Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Hydrodynamic limit of a disordered lattice gas
Download PDF
Download PDF
  • Published: 04 November 2003

Hydrodynamic limit of a disordered lattice gas

  • Alessandra Faggionato1 &
  • Fabio Martinelli2 

Probability Theory and Related Fields volume 127, pages 535–608 (2003)Cite this article

  • 145 Accesses

  • 25 Citations

  • Metrics details

Abstract.

We consider a model of lattice gas dynamics in ℤd in the presence of disorder. If the particle interaction is only mutual exclusion and if the disorder field is given by i.i.d. bounded random variables, we prove the almost sure existence of the hydrodynamical limit in dimension d≥3. The limit equation is a non linear diffusion equation with diffusion matrix characterized by a variational principle.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C., Scheffer, G.: Sur les inégalités de Sobolev logarithmiques. Société Mathématique de France, Paris, 2000

  2. Bernardin. C.: Regularity of the diffusion coefficient for a lattice gas reversible under Bernoulli measures. Stoch. Proc. their Appl. 101, 43–68 (2002)

    Article  Google Scholar 

  3. Benjamini, I., Ferrari, P., Landim, C.: Asymmetric conservative processes with random rates. Stochastic Processes Appl. 61 181–204 (1996)

    Google Scholar 

  4. Billingsley, P.: Convergence of probability Measures. John Wiley & Sons, New York, 1968

  5. Böttger, H., Bryksin. V.V.: Hopping conduction in solids. Akademie Verlag, Berlin, 1985

  6. Cancrini, N., Martinelli, F.: Comparison of finite volume canonical and grand canonical Gibbs measures under a mixing condition. Markov Proc. Rel. Fields 6 (1), 23–72 (2000)

    MATH  Google Scholar 

  7. Cancrini, N., Martinelli, F.: On the spectral gap of Kawasaki dynamics under a mixing condition revisited. J. Math. Phys. 41 (3), 1391–1423 (2000)

    Article  MATH  Google Scholar 

  8. Cancrini, N., Martinelli. F.: Diffusive scaling of the spectral gap for the dilute Ising lattice-gas dynamics below the percolation threshold. Probab. Theory Related Fields 120 (4), 497–534 (2001)

    MATH  Google Scholar 

  9. Cancrini, N., Cesi, F., Martinelli, F.: The spectral gap for the Kawasaki dynamics at low temperature. J. Statist. Phys. 95 (1-2), 215–271 (1999)

    Google Scholar 

  10. Caputo. P.: Uniform Poincaré inequalities for unbounded conservative spin systems: the non interacting case. Preprint 2002, to appear in Adv. Studies in Pure Math.

  11. Caputo, P., Martinelli, F.: Relaxation time of anisotropic simple exclusion processes and quantum Heisenberg models. To appear in Ann. Appl. Prob.

  12. Carlen, E., Caravalho, M.C., Loss, M.: Determination of the spectral gap for Kac’s master equation and related stochastic evolutions. Preprint, 2002

  13. De Masi, A., Ferrari, P.A., Goldstein, S., Wick, W.D.: An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Statistical Phys. 55 (3/4), 1989

  14. Faggionato, A.: Hydrodynamic limit of a disordered system. Ph.D. Thesis. Scuola Normale Superiore di Pisa. 2002 http://mpej.unige.ch/mparc/c/03/03-37.ps.gz

  15. Fritz, J.: Hydrodynamics in a symmetric random medium. Comm. Math. Phys. 125 (1), 13–25 (1989)

    MATH  Google Scholar 

  16. Gartner, P., Pitis, R.: Occupacy-correlation corrections in hopping. Phys. Rev. B. 45 (1992)

  17. Guo, M.Z., Papanicolaou, G.C., Varadhan, S.R.S.: Nonlinear diffusion limit for a system with nearest neighbor interactions. Comm. Math. Phys. 118, 31–59 (1988)

    MathSciNet  MATH  Google Scholar 

  18. Kehr, K.W., Paetzold, O.: Collective and tracer diffusion of lattice gases in lattices with site-energy disorder. Physica A 190, 1–12 (1992)

    Article  Google Scholar 

  19. Kehr, K.W., Wichmann, T.: in Diffusion processes: experiment, theory, simulations. A. Pekalski, (ed.), Lecture Notes in Physics Vol. 438 Springer, 1994, p. 179

