Hydrodynamical behavior of symmetric exclusion with slow bonds

Tertuliano Franco; Patrícia Gonçalves; Adriana Neumann

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 2, page 402-427
  • ISSN: 0246-0203

Abstract

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We consider the exclusion process in the one-dimensional discrete torus with N points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance N - β , with β ∈ [ 0 , ∞ ) . We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter β . If β ∈ [ 0 , 1 ) , the hydrodynamic limit is given by the usual heat equation. If β = 1 , it is given by a parabolic equation involving an operator d d x d d W , where W is the Lebesgue measure on the torus plus the sum of the Dirac measure supported on each macroscopic point related to the slow bond. If β ∈ ( 1 , ∞ ) , it is given by the heat equation with Neumann’s boundary conditions, meaning no passage through the slow bonds in the continuum.

How to cite

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Franco, Tertuliano, Gonçalves, Patrícia, and Neumann, Adriana. "Hydrodynamical behavior of symmetric exclusion with slow bonds." Annales de l'I.H.P. Probabilités et statistiques 49.2 (2013): 402-427. <http://eudml.org/doc/272066>.

@article{Franco2013,
abstract = {We consider the exclusion process in the one-dimensional discrete torus with $N$ points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance $N^\{-\beta \}$, with $\beta \in [0,\infty )$. We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter $\beta $. If $\beta \in [0,1)$, the hydrodynamic limit is given by the usual heat equation. If $\beta =1$, it is given by a parabolic equation involving an operator $\frac\{\mathrm \{d\}\}\{\mathrm \{d\}x\}\frac\{\mathrm \{d\}\}\{\mathrm \{d\}W\}$, where $W$ is the Lebesgue measure on the torus plus the sum of the Dirac measure supported on each macroscopic point related to the slow bond. If $\beta \in (1,\infty )$, it is given by the heat equation with Neumann’s boundary conditions, meaning no passage through the slow bonds in the continuum.},
author = {Franco, Tertuliano, Gonçalves, Patrícia, Neumann, Adriana},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {hydrodynamic limit; exclusion process; slow bonds},
language = {eng},
number = {2},
pages = {402-427},
publisher = {Gauthier-Villars},
title = {Hydrodynamical behavior of symmetric exclusion with slow bonds},
url = {http://eudml.org/doc/272066},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Franco, Tertuliano
AU - Gonçalves, Patrícia
AU - Neumann, Adriana
TI - Hydrodynamical behavior of symmetric exclusion with slow bonds
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 2
SP - 402
EP - 427
AB - We consider the exclusion process in the one-dimensional discrete torus with $N$ points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance $N^{-\beta }$, with $\beta \in [0,\infty )$. We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter $\beta $. If $\beta \in [0,1)$, the hydrodynamic limit is given by the usual heat equation. If $\beta =1$, it is given by a parabolic equation involving an operator $\frac{\mathrm {d}}{\mathrm {d}x}\frac{\mathrm {d}}{\mathrm {d}W}$, where $W$ is the Lebesgue measure on the torus plus the sum of the Dirac measure supported on each macroscopic point related to the slow bond. If $\beta \in (1,\infty )$, it is given by the heat equation with Neumann’s boundary conditions, meaning no passage through the slow bonds in the continuum.
LA - eng
KW - hydrodynamic limit; exclusion process; slow bonds
UR - http://eudml.org/doc/272066
ER -

References

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