Hydrodynamical scaling limits of simple
                    exclusion models

Jensen, Leif; Yau, Horug-Tzer

doi:10.1090/pcms/006/04

Cited by 6 publications

(17 citation statements)

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“…From this estimate we can obtain a result analogous to Theorem 2.4 ( [7] or section 6 of [4]) and thus the martingale decomposition Theorem 2.5. This was the estimate established in [2,7] for the fluctuation-dissipation equation.…”

Section: Step 4: Bounds On the Variancementioning

confidence: 88%

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Diffusive limit of a tagged particle in asymmetric simple exclusion processes

Sethuraman

Varadhan

Yau

2000

Comm Pure Appl Math

101

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The study of equilibrium fluctuations of a tagged particle in finite-range simple exclusion processes has a long history. The belief is that the scaled centered tagged particle motion behaves as some sort of homogenized random walk. In fact, invariance principles have been proved in all dimensions d ≥ 1 when the single particle jump rate is unbiased, in d ≥ 3 when the jump rate is biased, and in d = 1 when the jump rate is in addition nearest-neighbor.The purpose of this article is to give some partial results in the open cases in d ≤ 2. Namely, we show the tagged particle motion is "diffusive" in the sense that upper and lower bounds are given for the tagged particle variance at time t on order O(t) in d = 2 when the jump rate is biased, and also in d = 1 when in addition the jump rate is not nearest-neighbor. Also, a characterization of the tagged particle variance is given. The main methods are in analyzing H −1 norm variational inequalities.

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Section: Step 4: Bounds On the Variancementioning

confidence: 88%

“…This was the estimate established in [2,7] for the fluctuation-dissipation equation. In a sense these two estimates on the Green's functions can be substituted for each other in many contexts; see section 6 of the lecture [4].…”

Section: Step 4: Bounds On the Variancementioning

confidence: 99%

Diffusive limit of a tagged particle in asymmetric simple exclusion processes

Sethuraman

Varadhan

Yau

2000

Comm Pure Appl Math

101

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“…We discuss the history of this universality problem shortly. To highlight its significance, a general derivation of universal local fluctuations was asked for by Spohn [46], in the form of Conjecture II.3.6, and by Jensen-Yau [35], in the form of an open problem in Lecture 7; almost no progress has been made in the past few decades according to [26]. Additionally, since Spohn [46] and Jensen-Yau [35], there has been a surge of activity and interest in nonlinear KPZ statistics (KPZ=Kardar-Parisi-Zhang), in particular, as large-scale limits of fluctuations [53].…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

Fluctuations for some non-stationary interacting particle systems via Boltzmann-Gibbs Principle

Yang¹

2022

Preprint

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Conjecture II.3.6 of Spohn in [46] and Lecture 7 of Jensen-Yau in [35] ask for a general derivation of universal fluctuations of hydrodynamic limits in large-scale stochastic interacting particle systems. However, the past few decades have witnessed only minimal progress according to [26]. In this paper, we develop a general method for deriving the so-called Boltzmann-Gibbs principle for a general family of non-integrable and non-stationary interacting particle systems, thereby responding to Spohn and Jensen-Yau. Most importantly, our method depends mostly on local and dynamical, and thus more general/universal, features of the model. This contrasts with previous work [6,8,24,34], all of which rely on global and non-universal assumptions on invariant measures or initial measures of the model. As a concrete application of the method, we derive the KPZ equation as a large-scale limit of the height functions for a family of non-stationary and non-integrable exclusion processes with an environment-dependent asymmetry. This establishes a first result to Big Picture Question 1.6 in [53] for non-stationary and non-integrable "speed-change" models that have also been of interest beyond KPZ [18,22,23,38].

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“…Apart from stochastic systems on which some degree of stochastic integrability is present, up to our knowledge the only work dealing with non-equilibrium fluctuations is [11], where the authors consider the one-dimensional Ginzburg-Landau model in dimension d = 1. A general derivation of non-equilibrium fluctuations of conservative systems has remained largely open since then, and it is mentioned as Conjecture II.3.6 in [45], as a relevant open problem in Lecture 7 of [34] and "no progress has been made in the last 20 years" according to [28]. For more historical references, see Chapter 11 of [36].…”

Section: Introductionmentioning

confidence: 99%

Non-equilibrium Fluctuations of Interacting Particle Systems

Jara¹,

Menezes²

2018

Preprint

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We obtain the large scale limit of the fluctuations around its hydrodynamic limit of the density of particles of a weakly asymmetric exclusion process in dimension d ≤ 3. The proof is based upon a sharp estimate on the relative entropy of the law of the process with respect to product reference measures associated to the hydrodynamic limit profile, which holds in any dimension and is of independent interest. As a corollary of this entropy estimate, we obtain some quantitative bounds on the speed of convergence of the aforementioned hydrodynamic limit.

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Hydrodynamical scaling limits of simple exclusion models

Cited by 6 publications

References 0 publications

Diffusive limit of a tagged particle in asymmetric simple exclusion processes

Diffusive limit of a tagged particle in asymmetric simple exclusion processes

Fluctuations for some non-stationary interacting particle systems via Boltzmann-Gibbs Principle

Non-equilibrium Fluctuations of Interacting Particle Systems

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