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Hydrodynamic behavior of 1D subdiffusive exclusion processes with random conductances. (English) Zbl 1169.60326

Summary: Consider a system of particles performing nearest neighbor random walks on the lattice \({\mathbb{Z}}\) 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 \(\alpha \)-stable law, \(0 < \alpha < 1\). This exclusion process models conduction in strongly disordered 1D media. We prove that, when varying over the disorder and for a suitable slowly varying function \(L\), under the super-diffusive time scaling \(N ^{1 +1/\alpha } L(N)\), the density profile evolves as the solution of the random equation \({\partial_t \rho = \mathfrak{L}_W \rho}\), where \({\mathfrak{L}_W}\) is the generalized second-order differential operator \({\frac d{du} \frac d{dW}}\) in which \(W\) is a double-sided \(\alpha \)-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 \({\{\xi_{N,x} : x\in\mathbb{Z}\}}\) having same distribution and fulfilling an a.s. invariance principle. We also prove a law of large numbers for a tagged particle.

MSC:

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