Hydrodynamic behavior of one dimensional subdiffusive exclusion processes with random conductances

2007/09/03 by A. Faggionato, Faggionato, A., M. Jara +3
Mathematics · Physics and Astronomy · #60K35 #60K37 #82C44 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #math-ph #math.MP #math.PR #msc:60K35 #msc:60K37 #msc:82C44

paper · pdf · doi:10.48550/arxiv.0709.0306

arxiv created 2007/09/03 · arxiv updated 2009/12/01

Abstract

Consider a system of particles performing nearest neighbor random walks on the lattice \ZZ 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 \a--stable law, 0<\a<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 N^{1 + 1/α}L(N), the density profile evolves as the solution of the random equation ∂_t ρ= \mf L_W ρ, where \mf L_W is the generalized second-order differential operator \frac ddu \frac ddW in which W is a double sided \a--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 \ξ_N,x : x∈\bb Z\ having same distribution and fulfilling an a.s. invariance principle. We also prove a law of large numbers for a tagged particle.

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