arXiv subjects

Mitia Duerinckx

Publications and source records attributed to Mitia Duerinckx.

At least 37 records · Page 2Linked to original sources

arXiv2023

Non-perturbative approach to the Bourgain-Spencer conjecture in stochastic homogenization

In the context of stochastic homogenization, the Bourgain-Spencer conjecture states that the ensemble-averaged solution of a divergence-form linear elliptic equation with random coefficients admits an intrinsic description in terms of higher-order homogenized equations with an accuracy four times better than the almost sure solution itself. While previous rigorous results were restricted to a perturbative regime with small ellipticity ratio, we make the very first progress in a non-perturbative setting, establishing half of the conjectured optimal accuracy. The validity of the full conjecture remains an open question and might in fact fail in general. Our approach involves the construction of a new corrector theory in stochastic homogenization: while only a bounded number of correctors can be constructed as stationary random fields in a strong sense, we show that twice as many stationary correctors can be defined in a Schwartz-like distributional sense on the probability space. We focus on the Gaussian setting for the coefficient field, and the proof relies heavily on Malliavin calculus.

math.AP↗

arXiv2023

Homogenization of active suspensions and reduction of effective viscosity

We consider a suspension of active rigid particles (swimmers) in a steady Stokes flow, where particles are distributed according to a stationary ergodic random process, and we study its homogenization in the macroscopic limit. A key point in the model is that swimmers are allowed to adapt their propulsion to the surrounding fluid deformation: swimming forces are not prescribed a priori, but are rather obtained through the retroaction of the fluid. Qualitative homogenization of this nonlinear model requires an unusual proof that crucially relies on a semi-quantitative two-scale analysis. After introducing new correctors that accurately capture spatial oscillations created by swimming forces, we identify the contribution of the activity to the effective viscosity. In agreement with the physics literature, an analysis in the dilute regime shows that the activity of the particles can either increase or decrease the effective viscosity (depending on the swimming mechanism), which differs from the well-known case of passive suspensions.

math.AP↗

arXiv2022

Effective viscosity of semi-dilute suspensions

This review is devoted to the large-scale rheology of suspensions of rigid particles in Stokes fluid. After describing recent results on the definition of the effective viscosity of such systems in the framework of homogenization theory, we turn to our new results on the asymptotic expansion of the effective viscosity in the dilute regime. This includes an optimal proof of Einstein's viscosity formula for the first-order expansion, as well as the continuation of this expansion to higher orders. The essential difficulty originates in the long-range nature of hydrodynamic interactions: suitable renormalizations are needed and are captured by means of diagrammatic expansions.

math.AP↗

arXiv2022

The Clausius-Mossotti formula

In this note, we provide a short and robust proof of the Clausius-Mossotti formula for the effective conductivity in the dilute regime, together with an optimal error estimate. The proof makes no assumption on the underlying point process besides stationarity and ergodicity, and can be applied to dilute systems in many other contexts.

math.AP↗

arXiv2022

On Einstein's effective viscosity formula

In his PhD thesis, Einstein derived an explicit first-order expansion for the effective viscosity of a Stokes fluid with a suspension of small rigid particles at low density. His formal derivation relied on two implicit assumptions: (i) there is a scale separation between the size of the particles and the observation scale; and (ii) at first order, dilute particles do not interact with one another. In mathematical terms, the first assumption amounts to the validity of a homogenization result defining the effective viscosity tensor, which is now well understood. Next, the second assumption allowed Einstein to approximate this effective viscosity at low density by considering particles as being isolated. The rigorous justification is, in fact, quite subtle as the effective viscosity is a nonlinear nonlocal function of the ensemble of particles and as hydrodynamic interactions have borderline integrability. In the present memoir, we establish Einstein's effective viscosity formula in the most general setting. In addition, we pursue the low-density expansion to arbitrary order in form of a cluster expansion, where the summation of hydrodynamic interactions crucially requires suitable renormalizations. In particular, we justify a celebrated result by Batchelor and Green on the second-order correction and we explicitly describe all higher-order renormalizations for the first time. In some specific settings, we further address the summability of the whole cluster expansion. Our approach relies on a combination of combinatorial arguments, variational analysis, elliptic regularity, probability theory, and diagrammatic integration methods.

math.AP↗

arXiv2022

Cherenkov radiation with massive bosons and quantum friction

This work is devoted to several translation-invariant models in non-relativistic quantum field theory (QFT), describing a non-relativistic quantum particle interacting with a quantized relativistic field of bosons. In this setting, we aim at the rigorous study of Cherenkov radiation or friction effects at small disorder, which amounts to the metastability of the embedded mass shell of the free non-relativistic particle when the coupling to the quantized field is turned on. Although this problem is naturally approached by means of Mourre's celebrated commutator method, important regularity issues are known to be inherent to QFT models and restrict the application of this method. In this perspective, we introduce a novel non-standard construction procedure for Mourre conjugate operators, which differs from second quantization and allows to circumvent regularity issues. To show its versatility, we apply this construction to the Nelson model with massive bosons, to Fröhlich's polaron model, and to a quantum friction model with massless bosons introduced by Bruneau and De Bièvre: for each of these examples, we improve on previous results.

