The Annals of Probability

Bulk diffusion in a system with site disorder

Jeremy Quastel

Full-text: Open access


We consider a system of random walks in a random environment interacting via exclusion. The model is reversible with respect to a family of disordered Bernoulli measures. Assuming some weak mixing conditions, it is shown that, under diffusive scaling, the system has a deterministic hydrodynamic limit which holds for almost every realization of the environment. The limit is a nonlinear diffusion equation with diffusion coefficient given by a variational formula. The model is nongradient and the method used is the “long jump” variation of the standard nongradient method, which is a type of renormalization. The proof is valid in all dimensions.

Article information

Ann. Probab., Volume 34, Number 5 (2006), 1990-2036.

First available in Project Euclid: 14 November 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C44: Dynamics of disordered systems (random Ising systems, etc.)

Hydrodynamic limit disordered systems


Quastel, Jeremy. Bulk diffusion in a system with site disorder. Ann. Probab. 34 (2006), no. 5, 1990--2036. doi:10.1214/009117906000000322.

Export citation


  • Ambegaokar, V., Halperin, B. I. and Langer, J. S. (1971). Hopping conductivity in disordered systems. Phys. Rev. B 4 2612.
  • Brak, R. and Elliott, R. J. (1989). Correlated random walks with random hopping rates. J. Physics---Condensed Matter Dec 25 V1 N51 10299--10319.
  • Brak, R. and Elliott, R. J. (1989). Correlated tracer diffusion in a disordered medium. Materials Science and Engineering B---Solid State Materials for Advanced Technology Jul V3 N1-2 159--162.
  • Caputo, P. (2003). Uniform Poincare inequalities for unbounded conservative spin systems: The non-interacting case. Stochastic Process. Appl. 106 223--244.
  • Chang, C. C. and Yau, H. T. (1992). Fluctuations of one dimensional Ginzburg--Landau models in nonequilibrium. Comm. Math. Phys. 145 209--234.
  • Faggionato, A. and Martinelli, F. (2003). Hydrodynamic limit of a disordered lattice gas. Probab. Theory Related Fields 127 535--608.
  • Fritz, J. (1989). Hydrodynamics in a symmetric random medium. Comm. Math. Phys. 125 13--25.
  • Gartner, P. and Pitis, R. (1992). Occupancy-correlation corrections in hopping. Phys. Rev. B 45.
  • Guo, M. Z., Papanicolaou, G. C. and Varadhan, S. R. S. (1988). Nonlinear diffusion limit for a system with nearest neighbor interactions. Comm. Math. Phys. 118 31--53.
  • Kehr, K. W., Paetzold, O. and Wichmann, T. (1993). Collective diffusion of lattice gases on linear chain with site-energy disorder. Phys. Lett. A 182 135--139.
  • Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems. Springer, Berlin.
  • Kirkpatrick, S. (1971). Classical transport in disordered media: Scaling and effective-medium theories. Phys. Rev. Lett. 27 1722.
  • Lu, S. L. and Yau, H. T. (1993). Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics. Comm. Math. Phys. 156 399--433.
  • Miller, A. and Abrahams, E. (1960). Impurity conduction at low concentrations. Phys. Rev. 120 745.
  • Papanicolaou, G. C. and Varadhan, S. R. S. (1981). Boundary value problems with rapidly oscillating coefficients. In Random Fields (J. Fritz, J. L. Lebowitz and D. Szasz, eds.) 835--853. North-Holland, Amsterdam.
  • Quastel, J. (1996). Diffusion in disordered media. In Proceedings in Nonlinear Stochastic PDEs (T. Funaki and W. Woyczinky, eds.) 65--79. Springer, New York.
  • Quastel, J. (1992). Diffusion of color in the simple exclusion process. Comm. Pure Appl. Math. 45 623--679.
  • Quastel, J. and Yau, H.T. (2006). Bulk diffusion in a system with site disorder. Unpublished notes. Available at
  • Reed, M. and Simon, B. (1978). Methods of Modern Mathematical Physics IV. Analysis of Operators. Academic Press, San Diego.
  • Richards, P. M. (1977). Theory of one-dimensional hopping conductivity and diffusion. Phys. Rev. B 16 1393--1409.
  • Spohn, H. (1991). Large Scale Dynamics of Interacting Particles. Springer, Berlin.
  • Varadhan, S. R. S. (1990). Nonlinear diffusion limit for a system with nearest neighbor interactions. II. In Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals (K. D. Elworthy and N. Ikeda, eds.) 75--128. Wiley, New York.
  • Varadhan, S. R. S. and Yau, H.-T. (1997). Diffusive limit of lattice gas with mixing conditions. Asian J. Math. 1 623--678.
  • Wick, W. D. (1989). Hydrodynamic limit of non-gradient interacting particle process. J. Statist. Phys. 54 873--892.