    Google Scholar 

  20. Kehr, K.W., Wichmann, T.: Diffusion coefficients of single and many particles in lattices with different forms of disorder. cond-mat/9602121

  21. Kehr, K.W., Paetzold, O., Wichmann, T.: Collective diffusion of lattice gases on linear chains with site-energy disorder. Phys. Lett. A 182, 135–139 (1993)

    Article  Google Scholar 

  22. Kipnis, C., Landim, C.: Scaling limits of interacting particle systems. Grundleheren der mathematischen Wissenschaften 320, Springer Verlag, 1999

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

    Article  Google Scholar 

  24. Koukkous, A.: Hydrodynamic behavior of symmetric zero-range processes with random rates. Stochastic Processes Appl. 84, 297–312 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. Liggett, T.M.: Interacting particle systems. Springer-Verlag, New York, 1985

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

    MathSciNet  MATH  Google Scholar 

  27. Nagy, K.: Symmetric random walk in random environment in one dimension. Period. Math. Hungar. 45, 101–120 (2002)

    Article  MathSciNet  Google Scholar 

  28. Quastel, J.: Diffusion in disordered media. In: T. Funaki and W. Woyczinky, (eds.), Proceedings on stochastic method for nonlinear P.D.E., IMA volumes in Mathematics, 77, Springer Verlag, New York, 1995, pp. 65–79

  29. Quastel, J., Yau, H.T.: Bulk diffusion in a system with site disorder. Unpublished notes, pages 1–33

  30. Quastel, J., Yau, H.T.: Poincaré inequalities for inhomogeneous Bernoulli measures. Preprint, March 2003

  31. Quastel, J.: Diffusion of color in the simple exclusion process. Comm. Pure Appl. Math. 45 (6), 623–679 (1992)

    MATH  Google Scholar 

  32. Richards. P.M.: Theory of one-dimensional hopping conductivity and diffusion. Phys. Rev. B 16, 1393–1409 (1997)

    Google Scholar 

  33. Seppalainen, T.: Recent results and open problems on the hydrodynamics of disordered asymmetric exclusion and zero-range processes. Resenhas IME-USP 4, 1–15 (1999)

    Google Scholar 

  34. Spohn. H.: Large Scale Dynamics of Interacting Particles. Springer Verlag, Berlin, 1991

  35. Spohn, H., Yau, H.T.: Bulk diffusivity of lattice gases close to criticality. J. Stat. Phys. 79, 231–241 (1995)

    Google Scholar 

  36. Varadhan, S.R.S.: Nonlinear diffusion limit for a system with nearest neighbor interactions II. In K. D. Elworthy and N. Ikeda, (eds), Asymptotic Problems in Probability Theory: Stochastic Models and Diffusion on Fractals. Vol. 283 of Pitman Research Notes in Mathematics, John Wiley & Sons, New York, 75–128 (1994)

  37. Varadhan, S.R.S., Yau, H.T.: Diffusive limit of lattice gases with mixing conditions. Asian J. Math. 1 (4), 623–678 (1997)

    MATH  Google Scholar 

  38. Wick, W.D.: Hydrodynamic limit of non-gradient interacting particle process. J. Stat. Phys. 54, 832–892 (1989)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät II - Mathematik und Naturwissenschaften, Technische Universität Berlin, Strasse des 17, Juni 136, 10623, Berlin, Germany

    Alessandra Faggionato

  2. Dipartimento di Matematica, Universita’ di Roma Tre, L.go S. Murialdo 1, 00146, Roma, Italy

    Fabio Martinelli

Authors
  1. Alessandra Faggionato
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fabio Martinelli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fabio Martinelli.

Additional information

Mathematics Subject Classification (2000): 60K40, 60K35, 60J27, 82B10, 82B20

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Faggionato, A., Martinelli, F. Hydrodynamic limit of a disordered lattice gas. Probab. Theory Relat. Fields 127, 535–608 (2003). https://doi.org/10.1007/s00440-003-0305-z

Download citation

  • Received: 10 February 2003

  • Revised: 19 September 2003

  • Published: 04 November 2003

  • Issue Date: December 2003

  • DOI: https://doi.org/10.1007/s00440-003-0305-z

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Hydrodynamic limit
  • Disordered systems
  • Lattice gas dynamics
  • Exclusion process
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature