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Sylvia Serfaty

Publications and source records attributed to Sylvia Serfaty.

At least 55 records · Page 3Linked to original sources
arXiv2015

Branched Microstructures in the Ginzburg-Landau Model of Type-I Superconductors

We consider the Ginzburg-Landau energy for a type-I superconductor in the shape of an infinite three-dimensional slab, with two-dimensional periodicity, with an applied magnetic field which is uniform and perpendicular to the slab. We determine the optimal scaling law of the minimal energy in terms of the parameters of the problem, when the applied magnetic field is sufficiently small and the sample sufficiently thick. This optimal scaling law is proven via ansatz-free lower bounds and an explicit branching construction which refines further and further as one approaches the surface of the sample. Two different regimes appear, with different scaling exponents. In the first regime, the branching leads to an almost uniform magnetic field pattern on the boundary; in the second one the inhomogeneity survives up to the boundary.

math.AP↗
arXiv2014

Next Order Asymptotics and Renormalized Energy for Riesz Interactions

We study systems of $n$ points in the Euclidean space of dimension $d \ge 1$ interacting via a Riesz kernel $|x|^{-s}$ and confined by an external potential, in the regime where $d-2\le s<d$. We also treat the case of logarithmic interactions in dimensions $1$ and $2$. Our study includes and retrieves all cases previously studied in \cite{ss2d,ss1d,rs}. Our approach is based on the Caffarelli-Silvestre extension formula which allows to view the Riesz kernel as the kernel of a (inhomogeneous) local operator in the extended space $\mathbb{R}^{d+1}$. As $n \to \infty$, we exhibit a next to leading order term in $n^{1+s/d}$ in the asymptotic expansion of the total energy of the system, where the constant term in factor of $n^{1+s/d}$ depends on the microscopic arrangement of the points and is expressed in terms of a "renormalized energy." This new object is expected to penalize the disorder of an infinite set of points in whole space, and to be minimized by Bravais lattice (or crystalline) configurations. We give applications to the statistical mechanics in the case where temperature is added to the system, and identify an expected "crystallization regime." We also obtain a result of separation of the points for minimizers of the energy.

math-ph↗
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↗
arXiv2014

Bogoliubov spectrum of interacting Bose gases

We study the large-N limit of a system of N bosons interacting with a potential of intensity 1/N. When the ground state energy is to the first order given by Hartree's theory, we study the next order, predicted by Bogoliubov's theory. We show the convergence of the lower eigenvalues and eigenfunctions towards that of the Bogoliubov Hamiltonian (up to a convenient unitary transform). We also prove the convergence of the free energy when the system is sufficiently trapped. Our results are valid in an abstract setting, our main assumptions being that the Hartree ground state is unique and non-degenerate, and that there is complete Bose-Einstein condensation on this state. Using our method we then treat two applications: atoms with ''bosonic'' electrons on one hand, and trapped 2D and 3D Coulomb gases on the other hand.

math-ph↗
arXiv2014

Renormalized energy equidistribution and local charge balance in 2D Coulomb systems

We consider two related problems: the first is the minimization of the "Coulomb renormalized energy" of Sandier-Serfaty, which corresponds to the total Coulomb interaction of point charges in a uniform neutralizing background (or rather variants of it). The second corresponds to the minimization of the Hamiltonian of a two-dimensional "Coulomb gas" or "one-component plasma", a system of n point charges with Coulomb pair interaction, in a confining potential (minimizers of this energy also correspond to "weighted Fekete sets"). In both cases we investigate the microscopic structure of minimizers, i.e. at the scale corresponding to the interparticle distance. We show that in any large enough microscopic set, the value of the energy and the number of points are "rigid" and completely determined by the macroscopic density of points. In other words, points and energy are "equidistributed" in space (modulo appropriate scalings). The number of points in a ball is in particular known up to an error proportional to the radius of the ball. We also prove a result on the maximal and minimal distances between points. Our approach involves fully exploiting the minimality by reducing to minimization problems with fixed boundary conditions posed on smaller subsets.

