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Communications in Mathematical Physics

, Volume 125, Issue 1, pp 13–25 | Cite as

Hydrodynamics in a symmetric random medium

  • J. Fritz
Article

Abstract

We investigate the hydrodynamic behaviour of a one-dimensional Ginzburg-Landau model with conservation law in the presence of random conductivities. It is found that there is no interference between nonlinearity and randomness; the conductivities average out in the same way as they do in the case of the underlying random walk in the given medium. This means that we have an effective conductivity specified as the harmonic mean of microscopic conductivities. Some extensions including multidimensional systems in a small electric field are also discussed.

Keywords

Neural Network Statistical Physic Complex System Random Walk Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Boldrighini, C., Dobrushin, R.L., Suhov, Yu.M.: The hydrodynamic limit of a degenerate model of statistical physics. Usp. Mat. Nauk35, 152 (1980) (short communication, in Russian)Google Scholar
  2. 2.
    Boldrighini, C., Dobrushin, R.L., Suhov, Yu.M.: One-dimensional hard rod caricature of hydrodynamics. Stat. Phys.31, 577–616 (1983)Google Scholar
  3. 3.
    Dobrushin, R.L.: On the derivation of the equations of hydrodynamics. Lecture, Mathematical Institute, Budapest (1978)Google Scholar
  4. 4.
    Dobrushin, R.L., Siegmund-Schultze, R.: The hydrodynamic limit for systems of particles with independent evolution. Math. Nachr.105, 199–244 (1982)Google Scholar
  5. 5.
    Donsker, M.D., Varadhan, S.R.S.: Large deviations from a hydrodynamic scaling limit. Preprint 1988.Google Scholar
  6. 6.
    Fritz, J.: On the hydrodynamic limit of a one-dimensional Ginzburg-Landau lattice model. The a priori bounds. Stat. Phys.47, 551–572 (1987)Google Scholar
  7. 7.
    Fritz, J.: On the hydrodynamic limit of a Ginzburg-Landau lattice model. The law of large numbers in arbitrary dimensions. Preprint (1987). Probab. Theor. Rel. Fields81, 291–318 (1989)Google Scholar
  8. 8.
    Fritz, J., Maes, Ch.: Hydrodynamic behaviour in a small external field. Preprint. Stat. Phys.53, 1179–1205 (1988)Google Scholar
  9. 9.
    Funaki, T.: Derivation of the hydrodynamical equation for a 1-dimensional Ginzburg-Landau model. Preprint, to appear in Probab. Theor. Rel. Fields (1987)Google Scholar
  10. 10.
    Guo, M.Z. Papanicolau, G.C., Varadhan, S.R.S. (1988): Nonlinear diffusion limit for a system with nearest neighbor interactions. Commun. Math. Phys.118, 31–59 (1988)Google Scholar
  11. 11.
    Morrey, C.: On the derivation of the equations of hydrodynamics from statistical mechanics. Commun. Pure. Appl. Math.8, 279–327 (1955)Google Scholar
  12. 12.
    Papanicolau, G.C., Varadhan, S.R.S.: Boundary value problems with rapidly oscillating coefficients. In: Random Fields. Fritz, J., Lebowitz, J.L., Szász, D. (eds.) pp. 835–853. János Bólyai Mathematical Society and North Holland, Vol. II (1981)Google Scholar
  13. 13.
    Rost, H.: Non-equilibrium behaviour of a many particle system. Density profile and local equilibrium. Z. Wahrsch. Verw. Geb.58, 41–55 (1981)Google Scholar
  14. 14.
    Reed, M., Simon, B.: Methods of modern mathematical physics II. Fourier analysis, Self-Adjointness. New York: Academic Press 1975Google Scholar
  15. 15.
    Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton, NJ: Princeton University Press 1970Google Scholar
  16. 16.
    Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton, NJ: Princeton University Press 1971Google Scholar
  17. 17.
    Varadhan, S.R.S.: In preparationGoogle Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • J. Fritz
    • 1
    • 2
  1. 1.Mathematical Institute, H.A.S.BudapestHungary
  2. 2.Institut des Hautes Etudes ScientifiquesBures-sur-YvetteFrance

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