arXiv subjects

Antoine Gloria

Publications and source records attributed to Antoine Gloria.

At least 37 records · Page 2Linked to original sources

arXiv2022

Approximate normal forms via Floquet-Bloch theory: Nehorosev stability for linear waves in quasiperiodic media

We study the long-time behavior of the Schr{ö}dinger flow in a heterogeneous potential $λ$V with small intensity 0<$λ$$\ll$1 (or alternatively at high frequencies). The main new ingredient, which we introduce in the general setting of a stationary ergodic potential, is an approximate stationary Floquet--Bloch theory that is used to put the perturbed Schr{ö}dinger operator into approximate normal form. We apply this approach to quasiperiodic potentials and establish a Nehoro{\v s}ev-type stability result. In particular, this ensures asymptotic ballistic transport up to a stretched exponential timescale exp($λ$--1/s) for some s>0. More precisely, the approximate normal form leads to an accurate long-time description of the Schr{ö}dinger flow as an effective unitary correction of the free flow. The approach is robust and generically applies to linear waves. For classical waves, for instance, this allows to extend diffractive geometric optics to quasiperiodically perturbed media.

math.AP↗

arXiv2022

Continuum percolation in stochastic homogenization and the effective viscosity problem

This contribution is concerned with the effective viscosity problem, that is, the homogenization of the steady Stokes system with a random array of rigid particles, for which the main difficulty is the treatment of close particles. Standard approaches in the literature have addressed this issue by making moment assumptions on interparticle distances. Such assumptions however prevent clustering of particles, which is not compatible with physically-relevant particle distributions. In this contribution, we take a different perspective and consider moment bounds on the size of clusters of close particles. On the one hand, assuming such bounds, we construct correctors and prove homogenization (using a variational formulation and $Γ$-convergence to avoid delicate pressure issues). On the other hand, based on subcritical percolation techniques, these bounds are shown to hold for various mixing particle distributions with nontrivial clustering. As a by-product of the analysis, we also obtain similar homogenization results for compressible and incompressible linear elasticity with unbounded random stiffness.

math.AP↗

arXiv2021

Scaling limit of the homogenization commutator for Gaussian coefficient fields

Consider a linear elliptic partial differential equation in divergence form with a random coefficient field. The solution operator displays fluctuations around its expectation. The recently developed pathwise theory of fluctuations in stochastic homogenization reduces the characterization of these fluctuations to those of the so-called standard homogenization commutator. In this contribution, we investigate the scaling limit of this key quantity: starting from a Gaussian-like coefficient field with possibly strong correlations, we establish the convergence of the rescaled commutator to a fractional Gaussian field, depending on the decay of correlations of the coefficient field, and we investigate the (non)degeneracy of the limit. This extends to general dimension $d\ge1$ previous results so far limited to dimension $d=1$, and to the continuum setting with strong correlations recent results in the discrete iid case.

math.AP↗

arXiv2021

Quantitative homogenization theory for random suspensions in steady Stokes flow

This work develops a quantitative homogenization theory for random suspensions of rigid particles in a steady Stokes flow, and completes recent qualitative results. More precisely, we establish a large-scale regularity theory for this Stokes problem, and we prove moment bounds for the associated correctors and optimal estimates on the homogenization error; the latter further requires a quantitative ergodicity assumption on the random suspension. Compared to the corresponding quantitative homogenization theory for divergence-form linear elliptic equations, substantial difficulties arise from the analysis of the fluid incompressibility and the particle rigidity constraints. Our analysis further applies to the problem of stiff inclusions in (compressible or incompressible) linear elasticity and in electrostatics; it is also new in those cases, even in the periodic setting.

math.AP↗

arXiv2021

Enhancement of elasto-dielectrics by homogenization of active charges

We investigate the PDE system resulting from even electromechanical coupling in elastomers. Assuming a periodic microstructure and a periodic distribution of micro-charges of a prescribed order, we derive the homogenized system. The results depend crucially on periodicity (or adequate randomness) and on the type of microstructure under consideration. We also show electric enhancement if the charges are carefully tailored to the homogenized electric field and explicit that enhancement, as well as the corresponding electrostrictive enhancement in a dilute regime.