math-ph↗
arXiv2013

Quantum Hall phases and plasma analogy in rotating trapped Bose gases

A bosonic analogue of the fractional quantum Hall eff ect occurs in rapidly rotating trapped Bose gases: There is a transition from uncorrelated Hartree states to strongly correlated states such as the Laughlin wave function. This physics may be described by eff ective Hamiltonians with delta interactions acting on a bosonic N-body Bargmann space of analytic functions. In a previous paper [N. Rougerie, S. Serfaty, J. Yngvason, Phys. Rev. A 87, 023618 (2013)] we studied the case of a quadratic plus quartic trapping potential and derived conditions on the parameters of the model for its ground state to be asymptotically strongly correlated. This relied essentially on energy upper bounds using quantum Hall trial states, incorporating the correlations of the Bose-Laughlin state in addition to a multiply quantized vortex pinned at the origin. In this paper we investigate in more details the density of these trial states, thereby substantiating further the physical picture described in [N. Rougerie, S. Serfaty, J. Yngvason, Phys. Rev. A 87, 023618 (2013)], improving our energy estimates and allowing to consider more general trapping potentials. Our analysis is based on the interpretation of the densities of quantum Hall trial states as Gibbs measures of classical 2D Coulomb gases (plasma analogy). New estimates on the mean- field limit of such systems are presented.

math-ph↗
arXiv2013

Quantum Hall states of bosons in rotating anharmonic traps

We study a model of bosons in the lowest Landau level in a rotating trap where the confinement potential is a sum of a quadratic and a quartic term. The quartic term improves the stability of the system against centrifugal deconfinement and allows to consider rotation frequencies beyond the frequency of the quadratic part. The interactions between particles are modeled by a Dirac delta potential. We derive rigorously conditions for ground states of the system to be strongly correlated in the sense that they are confined to the kernel of the interaction operator, and thus contain the correlations of the Bose-Laughlin state. Rigorous angular momentum estimates and trial state arguments indicate a transition from a pure Laughlin state to a state containing in addition a giant vortex at the center of the trap (Laughlin quasi-hole). There are also indications of a second transition where the density changes from a flat profile in a disc or an annulus to a radial Gaussian confined to a thin annulus.

cond-mat.quant-gas↗
arXiv2013

The $\mathbf Γ$-limit of the two-dimensional Ohta-Kawasaki energy. I. Droplet density

This is the first in a series of two papers in which we derive a $Γ$-expansion for a two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion, also known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In that model, two phases appear, which interact via a nonlocal Coulomb type energy. We focus on the regime where one of the phases has very small volume fraction, thus creating small "droplets" of the minority phase in a "sea" of the majority phase. In this paper we show that an appropriate setting for $Γ$-convergence in the considered parameter regime is via weak convergence of the suitably normalized charge density in the sense of measures. We prove that, after a suitable rescaling, the Ohta-Kawasaki energy functional $Γ$-converges to a quadratic energy functional of the limit charge density generated by the screened Coulomb kernel. A consequence of our results is that minimizers (or almost minimizers) of the energy have droplets which are almost all asymptotically round, have the same radius and are uniformly distributed in the domain. The proof relies mainly on the analysis of the sharp interface version of the energy, with the connection to the original diffuse interface model obtained via matching upper and lower bounds for the energy. We thus also obtain a characterization of the limit charge density for the energy minimizers in the diffuse interface model.

math-ph↗
arXiv2013

Remarks on a constrained optimization problem for the Ginibre ensemble

We study the limiting distribution of the eigenvalues of the Ginibre ensemble conditioned on the event that a certain proportion lie in a given region of the complex plane. Using an equivalent formulation as an obstacle problem, we describe the optimal distribution and some of its monotonicity properties.

math.PR↗
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

Renormalized energy concentration in random matrices

We define a "renormalized energy" as an explicit functional on arbitrary point configurations of constant average density in the plane and on the real line. The definition is inspired by ideas of [SS1,SS3]. Roughly speaking, it is obtained by subtracting two leading terms from the Coulomb potential on a growing number of charges. The functional is expected to be a good measure of disorder of a configuration of points. We give certain formulas for its expectation for general stationary random point processes. For the random matrix $β$-sine processes on the real line (beta=1,2,4), and Ginibre point process and zeros of Gaussian analytic functions process in the plane, we compute the expectation explicitly. Moreover, we prove that for these processes the variance of the renormalized energy vanishes, which shows concentration near the expected value. We also prove that the beta=2 sine process minimizes the renormalized energy in the class of determinantal point processes with translation invariant correlation kernels.

math.PR↗
arXiv2012

The Gamma-limit of the two-dimensional Ohta-Kawasaki energy. II. Droplet arrangement at the sharp interface level via the renormalized energy

