Bulk diffusion in a system with site disorder

Abstract

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.Comment: Published at http://dx.doi.org/10.1214/009117906000000322 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

1D Mott Variable-Range Hopping with External Field

“…The mathematical analysis of the above exclusion process presents several technical challenges and has been performed only when ξ ≡ Z d (absence of geometric disorder) and with jumps restricted to nearest-neighbors (cf. [7,24]). This last assumption does not fit with the low temperature regime, where anomalous conductivity takes place.…”

Einstein relation and linear response in one-dimensional Mott variable-range hopping

“…We point out that the mathematical analysis of such an exclusion process is very demanding from a technical viewpoint due to site disorder. We refer to [13,42] for the derivation of the hydrodynamic limit when the impurities are localized at the sites of Z d and hopping is only between nearest-neighbor sites (from a physical viewpoint, the nearest-neighbor assumption leads to a good approximation of Mott variable-range hopping at not very small temperature). Due to the these technical difficulties, in the physical literature, in the regime of low density of conduction electrons the above exclusion process on {x i } is then approximated by independent continuous time random walks (hence one focuses on a single random walk), with probability rate r i,k for a jump from x i to x k = x i given by (1) times µ(η x i = 1 , η x k = 0).…”

“…The validity of this low density approximation has been indeed proved for the exclusion process with nearest-neighbor jumps on Z d (cf. [42,Thm. 1]).…”

Poincaré and logarithmic Sobolev inequality for Ginzburg-landau processes in random environment

“…Cancrini and Martinelli [4] proved a diffusive spectral gap and Cancrini and Roberto [6] obtained a bound for the logarithmic Sobolev constant. Quastel and Yau [21] proved a sharp estimate for the spectral gap of the symmetric exclusion process in random environment using the martingale approach.…”

“…The authors wish to thank J. Quastel and H. T. Yau for the private communication of the unpublished manuscript [21] …”