with random stationary coefficients, inProceedings of the 4th Japan-USSR Symposium on Probability Theory (Springer, Berlin, 1983), pp. 507–517.Google Scholar
  13. 13.
    H. Osada and T. Saitoh, An invariance principle for non-symmetric Markov processes and reflecting diffusions in random domains,Prob. Theory Related Fields (1995), to appear.Google Scholar
  14. 14.
    G. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating coefficients, inRandom Fields, J. Fritz, J. L. Lebowitz, and D. Szász, eds. (North-Holland, Amsterdam, 1981), pp. 835–853.Google Scholar
  15. 15.
    D. Revuz and M. Yor,Continuous Martingales and Brownian Motion, (Springer, Berlin, 1991).Google Scholar
  16. 16.
    H. Spohn,Large Scale Dynamics of Interacting Particles (Springer, Berlin, 1991).Google Scholar
  17. 17.
    V. V. Zhikov, S. M. Kozlov, and O. A. Oleinik,Homogenization of Differential Operators and Integral Functionals (Springer, Berlin, 1994).Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • Reinhard Lang
    • 1
  1. 1.Institut für Angewandte MathematikHeidelbergGermany

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