This is the second in a series of papers in which we derive a $Γ$-expansion for the two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In this model, two phases appear, which interact via a nonlocal Coulomb type energy. Here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction, thus creating small "droplets" of the minority phase in a "sea" of the majority phase. In our previous paper, we computed the $Γ$-limit of the leading order energy, which yields the averaged behavior for almost minimizers, namely that the density of droplets should be uniform. Here we go to the next order and derive a next order $Γ$-limit energy, which is exactly the Coulombian renormalized energy obtained by Sandier and Serfaty as a limiting interaction energy for vortices in the magnetic Ginzburg-Landau model. The derivation is based on the abstract scheme of Sandier-Serfaty that serves to obtain lower bounds for 2-scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem. Without thus appealing to the Euler-Lagrange equation, we establish for all configurations which have "almost minimal energy" the asymptotic roundness and radius of the droplets, and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense. Via a kind of $Γ$-equivalence, the obtained results also yield an expansion of the minimal energy for the original Ohta-Kawasaki energy. This leads to expecting to see triangular lattices of droplets as energy minimizers.

math.AP↗
arXiv2012

Hydrodynamic Limit of Nonlinear Diffusions with Fractional Laplacian Operators

In the limit of a nonlinear diffusion model involving the fractional Laplacian we get a "mean field" equation arising in superconductivity and superfluidity. For this equation, we obtain uniqueness, universal bounds and regularity results. We also show that solutions with finite second moment and radial solutions admit an asymptotic large time limiting profile which is a special self-similar solution: the "elementary vortex patch".

math.AP↗
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↗
arXiv2011

Large vorticity stable solutions to the Ginzburg-Landau equations

We construct local minimizers to the Ginzburg-Landau functional of superconductivity whose number of vortices N is prescribed and blows up as the parameter epsilon, inverse of the Ginzburg-Landau parameter kappa, tends to zero. We treat the case of N as large as log epsilon, and a wide range of intensity of external magnetic field. The vortices of our solutions arrange themselves with uniform density over a subregion of the domain bounded by a "free boundary" determined via an obstacle problem, and asymptotically tend to minimize the "Coulombian renormalized energy" W introduced in [14]. The method, inspired by [22], consists in minimizing the energy over a suitable subset of the functional space, and in showing that the minimum is achieved in the interior of the subset. It also relies heavily on refined asymptotic estimates for the Ginzburg-Landau energy obtained in [14].

math.AP↗
arXiv2011

Energy estimates and cavity interaction for a critical-exponent cavitation model

We consider the minimization of $\int_{Ω_{\ep}} |D\vec u|^p \dd\vec x$ in a perforated domain $Ω_{\ep}:= Ω\setminus \bigcup_{i=1}^M B_{\ep}(\vec a_i)$ of $\R^n$, among maps $\vec u \in W^{1,p}(Ω_{\ep}, \R^n)$ that are incompressible ($\det D\vec u\equiv 1$), invertible, and satisfy a Dirichlet boundary condition $\vec u= \vec g$ on $\partial Ω$. If the volume enclosed by $\vec g (\partial Ω)$ is greater than $|Ω|$, any such deformation $\vec u$ is forced to map the small holes $B_{\ep}(\vec a_i)$ onto macroscopically visible cavities (which do not disappear as $\ep\to 0$). We restrict our attention to the critical exponent $p=n$, where the energy required for cavitation is of the order of $\sum_{i=1}^M v_i |\log \ep|$ and the model is suited, therefore, for an asymptotic analysis ($v_1,..., v_M$ denote the volumes of the cavities). In the spirit of the analysis of vortices in Ginzburg-Landau theory, we obtain estimates for the "renormalized" energy $\frac{1}{n}\int_{Ω_{\ep}} |\frac{D\vec u}{\sqrt{n-1}}|^p \dd\vec x - \sum_i v_i |\log \ep|$, showing its dependence on the size and the shape of the cavities, on the initial distance between the cavitation points $\vec a_1,..., \vec a_M$, and on the distance from these points to the outer boundary $\partial Ω$. Based on those estimates we conclude, for the case of two cavities, that either the cavities prefer to be spherical in shape and well separated, or to be very close to each other and appear as a single equivalent round cavity. This is in agreement with existing numerical simulations, and is reminiscent of the interaction between cavities in the mechanism of ductile fracture by void growth and coalescence.

math.AP↗
arXiv2011

Lorentz space estimates for the Coulombian renormalized energy

In this paper we obtain optimal estimates for the "currents" associated to point masses in the plane, in terms of the Coulombian renormalized energy of Sandier-Serfaty \cite{ss1,ss3}. To derive the estimates, we use a technique that we introduced in \cite{st}, which couples the "ball construction method" to estimates in the Lorentz space $L^{2,\infty}$.

math.AP↗