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Etienne Sandier

Publications and source records attributed to Etienne Sandier.

18 recordsLinked to original sources
arXiv2025

Vortex lines interaction in the three-dimensional magnetic Ginzburg--Landau model

We complete our study of the three dimensional Ginzburg--Landau functional with magnetic field, in the asymptotic regime of a small inverse Ginzburg--Landau parameter $\varepsilon$, and near the first critical field $H_{c_1}$ for which the first vortex filaments appear in energy minimizers. Under a nondegeneracy condition, we show a next order asymptotic expansion of $H_{c_1}$ as $\varepsilon \to 0$, and exhibit a sequence of transitions, with vortex lines appearing one by one as the intensity of the applied magnetic field is increased: passing $H_{c_1}$ there is one vortex, then increasing $H_{c_1}$ by an increment of order $\log |\log\varepsilon|$ a second vortex line appears, etc. These vortex lines accumulate near a special curve $\Gamma_0$, solution to an isoflux problem. We derive a next order energy that the vortex lines must minimize in the asymptotic limit, after a suitable horizontal blow-up around $\Gamma_0$. This energy is the sum of terms where penalizations of the length of the lines, logarithmic repulsion between the lines and magnetic confinement near $\Gamma_0$ compete. This elucidates the shape of vortex lines in superconductors.

math.AP↗
arXiv2025

A Ginzburg-Landau problem on a circular cone

We carry out an asymptotic analysis for a Ginzburg-Landau type model for tangent vector fields defined on a cone. The results, in the spirit of Brezis, Bethuel and Helein, establish the degree and asymptotic location of vortices, one of which must be situated at the tip of the cone.

math.AP↗
arXiv2024

Bifurcating solitonic vortices in a strip

The specific geometry of a strip provides connections between solitons and solitonic vortices, which are vortices with a solitonic behaviour in the infinite direction of the strip. We show that there exist stationary solutions to the Gross-Pitaevskii equation with k vortices on a transverse line, which bifurcate from the soliton solution as the width of the strip is increased. After decomposing into Fourier series with respect to the transverse variable, the construction of these solitonic vortices is achieved by relying on a careful analysis of the linearized operator around the soliton solution: we apply a fixed point argument to solve the equation in the directions orthogonal to the kernel of the linearized operator, and then handle the direction corresponding to the kernel by an inverse function theorem.

math.AP↗
arXiv2024

Enhancing Accuracy in Deep Learning Using Random Matrix Theory

We explore the applications of random matrix theory (RMT) in the training of deep neural networks (DNNs), focusing on layer pruning that is reducing the number of DNN parameters (weights). Our numerical results show that this pruning leads to a drastic reduction of parameters while not reducing the accuracy of DNNs and CNNs. Moreover, pruning the fully connected DNNs actually increases the accuracy and decreases the variance for random initializations. Our numerics indicate that this enhancement in accuracy is due to the simplification of the loss landscape. We next provide rigorous mathematical underpinning of these numerical results by proving the RMT-based Pruning Theorem. Our results offer valuable insights into the practical application of RMT for the creation of more efficient and accurate deep-learning models.

cs.LG↗
arXiv2022

Solitons and solitonic vortices in a strip

We study the ground state of the Gross Pitaveskii energy in a strip, with a phase imprinting condition, motivated by recent experiments on matter waves solitons. We prove that when the width of the strip is small, the ground state is a one dimensional soliton. On the other hand, when the width is large, the ground state is a solitonic vortex. We provide an explicit expression for the limiting phase of the solitonic vortex as the size of the strip is large: it has the same behaviour as the soliton in the infinite direction and decays exponentially due to the geometry of the strip, instead of algebraically as vortices in the whole space.

math.AP↗
arXiv2022

The Ginzburg-Landau energy with a pinning term oscillating faster than the coherence length

The aim of this article is to study the magnetic Ginzburg-Landau functional with an oscillating pinning term. We consider here oscillations of the pinning term that are much faster than the coherence length \(\varepsilon>0\) which is also the inverse of the Ginzburg-Landau parameter. We study both the case of a periodic potential and of a random stationary ergodic one. We prove that we can reduce the study of the problem to the case where the pinning term is replaced by its average, in the periodic case, and by its expectation with respect to the random parameter in the random case. In order to do that we use a decoupling of the energy due to Lassoued-Mironescu. This leads us to the study of the convergence of a scalar positive minimizer of the Ginzburg-Landau energy with pinning term and with homogeneous Neumann boundary conditions. We prove uniform convergence of this minimizer towards the mean value of the pinning term by using a blow-up argument and a Liouville type result for non-vanishing entire solutions of the real Ginzburg-Landau/Allen-Cahn equation, due to Farina.

