Hydrodynamic Behavior of One Dimensional Subdiffusive Exclusion Processes with Random Conductances

Abstract. Consider a system of particles performing nearest neighbor random walks on the lattice Z under hard–core interaction. The rate for a jump over a given bond is direction–independent and the inverse of the jump rates are i.i.d. random variables belonging to the domain of attraction of an α– stable law, 0 < α < 1. This exclusion process models conduction in strongly disordered one-dimensional media. We prove that, when varying over the disorder and for a suitable slowly varying function L, under the super-diffusive time scaling N1+1/αL(N), the density profile evolves as the solution of the random equation ∂tρ = LW ρ, where LW is the generalized second-order differential operator d du d dW in which W is a double sided α–stable subordinator. This result follows from a quenched hydrodynamic limit in the case that the i.i.d. jump rates are replaced by a suitable array {ξN,x : x ∈ Z} having same distribution and fulfilling an a.s. invariance principle. We also prove a law of large numbers for a tagged particle.