Skip to main content

Diffusion in Disordered Media

  • Conference paper
Nonlinear Stochastic PDEs

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 77))

Abstract

We study the transport properties of a large system of interacting parti­cles moving on an integer lattice in the presence of a random field. This article contains a description of the problem, a survey of results, and an outline of our method. The results include variational formulas for the transport coefficients. The details of the proof of the hydrodynamic limit can be found in [15].

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ambegaokar, V., Halperin, B. I., and Langer, J. S.Hopping conductivity in disordered systems, Phys. Rev. B 4 (1971) 2612.

    Article  Google Scholar 

  2. Brak, R. and Elliott, R. J.Correlated random walks with random hopping rates, Journal of Physics - Condensed Matter, 1989 Dec 25, 1 (51): 10299-­10319.

    Google Scholar 

  3. Brak, R. and Elliott, R. J. Correlated tracer diffusion in a disordered medium, Materials Science and Engineering B - Solid State Materials for Advanced Technology, 1989 Jul, 3 (1–2): 159–162.

    Google Scholar 

  4. Chang, C. C. and YauH. T. Fluctuations of one dimensional Ginzburg-Landau models in nonequilibrium, Comm. Math. Phys. 145 (1992) 209–234.

    Article  MATH  MathSciNet  Google Scholar 

  5. Donsker, M. and Varadhan, S. R. S. Large deviations from a hydrodynamic scaling limit, Comm Pure Appl. Math. 42 (1989), 243–270.

    Article  MathSciNet  Google Scholar 

  6. Fritz, J.Hydrodynamics in a symmetric random medium,Comm. Math. Phys. 125 (1989) 13–25.

    Article  MATH  MathSciNet  Google Scholar 

  7. Gartner, P. and Pitis, R., Occupancy-correlation corrections in hopping, Phys. Rev. B 45 (1992).

    Google Scholar 

  8. Guo, M. Z., Papanicolaou, G. C., and Varadhan, S. R. S. Nonlinear diffusion limit for a system with nearest neighbor interactions, Comm Math. Phys. 118 (1988) 31–53.

    Article  MATH  MathSciNet  Google Scholar 

  9. Kehr, K. W., Paetzold, O., and Wichmann, T., Collective diffusion of lattice gases on linear chain with site-energy disorder, to appear in Phys. Lett. A.

    Google Scholar 

  10. Kipnis, C. and Varadhan, S. R. S., Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusion, Comm. Math. Phys. 106 (1986).

    Google Scholar 

  11. Kirkpatrick, S., Classical transport in disordered media: Scaling and effective-medium theories, Phys. Rev. Lett. 27 (1971) 1722.

    Article  Google Scholar 

  12. Lu, S. L. and Yau, H. T., Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics, Comm. Math. Phys. 156 (1993) 399–433.

    Article  MATH  MathSciNet  Google Scholar 

  13. Miller, A. and Abrahams, E., Impurity conduction at low concentrations, Phys. Rev. 120(1960) 745.

    Article  MATH  Google Scholar 

  14. Papanicolaou, G. C. and Varadhan, S. R. S., Boundary value problems with rapidly oscillating coefficients, In: Random Fields, ed. by J. Fritz, J. L. Lebowitz, and D. Szasz pp. 835–853. North-Holland, Amsterdam (1981).

    Google Scholar 

  15. Quastel, J. and Yau, H. T., Bulk diffusion in a system with site disorder, in preparation.

    Google Scholar 

  16. Quastel, J., Large deviations from a hydrodynamic scaling limit for a nongradient system, preprint.

    Google Scholar 

  17. Quastel, J., Diffusion of color in the simple exclusion process, Comm. Pure Appl. Math. 45 (1992), 623–679.

    Article  MATH  MathSciNet  Google Scholar 

  18. Richards, P. M., Theory of one-dimensional hopping conductivity and diffusion, Phys. Rev. B 16 (1977) 1393–1409.

    Article  MathSciNet  Google Scholar 

  19. Spohn, H. Large scale dynamics of interacting particles, Springer-Verlag (1991).

    MATH  Google Scholar 

  20. Varadhan, S. R. S. Nonlinear diffusion limit for a system with nearest neighbor interactions II, Proc. Tanaguchi Symp., 1990, Kyoto.

    Google Scholar 

  21. Wick, W. D. Hydrodynamic limit of non-gradient interacting particle process, J. Stat. Phys. 54 (1989) 873–892.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag New York, Inc.

About this paper

Cite this paper

Quastel, J. (1996). Diffusion in Disordered Media. In: Funaki, T., Woyczynski, W.A. (eds) Nonlinear Stochastic PDEs. The IMA Volumes in Mathematics and its Applications, vol 77. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8468-7_4

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-8468-7_4

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-8470-0

  • Online ISBN: 978-1-4613-8468-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics