Bulk diffusion in a system with site disorder. (English) Zbl 1104.60066

Summary: 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.


60K37 Processes in random environments
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
Full Text: DOI arXiv


[1] Ambegaokar, V., Halperin, B. I. and Langer, J. S. (1971). Hopping conductivity in disordered systems. Phys. Rev. B 4 2612.
[2] Brak, R. and Elliott, R. J. (1989). Correlated random walks with random hopping rates. J. Physics—Condensed Matter Dec 25 V1 N51 10299–10319.
[3] 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.
[4] Caputo, P. (2003). Uniform Poincare inequalities for unbounded conservative spin systems: The non-interacting case. Stochastic Process. Appl. 106 223–244. · Zbl 1075.60581 · doi:10.1016/S0304-4149(03)00044-9
[5] Chang, C. C. and Yau, H. T. (1992). Fluctuations of one dimensional Ginzburg–Landau models in nonequilibrium. Comm. Math. Phys. 145 209–234. · Zbl 0754.76006 · doi:10.1007/BF02099137
[6] Faggionato, A. and Martinelli, F. (2003). Hydrodynamic limit of a disordered lattice gas. Probab. Theory Related Fields 127 535–608. · Zbl 1052.60083 · doi:10.1007/s00440-003-0305-z
[7] Fritz, J. (1989). Hydrodynamics in a symmetric random medium. Comm. Math. Phys. 125 13–25. · Zbl 0682.76001 · doi:10.1007/BF01217766
[8] Gartner, P. and Pitis, R. (1992). Occupancy-correlation corrections in hopping. Phys. Rev. B 45 .
[9] 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. · Zbl 0652.60107 · doi:10.1007/BF01218476
[10] 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.
[11] Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems. Springer, Berlin. · Zbl 0927.60002
[12] Kirkpatrick, S. (1971). Classical transport in disordered media: Scaling and effective-medium theories. Phys. Rev. Lett. 27 1722.
[13] 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. · Zbl 0779.60078 · doi:10.1007/BF02098489
[14] Miller, A. and Abrahams, E. (1960). Impurity conduction at low concentrations. Phys. Rev. 120 745. · Zbl 0093.22403 · doi:10.1103/PhysRev.120.745
[15] 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. · Zbl 0499.60059
[16] Quastel, J. (1996). Diffusion in disordered media. In Proceedings in Nonlinear Stochastic PDEs (T. Funaki and W. Woyczinky, eds.) 65–79. Springer, New York. · Zbl 0840.60093
[17] Quastel, J. (1992). Diffusion of color in the simple exclusion process. Comm. Pure Appl. Math. 45 623–679. · Zbl 0769.60097 · doi:10.1002/cpa.3160450602
[18] Quastel, J. and Yau, H.T. (2006). Bulk diffusion in a system with site disorder. Unpublished notes. Available at http://arxiv.org/abs/math/0601124. · Zbl 1104.60066 · doi:10.1214/009117906000000322
[19] Reed, M. and Simon, B. (1978). Methods of Modern Mathematical Physics IV . Analysis of Operators . Academic Press, San Diego. · Zbl 0401.47001
[20] Richards, P. M. (1977). Theory of one-dimensional hopping conductivity and diffusion. Phys. Rev. B 16 1393–1409.
[21] Spohn, H. (1991). Large Scale Dynamics of Interacting Particles . Springer, Berlin. · Zbl 0742.76002
[22] 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. · Zbl 0793.60105
[23] Varadhan, S. R. S. and Yau, H.-T. (1997). Diffusive limit of lattice gas with mixing conditions. Asian J. Math. 1 623–678. · Zbl 0947.60089
[24] Wick, W. D. (1989). Hydrodynamic limit of non-gradient interacting particle process. J. Statist. Phys. 54 873–892. · doi:10.1007/BF01019779
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.