Quantitative homogenization of interacting particle systems

Giunti, Arianna; Gu, Chenlin; Mourrat, Jean-Christophe

doi:10.1214/22-aop1573

Cited by 2 publications

(2 citation statements)

References 73 publications

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“…Motivated by applications to homogenization of particle systems [5], Giunti, Gu, Mourrat, and Nitzschner recently addressed a related problem in a different setting, and proved the Gevrey regularity of λ → a λ in [6] (a variant of λ → A λ ). Their approach is based on Poisson calculus (cf.…”

Section: Contextmentioning

confidence: 99%

“…First, adapting the proof of [4, Proposition 5.2] by using Lemma 4 (as we did above for [4, Proposition 4.6]), and replacing [4, Proposition 4.6] by Proposition 3, we directly obtain the uniform bounds (7). In order to use this bound to prove regularity based on the qualitative convergence (5) and the regularity of p → A (p)…”

Section: Proof Of Gevrey Regularitymentioning

confidence: 99%

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A short proof of Gevrey regularity for homogenized coefficients of the Poisson point process

Duerinckx¹,

Gloria²

2022

Comptes Rendus. Mathématique

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In this short note we capitalize on and complete our previous results on the regularity of the homogenized coefficients for Bernoulli perturbations by addressing the case of the Poisson point process, for which the crucial uniform local finiteness assumption fails. In particular, we strengthen the qualitative regularity result first obtained in this setting by the first author to Gevrey regularity of order 2. The new ingredient is a fine application of properties of Poisson point processes, in a form recently used by Giunti, Gu, Mourrat, and Nitzschner.

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Section: Contextmentioning

confidence: 99%

Section: Proof Of Gevrey Regularitymentioning

confidence: 99%

A short proof of Gevrey regularity for homogenized coefficients of the Poisson point process

Duerinckx¹,

Gloria²

2022

Comptes Rendus. Mathématique

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Homogenization of sound-soft and high-contrast acoustic metamaterials in subcritical regimes

Feppon

Ammari

2023

ESAIM: M2AN

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We propose a quantitative effective medium theory for two types of acoustic metamaterials constituted of a large number N of small heterogeneities of characteristic size s, randomly and independently distributed in a bounded domain. We first consider a “sound-soft” material, in which the total wave field satisfies a Dirichlet boundary condition on the acoustic obstacles. In the “sub-critical” regime sN = O(1), we obtain that the effective medium is governed by a dissipative Lippmann-Schwinger equation which approximates the total field with a relative mean-square error of order O(max((sN)^2N^(-1/3),N^(-1/2))). We retrieve the critical size s ∼ 1/N of the literature at which the effects of the obstacles can be modelled by a “strange term” added to the Helmholtz equation. Second, we consider high-contrast acoustic metamaterials, in which each of the N heterogeneities are packets of K inclusions filled with a material of density much lower than the one of the background medium. As the contrast parameter vanishes, δ → 0, the effective medium admits K resonant characteristic sizes (si (δ))1≤i≤K and is governed by a Lippmann-Schwinger equation, which is diffusive or dispersive (with negative refractive index) for frequencies ω respectively slightly larger or slightly smaller than the corresponding K resonant frequencies (ωi (δ))1≤i≤K . These conclusions are obtained under the condition that (i) the resonance is of monopole type, and (ii) lies in the “subcritical regime” where the contrast parameter is small enough, i.e. δ = o(N^(-2)), while the considered frequency is “not too close” to the resonance, i.e. N δ 2 = O(|1 − s/si (δ)|). Our mathematical analysis and the current literature allow us to conjecture that “solidification” phenomena are expected to occur in the “super-critical” regime N δ 2 |1 − s/si (δ)|−1 → +∞.

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Quantitative homogenization of interacting particle systems

Cited by 2 publications

References 73 publications

A short proof of Gevrey regularity for homogenized coefficients of the Poisson point process

A short proof of Gevrey regularity for homogenized coefficients of the Poisson point process

Homogenization of sound-soft and high-contrast acoustic metamaterials in subcritical regimes

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