math.AP↗

arXiv2021

Sedimentation of random suspensions and the effect of hyperuniformity

This work is concerned with the mathematical analysis of the bulk rheology of random suspensions of rigid particles settling under gravity in viscous fluids. Each particle generates a fluid flow that in turn acts on other particles and hinders their settling. In an equilibrium perspective, for a given ensemble of particle positions, we analyze both the associated mean settling speed and the velocity fluctuations of individual particles. In the 1970s, Batchelor gave a proper definition of the mean settling speed, a 60-year-old open problem in physics, based on the appropriate renormalization of long-range particle contributions. In the 1980s, a celebrated formal calculation by Caflisch and Luke suggested that velocity fluctuations in dimension $d=3$ should diverge with the size of the sedimentation tank, contradicting both intuition and experimental observations. The role of long-range self-organization of suspended particles in form of hyperuniformity was later put forward to explain additional screening of this divergence in steady-state observations. In the present contribution, we develop the first rigorous theory that allows to justify all these formal calculations of the physics literature.

math.AP↗

arXiv2020

Corrector equations in fluid mechanics: Effective viscosity of colloidal suspensions

Consider a colloidal suspension of rigid particles in a steady Stokes flow. In a celebrated work, Einstein argued that in the regime of dilute particles the system behaves at leading order like a Stokes fluid with some explicit effective viscosity. In the present contribution, we rigorously define a notion of effective viscosity, regardless of the dilute regime assumption. More precisely, we establish a homogenization result when particles are distributed according to a given stationary and ergodic random point process. The main novelty is the introduction and analysis of suitable corrector equations.

math.AP↗

arXiv2020

A scalar version of the Caflisch-Luke paradox

Consider an infinite cloud of hard spheres sedimenting in a Stokes flow in the whole space $\mathbb R^d$. Despite many contributions in fluid mechanics and applied mathematics, there is so far no rigorous definition of the associated effective sedimentation velocity. Calculations by Caflisch and Luke in dimension $d=3$ suggest that the effective velocity is well-defined for hard spheres distributed according to a weakly correlated and dilute point process, and that the variance of the sedimentation speed is infinite. This constitutes the Caflisch-Luke paradox. In this contribution, we consider a scalar version of this problem that displays the same difficulties in terms of interaction between the differential operator and the randomness, but is simpler in terms of PDE analysis. For a class of hardcore point processes we rigorously prove that the effective velocity is well-defined in dimensions $d>2$, and that the variance is finite in dimensions $d>4$, confirming the formal calculations by Caflisch and Luke, and opening a way to the systematic study of such problems.

math.AP↗

arXiv2019

Robustness of the pathwise structure of fluctuations in stochastic homogenization

We consider a linear elliptic system in divergence form with random coefficients and study the random fluctuations of large-scale averages of the field and the flux of the solution operator. In the context of the random conductance model, we developed in a previous work a theory of fluctuations based on the notion of homogenization commutator: we proved that the two-scale expansion of this special quantity is accurate at leading order in the fluctuation scaling when averaged on large scales (as opposed to the two-scale expansion of the solution operator taken separately) and that the large-scale fluctuations of the field and the flux of the solution operator can be recovered from those of the commutator. This implies that the large-scale fluctuations of the commutator of the corrector drive all other large-scale fluctuations to leading order, which we refer to as the pathwise structure of fluctuations in stochastic homogenization. In the present contribution we extend this result in two directions: we treat continuum elliptic (possibly non-symmetric) systems and allow for strongly correlated coefficient fields (Gaussian-like with a covariance function that can display an arbitrarily slow algebraic decay at infinity). Our main result shows in this general setting that the two-scale expansion of the homogenization commutator is still accurate to leading order when averaged on large scales, which illustrates the robustness of the pathwise structure of fluctuations.

math.AP↗

arXiv2019

Quantitative estimates in stochastic homogenization for correlated coefficient fields

This paper is about the homogenization of linear elliptic operators in divergence form with stationary random coefficients that have only slowly decaying correlations. It deduces optimal estimates of the homogenization error from optimal growth estimates of the (extended) corrector. In line with the heuristics, there are transitions at dimension $d=2$, and for a correlation-decay exponent $β=2$; we capture the correct power of logarithms coming from these two sources of criticality. The decay of correlations is sharply encoded in terms of a multiscale logarithmic Sobolev inequality (LSI) for the ensemble under consideration --- the results would fail if correlation decay were encoded in terms of an $α$-mixing condition. Among other ensembles popular in modelling of random media, this class includes coefficient fields that are local transformations of stationary Gaussian fields. The optimal growth of the corrector $ϕ$ is derived from bounding the size of spatial averages $F=\int g\cdot\nablaϕ$ of its gradient. This in turn is done by a (deterministic) sensitivity estimate of $F$, that is, by estimating the functional derivative $\frac{\partial F}{\partial a}$ of $F$ w.~r.~t.~the coefficient field $a$. Appealing to the LSI in form of concentration of measure yields a stochastic estimate on $F$. The sensitivity argument relies on a large-scale Schauder theory for the heterogeneous elliptic operator $-\nabla\cdot a\nabla$. The treatment allows for non-symmetric $a$ and for systems like linear elasticity.

