Abstract
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 α-stable law, 0 < α < 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/α 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 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 \({\{\xi_{N,x} : x\in\mathbb{Z}\}}\) having same distribution and fulfilling an a.s. invariance principle. We also prove a law of large numbers for a tagged particle.
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Faggionato, A., Jara, M. & Landim, C. Hydrodynamic behavior of 1D subdiffusive exclusion processes with random conductances. Probab. Theory Relat. Fields 144, 633–667 (2009). https://doi.org/10.1007/s00440-008-0157-7
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DOI: https://doi.org/10.1007/s00440-008-0157-7
Keywords
- Interacting particle system
- Hydrodynamic limit
- α-stable subordinator
- Random environment
- Subdiffusion
- Quasi-diffusion