Hydrodynamic limit of a d-dimensional exclusion process with conductances

Valentim, Fábio

doi:10.1214/10-aihp397

Cited by 9 publications

(24 citation statements)

References 8 publications

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“…This operator has the properties stated in Theorem 2.1 in [11]. We now outline the main steps to prove it.…”

Section: Proposition 34 (Energy Estimates For λ > 0) Let F ∈ L 2 (Tmentioning

confidence: 90%

“…Then f = d k=1 f k belongs to D W and is an eigenvector of L W with eigenvalue d k=1 λ k . Moreover, [11] proved the following result:…”

Section: W -Sobolev Spacesmentioning

confidence: 93%

“…Furthermore, they also proved that these operators have a countable complete orthonormal system of eigenvectors, which we denote by A W k . Then, following [11],…”

Section: W -Sobolev Spacesmentioning

confidence: 99%

“…However, the latter covers more general situations. For instance, [5] and [11] argue that these equations may be used to model a diffusion of particles within a region with membranes induced by the discontinuities of the functions W i . Unfortunately, the standard Sobolev spaces are not suitable for being used as the space of weak solutions of equations in the form of (2).…”

Section: Introductionmentioning

confidence: 99%

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W-Sobolev spaces

Simas

Valentim

2011

Journal of Mathematical Analysis and Applications

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Fix strictly increasing right continuous functions with left limits and periodic increments,. Several properties, that are analogous to classical results on Sobolev spaces, are obtained. Existence and uniqueness results for W -generalized elliptic equations, and uniqueness results for W -generalized parabolic equations are also established. Finally, an application of this theory to stochastic homogenization is presented.

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“…This operator has the properties stated in Theorem 2.1 in [11]. We now outline the main steps to prove it.…”

Section: Proposition 34 (Energy Estimates For λ > 0) Let F ∈ L 2 (Tmentioning

confidence: 90%

“…Then f = d k=1 f k belongs to D W and is an eigenvector of L W with eigenvalue d k=1 λ k . Moreover, [11] proved the following result:…”

Section: W -Sobolev Spacesmentioning

confidence: 93%

“…Furthermore, they also proved that these operators have a countable complete orthonormal system of eigenvectors, which we denote by A W k . Then, following [11],…”

Section: W -Sobolev Spacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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W-Sobolev spaces

Simas

Valentim

2011

Journal of Mathematical Analysis and Applications

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“…The techniques used in those papers were strongly based on theorems about the convergence of one-dimensional continuous-time stochastic processes. In fact, even the d-dimensional case treated in [10] considered a class of nonhomogeneous environments that could be decomposed, in a proper sense, into d one-dimensional cases. Recently, different approaches have been examined to deal with d-dimensional environments; see [4] and [7].…”

Section: T Franco Et Almentioning

confidence: 99%

Hydrodynamic Limit for a Type of Exclusion Process with Slow Bonds in Dimension d ≥ 2

2011

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Let be a connected closed region with smooth boundary contained in the d-dimensional continuous torus T d . In the discrete torus N −1 T d N , we consider a nearest-neighbor symmetric exclusion process where occupancies of neighboring sites are exchanged at rates depending on in the following way: if both sites are in or c , the exchange rate is 1; if one site is in and the other site is in c , and the direction of the bond connecting the sites is e j , then the exchange rate is defined as N −1 times the absolute value of the inner product between e j and the normal exterior vector to ∂ . We show that this exclusiontype process has a nontrivial hydrodynamical behavior under diffusive scaling and, in the continuum limit, particles are not blocked or reflected by ∂ . Thus, the model represents a system of particles under hard-core interaction in the presence of a permeable membrane which slows down the passage of particles between two complementary regions.

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Slowed Exclusion Process: Hydrodynamics, Fluctuations and Phase Transitions

Franco¹,

Gonçalves

Neumann

2014

Springer Proceedings in Mathematics &Amp; Statistics

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This is a short survey on recent results obtained by the authors on dynamical phase transitions of interacting particle systems. We consider particle systems with exclusion dynamics, but it is conjectured that our results should hold for a general class of particle systems. The parameter giving rise to the phase transition is the "slowness" of a single bond in the discrete lattice. The phase transition is verified not only in the hydrodynamics, but also in the fluctuations of the density, the current and the tagged particle. Moreover, we found a phase transition in the continuum, that is, at the level of the hydrodynamic equations, in agreement with the dynamical phase transition for the particle systems.

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Hydrodynamic limit of a d-dimensional exclusion process with conductances

Cited by 9 publications

References 8 publications

W-Sobolev spaces

W-Sobolev spaces

Hydrodynamic Limit for a Type of Exclusion Process with Slow Bonds in Dimension d ≥ 2

Slowed Exclusion Process: Hydrodynamics, Fluctuations and Phase Transitions

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