Hydrodynamic limit for attractive particle systems on 417-1417-1417-1

Rezakhanlou, Fraydoun

doi:10.1007/bf02099130

Cited by 187 publications

(221 citation statements)

References 11 publications

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“…We follow the scheme of [39] by deriving a mv version of entropy inequality (9) for the particle system, and using uniqueness results of Theorems 2.1 and 2.3. The key novelty is to obtain the boundary term of [46] from a proper coupling of open systems.…”

Section: Proofs Of Theorems 22 and 24mentioning

confidence: 99%

“…To deduce convergence of the empirical measure is a standard technical step (see e.g. [39,22]). Similarly, Corollary 3.1 combined with Theorem 2.3 implies Theorem 2.4.…”

Section: Average Entropy Inequalitymentioning

confidence: 99%

“…The latter uses uniqueness of stationary solutions, proved in Section 4, together with asymptotic stability (Theorem 2.3). For simplicity, we prove the hydrodynamic and hydrostatic limits in detail for the Misanthrope's process, formerly treated by [39] in Z d . However, as explained after Theorem 2.4, the scope of our results is best illustrated by the exclusion process with overtaking, a model treated in Appendix A.…”

Section: Introductionmentioning

confidence: 99%

“…The celebrated result of [39] shows that, under Euler time scaling, the hydrodynamic limit of attractive particle systems on Z d with product invariant measures, is given by the entropy solution to a scalar conservation law. We extend this result to systems living in an open subset of R d .…”

Section: Introductionmentioning

confidence: 99%

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Hydrodynamics and Hydrostatics for a Class of Asymmetric Particle Systems with Open Boundaries

Bahadoran

2012

Commun. Math. Phys.

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We consider attractive particle systems in Z d with product invariant measures. We prove that when particles are restricted to a subset of Z d , with birth and death dynamics at the boundaries, the hydrodynamic limit is given by the unique entropy solution of a conservation law, with boundary conditions in the sense of Bardos et al. ([7]). For the hydrostatic limit between parallel hyperplanes, we prove a multidimensional version of the phase diagram conjectured in [38], and show that it is robust with respect to perturbations of the boundaries.AMS 2000 subject classifications. 60K35, 82C22, 82C26; 35L65, 35L67, 35L50.

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Section: Proofs Of Theorems 22 and 24mentioning

confidence: 99%

“…To deduce convergence of the empirical measure is a standard technical step (see e.g. [39,22]). Similarly, Corollary 3.1 combined with Theorem 2.3 implies Theorem 2.4.…”

Section: Average Entropy Inequalitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hydrodynamics and Hydrostatics for a Class of Asymmetric Particle Systems with Open Boundaries

Bahadoran

2012

Commun. Math. Phys.

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“…Since [3,18], it is known that the empirical measures (see Section 2.2) associated to these processes, converge (in probability) to a deterministic measure whose density is the unique entropy solution of a hyperbolic conservation law with concave flux as given in (2.3). The hydrodynamic limit for these processes was set under two different approaches: for both processes in [18] using the Entropy method and for a general set of initial measures associated to a profile; and for the asep in [12] using the Relative Entropy method, under a more restrictive set of initial measures. We notice that it is not difficult to show the hydrodynamic limit for the azrp invoking the same arguments as in [12].…”

Section: Introductionmentioning

confidence: 99%

On the Asymmetric Zero-Range in the Rarefaction Fan

Gonçalves

2013

J Stat Phys

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ABSTRACT. We consider one-dimensional asymmetric zero-range processes starting from a step decreasing profile leading, in the hydrodynamic limit, to the rarefaction fan of the associated hydrodynamic equation. Under that initial condition, and for totally asymmetric jumps, we show that the weighted sum of joint probabilities for second class particles sharing the same site is convergent and we compute its limit. For partially asymmetric jumps, we derive the Law of Large Numbers for a second class particle, under the initial configuration in which all positive sites are empty, all negative sites are occupied with infinitely many first class particles and there is a single second class particle at the origin. Moreover, we prove that among the infinite characteristics emanating from the position of the second class particle it picks randomly one of them. The randomness is given in terms of the weak solution of the hydrodynamic equation, through some sort of renormalization function. By coupling the constant-rate totally asymmetric zero-range with the totally asymmetric simple exclusion, we derive limiting laws for more general initial conditions.

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Entropy, Large Deviations, and Scaling Limits

Varadhan

2013

Comm Pure Appl Math

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We often model the evolution of a complex system at the lowest or micro level because that is what makes the most physical sense. But if the system is large, we invariably want answers to questions posed on the larger macro or global scale. This then involves the study of equations with a large number of variables and extracting useful information from their solutions. The macroscopic quantities of interest may not contain complete information about the microscopic variables that drive their evolution. Something has to be done to obtain a closed system of equations for quantities of interest. Different contexts require different approaches. We are concerned here with the evolution of large systems of interacting particles or field variables that have some built-in noise and outline some of the work done at the Institute during the last 25 years.

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Hydrodynamic limit for attractive particle systems on 417-1417-1417-1

Cited by 187 publications

References 11 publications

Hydrodynamics and Hydrostatics for a Class of Asymmetric Particle Systems with Open Boundaries

Hydrodynamics and Hydrostatics for a Class of Asymmetric Particle Systems with Open Boundaries

On the Asymmetric Zero-Range in the Rarefaction Fan

Entropy, Large Deviations, and Scaling Limits

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