math-ph↗

arXiv2022

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

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 the independence of Poisson point processes, in a form recently used by Giunti, Gu, Mourrat, and Nitzschner.

math.PR↗

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

On the size of chaos via Glauber calculus in the classical mean-field dynamics

We consider a system of classical particles, interacting via a smooth, long-range potential, in the mean-field regime, and we optimally analyze the propagation of chaos in form of sharp estimates on many-particle correlation functions. While approaches based on the BBGKY hierarchy are doomed by uncontrolled losses of derivatives, we propose a novel non-hierarchical approach that focusses on the empirical measure of the system and exploits a Glauber type calculus with respect to initial data in form of higher-order Poincaré inequalities for cumulants. This main result allows to rigorously truncate the BBGKY hierarchy to an arbitrary precision on the mean-field timescale, thus justifying the Bogolyubov corrections to mean field. As corollaries, we also deduce a quantitative central limit theorem for fluctuations of the empirical measure, and we partially justify the Lenard-Balescu limit for a spatially homogeneous system away from thermal equilibrium.

math.AP↗

arXiv2021

A new spectral analysis of stationary random Schrödinger operators

Motivated by the long-time transport properties of quantum waves in weakly disordered media, the present work puts random Schrödinger operators into a new spectral perspective. Based on a stationary random version of a Floquet type fibration, we reduce the description of the quantum dynamics to a fibered family of abstract spectral perturbation problems on the underlying probability space. We state a natural resonance conjecture for these fibered operators: in contrast with periodic and quasiperiodic settings, this would entail that Bloch waves do not exist as extended states, but rather as resonant modes, and this would justify the expected exponential decay of time correlations. Although this resonance conjecture remains open, we develop new tools for spectral analysis on the probability space, and in particular we show how ideas from Malliavin calculus lead to rigorous Mourre type results: we obtain an approximate dynamical resonance result and the first spectral proof of the decay of time correlations on the kinetic timescale. This spectral approach suggests a whole new way of circumventing perturbative expansions and renormalization techniques.

math-ph↗

arXiv2021

Lenard-Balescu correction to mean-field theory

In the mean-field regime, the evolution of a gas of $N$ interacting particles is governed in first approximation by a Vlasov type equation with a self-induced force field. This equation is conservative and describes return to equilibrium only in the very weak sense of Landau damping. However, the first correction to this approximation is given by the Lenard-Balescu operator, which dissipates entropy on the very long timescale $O(N)$. In this paper, we show how one can derive rigorously this correction on intermediate timescales (of order $O(N^r)$ for $r<1$), close to equilibrium.

math-ph↗

arXiv2021

Well-posedness of the Lenard-Balescu equation with smooth interactions

The Lenard-Balescu equation was formally derived in the 1960s as the fundamental description of the collisional process in a spatially homogeneous system of interacting particles. It can be viewed as correcting the standard Landau equation by taking into account collective screening effects. Due to the reputed complexity of the Lenard-Balescu equation in case of Coulomb interactions, its mathematical theory has remained void apart from the linearized setting. In this contribution, we focus on the case of smooth interactions and we show that dynamical screening effects can then be handled perturbatively. Taking inspiration from the Landau theory, we establish global well-posedness close to equilibrium, local well-posedness away from equilibrium, and we discuss the convergence to equilibrium and the validity of the Landau approximation.

math.AP↗

arXiv2021

Effective viscosity of random suspensions without uniform separation

This work is devoted to the definition and the analysis of the effective viscosity associated with a random suspension of small rigid particles in a steady Stokes fluid. While previous works on the topic have been conveniently assuming that particles are uniformly separated, we relax this restrictive assumption in form of mild moment bounds on interparticle distances.

math.AP↗

arXiv2021

Eigenvalue fluctuations for random elliptic operators in homogenization regime

This work is devoted to the asymptotic behavior of eigenvalues of an elliptic operator with rapidly oscillating random coefficients on a bounded domain with Dirichlet boundary conditions. A sharp convergence rate is obtained for isolated eigenvalues towards eigenvalues of the homogenized problem, as well as a quantitative two-scale expansion result for eigenfunctions. Next, a quantitative central limit theorem is established for eigenvalue fluctuations; more precisely, a pathwise characterization of eigenvalue fluctuations is obtained in terms of the so-called homogenization commutator, in parallel with the recent fluctuation theory for the solution operator.

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

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↗