math.AP↗
arXiv2021

Bounded vorticity for the 3D Ginzburg-Landau model and an isoflux problem

We consider the full three-dimensional Ginzburg-Landau model of superconductivity with applied magnetic field, in the regime where the intensity of the applied field is close to the "first critical field" $H_{c_1}$ at which vortex filaments appear, and in the asymptotics of a small inverse Ginzburg-Landau parameter $\varepsilon$. This onset of vorticity is directly related to an "isoflux problem" on curves (finding a curve that maximizes the ratio of a magnetic flux by its length), whose study was initiated in [Rom\'an, C. On the First Critical Field in the Three Dimensional Ginzburg-Landau Model of Superconductivity. Commun. Math. Phys. 367, 317-349 (2019). https://doi.org/10.1007/s00220-019-03306-w] and which we continue here. By assuming a nondegeneracy condition for this isoflux problem, which we show holds at least for instance in the case of a ball, we prove that if the intensity of the applied field remains below $H_{c_1}+ C \log |\log \varepsilon|$, the total vorticity remains bounded independently of $\varepsilon$, with vortex lines concentrating near the maximizer of the isoflux problem, thus extending to the three-dimensional setting a two-dimensional result of [Sandier, E., Serfaty, S. Ginzburg-Landau minimizers near the first critical field have bounded vorticity. Cal Var 17, 17-28 (2003). https://doi.org/10.1007/s00526-002-0158-9]. We finish by showing an improved estimate on the value of $H_{c_1}$ in some specific simple geometries.

math.AP↗
arXiv2019

Vortex patterns and sheets in segregated two component Bose-Einstein condensates

We study minimizers of a Gross-Pitaevskii energy describing a two-component Bose-Einstein condensate set into rotation. We consider the case of segregation of the components in the Thomas-Fermi regime, where a small parameter $ε$ conveys a singular perturbation. We estimate the energy as a term due to a perimeter minimization and a term due to rotation. In particular, we prove a new estimate concerning the error of a Modica Mortola type energy away from the interface. For large rotations, we show that the interface between the components gets long, which is a first indication towards vortex sheets.

math.AP↗
arXiv2018

Renormalized Energy and Asymptotic Expansion of Optimal Logarithmic Energy on the Sphere

We study the Hamiltonian of a two-dimensional log-gas with a confining potential $V$ satisfying the weak growth assumption -- $V$ is of the same order than $2\log|x|$ near infinity -- considered by Hardy and Kuijlaars [J. Approx. Theory, 170(0) : 44-58, 2013]. We prove an asymptotic expansion, as the number $n$ of points goes to infinity, for the minimum of this Hamiltonian using the Gamma-Convergence method of Sandier and Serfaty [Ann. Proba., to appear, 2015]. We show that the asymptotic expansion as $n\to +\infty$ of the minimal logarithmic energy of $n$ points on the unit sphere in $\mathbb{R}^3$ has a term of order $n$ thus proving a long standing conjecture of Rakhmanov, Saff and Zhou [Math. Res. Letters, 1:647-662, 1994]. Finally we prove the equivalence between the conjecture of Brauchart, Hardin and Saff [Contemp. Math., 578:31-61,2012] about the value of this term and the conjecture of Sandier and Serfaty [Comm. Math. Phys., 313(3):635-743, 2012] about the minimality of the triangular lattice for a "renormalized energy" $W$ among configurations of fixed asymptotic density.

math.AP↗
arXiv2017

A Two scale $Γ$-convergence Approach for Random Non-Convex Homogenization

We propose an abstract framework for the homogenization of random functionals which may contain non-convex terms, based on a two-scale $Γ$-convergence approach and a definition of Young measures on micropatterns which encodes the profiles of the oscillating functions and of functionals. Our abstract result is a lower bound for such energies in terms of a cell problem (on large expanding cells) and the $Γ$-limits of the functionals at the microscale. We show that our method allows to retrieve the results of Dal Maso and Modica in the well-known case of the stochastic homogenization of convex Lagrangians. As an application, we also show how our method allows to stochastically homogenize a variational problem introduced and studied by Alberti and Müller, which is a paradigm of a problem where an additional mesoscale arises naturally due to the non-convexity of the singular perturbation (lower order) terms in the functional.

math.AP↗
arXiv2017

Small energy Ginzburg-Landau minimizers in ${\mathbb R}^3$

We prove that a local minimizer of the Ginzburg-Landau energy in ${\mathbb R}^3$ satisfying the condition $\liminf_{R\to\infty}E(u;B_R)/RlnR < 2π$ must be constant. The main tool is a new sharp eta-ellipticity result for minimizers in dimension three that might be of independent interest.

math.AP↗
arXiv2014

1D Log Gases and the Renormalized Energy: Crystallization at Vanishing Temperature

We study the statistical mechanics of a one-dimensional log gas with general potential and arbitrary beta, the inverse of temperature, according to the method we introduced for two-dimensional Coulomb gases in [SS2]. Such ensembles correspond to random matrix models in some particular cases. The formal limit beta infinite corresponds to "weighted Fekete sets" and is also treated. We introduce a one-dimensional version of the "renormalized energy" of [SS1], measuring the total logarithmic interaction of an infinite set of points on the real line in a uniform neutralizing background. We show that this energy is minimized when the points are on a lattice. By a suitable splitting of the Hamiltonian we connect the full statistical mechanics problem to this renormalized energy W, and this allows us to obtain new results on the distribution of the points at the microscopic scale: in particular we show that configurations whose W is above a certain threshhold (which tends to min W as beta tends to infinity) have exponentially small probability. This shows that the configurations have increasing order and crystallize as the temperature goes to zero.