math.AP↗

arXiv2019

Multiscale functional inequalities in probability: Concentration properties

In a companion article we have introduced a notion of multiscale functional inequalities for functions $X(A)$ of an ergodic stationary random field $A$ on the ambient space $\mathbb R^d$. These inequalities are multiscale weighted versions of standard Poincaré, covariance, and logarithmic Sobolev inequalities. They hold for all the examples of fields $A$ arising in the modelling of heterogeneous materials in the applied sciences whereas their standard versions are much more restrictive. In this contribution we first investigate the link between multiscale functional inequalities and more standard decorrelation or mixing properties of random fields. Next, we show that multiscale functional inequalities imply fine concentration properties for nonlinear functions $X(A)$. This constitutes the main stochastic ingredient to the quenched large-scale regularity theory for random elliptic operators by the second author, Neukamm, and Otto, and to the corresponding quantitative stochastic homogenization results.

math.PR↗

arXiv2019

Multiscale functional inequalities in probability: Constructive approach

Consider an ergodic stationary random field $A$ on the ambient space $\mathbb R^d$. In order to establish concentration properties for nonlinear functions $Z(A)$, it is standard to appeal to functional inequalities like Poincaré or logarithmic Sobolev inequalities in the probability space. These inequalities are however only known to hold for a restricted class of laws (product measures, Gaussian measures with integrable covariance, or more general Gibbs measures with nicely behaved Hamiltonians). In this contribution, we introduce variants of these inequalities, which we refer to as multiscale functional inequalities and which still imply fine concentration properties, and we develop a constructive approach to such inequalities. We consider random fields that can be viewed as transformations of a product structure, for which the question is reduced to devising approximate chain rules for nonlinear random changes of variables. This approach allows us to cover most examples of random fields arising in the modelling of heterogeneous materials in the applied sciences, including Gaussian fields with arbitrary covariance function, Poisson random inclusions with (unbounded) random radii, random parking and Matérn-type processes, as well as Poisson random tessellations. The obtained multiscale functional inequalities, which we primarily develop here in view of their application to concentration and to quantitative stochastic homogenization, are of independent interest.

math.PR↗

arXiv2019

Multiscale second-order Poincaré inequalities in probability

Consider an ergodic stationary random field $A$ on the ambient space $\mathbb R^d$. In a companion article, we introduced the notion of multiscale (first-order) functional inequalities, which extend standard functional inequalities like Poincaré, covariance, and logarithmic Sobolev inequalities in the probability space, while still ensuring strong concentration properties. We also developed a constructive approach to these functional inequalities, proving their validity for prototypical examples including Gaussian fields, Poisson random tessellations, and random sequential adsorption (RSA) models, which do not satisfy standard functional inequalities. In the present contribution, we turn to second-order Poincaré inequalities à la Chatterjee: while first-order inequalities quantify the distance to constants for nonlinear functions $Z(A)$ in terms of their local dependence on the random field $A$, second-order inequalities quantify their distance to normality. For the above-mentioned examples, we prove the validity of suitable multiscale second-order Poincaré inequalities. In particular, applied to RSA models, these functional inequalities allow to complete and improve previous results by Schreiber, Penrose, and Yukich on the jamming limit, and to propose and fully analyze a more efficient algorithm to approximate the latter.

math.PR↗

arXiv2019

A regularity theory for random elliptic operators

Since the seminal results by Avellaneda \& Lin it is known that elliptic operators with periodic coefficients enjoy the same regularity theory as the Laplacian on large scales. In a recent inspiring work, Armstrong \& Smart proved large-scale Lipschitz estimates for such operators with random coefficients satisfying a finite-range of dependence assumption. In the present contribution, we extend the \emph{intrinsic large-scale} regularity of Avellaneda \& Lin (namely, intrinsic large-scale Schauder and Calderéron-Zygmund estimates) to elliptic systems with random coefficients. The scale at which this improved regularity kicks in is characterized by a stationary field $r_*$ which we call the minimal radius. This regularity theory is \textit{qualitative} in the sense that $r_*$ is almost surely finite (which yields a new Liouville theorem) under mere ergodicity, and it is \textit{quantifiable} in the sense that $r_*$ has high stochastic integrability provided the coefficients satisfy quantitative mixing assumptions. We illustrate this by establishing \emph{optimal} moment bounds on $r_*$ for a class of coefficient fields satisfying a multiscale functional inequality, and in particular for Gaussian-type coefficient fields with arbitrary slow-decaying correlations.

