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Hydrodynamic behavior of 1D subdiffusive exclusion processes with random conductances
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  • Published: 03 June 2008

Hydrodynamic behavior of 1D subdiffusive exclusion processes with random conductances

  • A. Faggionato1,
  • M. Jara2 &
  • C. Landim2,3 

Probability Theory and Related Fields volume 144, pages 633–667 (2009)Cite this article

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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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Authors and Affiliations

  1. Dipartimento di Matematica “G. Castelnuovo”, Università “La Sapienza”, P.le Aldo Moro 2, 00185, Rome, Italy

    A. Faggionato

  2. IMPA, Estrada Dona Castorina 110, CEP 22460, Rio de Janeiro, Brazil

    M. Jara & C. Landim

  3. CNRS UMR 6085, Université de Rouen, UMR 6085, Avenue de l’Université, Technopôle du Madrillet, BP.12, 76801, Saint-Étienne-du-Rouvray, France

    C. Landim

Authors
  1. A. Faggionato
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  2. M. Jara
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  3. C. Landim
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Correspondence to C. Landim.

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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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  • Received: 04 September 2007

  • Revised: 17 April 2008

  • Published: 03 June 2008

  • Issue Date: July 2009

  • DOI: https://doi.org/10.1007/s00440-008-0157-7

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Keywords

  • Interacting particle system
  • Hydrodynamic limit
  • α-stable subordinator
  • Random environment
  • Subdiffusion
  • Quasi-diffusion

Mathematics Subject Classification (2000)

  • Primary: 60K35
  • 60K37
  • Secondary: 82C44
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