math.PR↗
arXiv2013

On lattices with finite Coulombian interaction energy in the plane

Given a discrete set $Λ$ in the plane (we will also say a {\em lattice}) and a real number $m\ge 0$, the renormalized energy introduced in \cite{ss1} heuristically describes the interaction energy of unit charges placed at the points of $Λ$ with a uniform negative background of density $m\in\R$. It is defined in several steps, following mostly \cite{ss1}.

math-ph↗
arXiv2013

2D Coulomb Gases and the Renormalized Energy

We study the statistical mechanics of classical two-dimensional "Coulomb gases" with general potential and arbitrary β, the inverse of the temperature. Such ensembles also correspond to random matrix models in some particular cases. The formal limit case β=\infty corresponds to "weighted Fekete sets" and also falls within our analysis. It is known that in such a system points should be asymptotically distributed according to a macroscopic "equilibrium measure," and that a large deviations principle holds for this, as proven by Ben Arous and Zeitouni. By a suitable splitting of the Hamiltonian, we connect the problem to the "renormalized energy" W, a Coulombian interaction for points in the plane introduced in our prior work, which is expected to be a good way of measuring the disorder of an infinite configuration of points in the plane. By so doing, we are able to examine the situation at the microscopic scale, and obtain several new results: a next order asymptotic expansion of the partition function, estimates on the probability of fluctuation from the equilibrium measure at microscale, and a large deviations type result, which states that configurations above a certain threshhold of W have exponentially small probability. When β\to \infty, the estimate becomes sharp, showing that the system has to "crystallize" to a minimizer of W. In the case of weighted Fekete sets, this corresponds to saying that these sets should microscopically look almost everywhere like minimizers of W, which are conjectured to be "Abrikosov" triangular lattices.

math-ph↗
arXiv2012

From the Ginzburg-Landau model to vortex lattice problems

We study minimizers of the two-dimensional Ginzburg-Landau energy with applied magnetic field, between the first and second critical fields. In this regime, minimizing configurations exhibit densely packed hexagonal vortex lattices, called Abrikosov lattices. We derive, in some asymptotic regime, a limiting interaction energy between points in the plane, $W$, which we prove has to be minimized by limits of energy-minimizing configurations, once blown-up at a suitable scale. This is a next order effect compared to the mean-field type results we previously established. The limiting "Coulombian renormalized energy" $W$ is a logarithmic type of interaction, computed by a "renormalization," and we believe it should be rather ubiquitous. We study various of its properties, and show in particular, using results from number theory, that among lattice configurations the hexagonal lattice is the unique minimizer, thus providing a first rigorous hint at the Abrikosov lattice. Its minimization in general remains open. The derivation of $W$ uses energy methods: the framework of $Γ$-convergence, and an abstract scheme for obtaining lower bounds for "2-scale energies" via the ergodic theorem.

math.AP↗
arXiv2010

Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement

We prove some improved estimates for the Ginzburg-Landau energy (with or without magnetic field) in two dimensions, relating the asymptotic energy of an arbitrary configuration to its vortices and their degrees, with possibly unbounded numbers of vortices. The method is based on a localisation of the ``ball construction method" combined with a mass displacement idea which allows to compensate for negative errors in the ball construction estimates by energy ``displaced" from close by. Under good conditions, our main estimate allows to get a lower bound on the energy which includes a finite order ``renormalized energy" of vortex interaction, up to the best possible precision i.e. with only a $o(1)$ error per vortex, and is complemented by local compactness results on the vortices. This is used crucially in a forthcoming paper relating minimizers of the Ginzburg-Landau energy with the Abrikosov lattice. It can also serve to provide lower bounds for weighted Ginzburg-Landau energies.

math.AP↗
arXiv2000

Pinning phenomena in the Ginzburg-Landau Model of Superconductivity

We study the Ginzburg-Landau energy of superconductors with a term $a_\ep$ modelling the pinning of vortices by impurities in the limit of a large Ginzburg-Landau parameter $κ=1/\ep$. The function $a_\ep$ is oscillating between 1/2 and 1 with a scale which may tend to 0 as $κ$ tends to infinity. Our aim is to understand that in the large $κ$ limit, stable configurations should correspond to vortices pinned at the minimum of $a_\ep$ and to derive the limiting homogenized free-boundary problem which arises for the magnetic field in replacement of the London equation. The method and techniques that we use are inspired from those of Sandier-Serfaty (in which the case $a_\ep \equiv 1$ was treated) and based on energy estimates, convergence of measures and construction of approximate solutions. Because of the term $a_\ep(x)$ in the equations, we also need homogenization theory to describe the fact that the impurities, hence the vortices, form a homogenized medium in the material.

cond-mat.supr-con↗