math.AP↗

arXiv2019

The structure of fluctuations in stochastic homogenization

Four quantities are fundamental in homogenization of elliptic systems in divergence form and in its applications: the field and the flux of the solution operator (applied to a general deterministic right-hand side), and the field and the flux of the corrector. Homogenization is the study of the large-scale properties of these objects. In case of random coefficients, these quantities fluctuate and their fluctuations are a priori unrelated. Depending on the law of the coefficient field, and in particular on the decay of its correlations on large scales, these fluctuations may display different scalings and different limiting laws (if any). In this contribution, we identify another crucial intrinsic quantity, motivated by H-convergence, which we refer to as the \emph{homogenization commutator} and is related to variational quantities first considered by Armstrong and Smart. In the simplified setting of the random conductance model, we show what we believe to be a general principle, namely that the homogenization commutator drives at leading order the fluctuations of each of the four other quantities in a strong norm in probability, which is expressed in form of a suitable two-scale expansion and reveals the \emph{pathwise structure} of fluctuations in stochastic homogenization. In addition, we show that the (rescaled) homogenization commutator converges in law to a Gaussian white noise, and we analyze to which precision the covariance tensor that characterizes the latter can be extracted from the representative volume element method. This collection of results constitutes a new theory of fluctuations in stochastic homogenization that holds in any dimension and yields optimal rates. Extensions to the (non-symmetric) continuum setting are also discussed, the details of which are postponed to forthcoming works.

math.AP↗

arXiv2019

A remark on a surprising result by Bourgain in homogenization

In a recent work, Bourgain gave a fine description of the expectation of solutions of discrete linear elliptic equations on $\mathbb Z^d$ with random coefficients in a perturbative regime using tools from harmonic analysis. This result is surprising for it goes beyond the expected accuracy suggested by recent results in quantitative stochastic homogenization. In this short article we reformulate Bourgain's result in a form that highlights its interest to the state-of-the-art in homogenization (and especially the theory of fluctuations), and we state several related conjectures.

math.AP↗

arXiv2018

Long-time homogenization and asymptotic ballistic transport of classical waves

Consider an elliptic operator in divergence form with symmetric coefficients.If the diffusion coefficients are periodic, the Bloch theorem allows one to diagonalize the elliptic operator, which is key to the spectral properties of the elliptic operator and the usual starting point for the study of its long-time homogenization.When the coefficients are not periodic (say, quasi-periodic, almost periodic, or random with decaying correlations at infinity), the Bloch theorem does not hold and both the spectral properties and the long-time behavior of the associatedoperator are unclear.At low frequencies, we may however consider a formal Taylor expansion of Bloch waves (whether they exist or not) based on correctors in elliptic homogenization.The associated Taylor-Bloch waves diagonalize the elliptic operator up to an error term (an "eigendefect"), which we express with the help of a new family of extended correctors.We use the Taylor-Bloch waves with eigendefects to quantify the transport properties and homogenization error over large timesfor the wave equation in terms of the spatial growth of these extended correctors.On the one hand, this quantifies the validity of homogenization over large times (both for the standard homogenized equation and higher-order versions).On the other hand, this allows us to prove asymptotic ballistic transport of classical waves at low energies for almost periodic and random operators.

math.AP↗

arXiv2018

From statistical polymer physics to nonlinear elasticity

A polymer-chain network is a collection of interconnected polymer-chains, made themselves of the repetition of a single pattern called a monomer. Our first main result establishes that, for a class of models for polymer-chain networks, the thermodynamic limit in the canonical ensemble yields a hyperelastic model in continuum mechanics. In particular, the discrete Helmholtz free energy of the network converges to the infimum of a continuum integral functional (of an energy density depending only on the local deformation gradient) and the discrete Gibbs measure converges (in the sense of a large deviation principle) to a measure supported on minimizers of the integral functional. Our second main result establishes the small temperature limit of the obtained continuum model (provided the discrete Hamiltonian is itself independent of the temperature), and shows that it coincides with the $Γ$-limit of the discrete Hamiltonian, thus showing that thermodynamic and small temperature limits commute. We eventually apply these general results to a standard model of polymer physics from which we derive nonlinear elasticity. We moreover show that taking the $Γ$-limit of the Hamiltonian is a good approximation of the thermodynamic limit at finite temperature in the regime of large number of monomers per polymer-chain (which turns out to play the role of an effective inverse temperature in the analysis).

math-ph↗