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We address in this work the question of the discretization of two-dimensional periodic Dirac Hamiltonians. Standard finite differences methods on rectangular grids are plagued with the so-called Fermion doubling problem, which creates spurious unphysical modes. The classical way around the difficulty used in the physics community is to work in the Fourier space, with the inconvenience of having to compute the Fourier decomposition of the coefficients in the Hamiltonian and related convolutions. We propose in this work a simple real-space method immune to the Fermion doubling problem and applicable to all two-dimensional periodic lattices. The method is based on spectral differentiation techniques. We apply our numerical scheme to the study of flat bands in graphene subject to periodic magnetic fields and in twisted bilayer graphene.

This work is devoted to the analysis of the quantum drift-diffusion model derived by Degond et al. The model is obtained as the diffusive limit of the quantum Liouville-BGK equation, where the collision term is defined after a local quantum statistical equilibrium. The corner stone of the model is the closure relation between the density and the current, which is nonlinear and nonlocal, and is the main source of the mathematical difficulties. The question of the existence of solutions has been open since the derivation of the model, and we provide here a first result in a one-dimensional periodic setting. The proof is based on an approximation argument, and exploits some properties of the minimizers of an appropriate quantum free energy. We investigate as well the long time behavior, and show that the solutions converge exponentially fast to the equilibrium. This is done by deriving a non-commutative logarithmic Sobolev inequality for the local quantum statistical equilibrium., Minor fix compared to previous version

The problem considered here is motivated by a work by Nachtergaele and Yau where the Euler equations of fluid dynamics are derived from many-body quantum mechanics, see (Commun Math Phys 243(3):485–540, 2003). A crucial concept in their work is that of local quantum Gibbs states, which are quantum statistical equilibria with prescribed particle, current, and energy densities at each point of space (here $${\mathbb {R}}^d$$ , $$d \ge 1$$ ). They assume that such local Gibbs states exist, and show that if the quantum system is initially in a local Gibbs state, then the system stays, in an appropriate asymptotic limit, in a Gibbs state with particle, current, and energy densities now solutions to the Euler equations. Our main contribution in this work is to prove that such local quantum Gibbs states can be constructed from prescribed densities under mild hypotheses, in both the fermionic and bosonic cases. The problem consists in minimizing the von Neumann entropy in the quantum grand canonical picture under constraints of local particle, current, and energy densities. The main mathematical difficulty is the lack of compactness of the minimizing sequences to pass to the limit in the constraints. The issue is solved by defining auxiliary constrained optimization problems, and by using some monotonicity properties of equilibrium entropies.

International audience; An algorithm for the reconstruction of the magnetization inside an iron sheet is proposed. The magnetization distribution is reconstructed from magnetic measurements made on sensors placed in the surrounding air region. The problem consists in identifying magnetic charges thanks to Bayesian a priori information. The charges derive from the divergence of the flux density through element facets. The efficiency of the method is demonstrated with a numerical application.

As underwater structure, ships are protected by a suitable cathodic protection system to prevent and reduce significantly corrosion phenomenon. This protective system, called Impressed Current Cathodic Protection (ICCP), creates an underwater electrical current. Then, this current generates an electromagnetic field also referred as electrical signature that could be detectable by sensors [1-2]. Therefore, this work refers to the electrochemical characterization of the materials used in naval framework in addition to a better comprehension of the corrosion phenomenon. The final goal is to predict electrical signatures of ships under different corrosion conditions as illustrated on figure 1: Electrical fields induce by natural corrosion or ICCP. These predictions will be done by including the obtained models of material in numerical simulation tools based on the boundary element method. In a first time, polarization laws of different materials present on the immersed hull and electrically connected are obtained: the hull material (coated and uncoated), the propeller material and zinc for sacrificial anodes to complete the cathodic protection system. One part of this project is to be able to model the metal/electrolyte interface in different operating conditions: speed, temperature, polarization state, ageing of the coating and galvanic coupling between the different metals. The second part of this project is to integrate these laboratory data in simulation tools to optimize it. In these simulation tools, the hull is discretized by a surface mesh. Each element of the mesh is being associated to local polarization laws obtained during the laboratory test. Then laboratory models and simulation tools will be tested at a bigger scale on a mockup. To achieve these goals, it’s important to understand what happens physically at the interface and modeling the different corrosion phenomena which take place at the interface. To model the interface experimentally, Electrical Equivalent Circuits are used. These Electrical Equivalent Circuits, commonly called EEC, are deduced considering physical phenomenon happening at the interface by fitting Electrochemical Impedance Spectroscopy (EIS) data. To complete EIS investigations and confirm the different models, potentiodynamic curves and Scanning Electron Microscopy with EDX analyses were also performed. From the first results, Electrical Equivalent Circuits (EEC) of the metallic interface behavior (hull) are proposed for different polarizations conditions: anodic, corrosion potential and cathodic (ICCP potential: -0,8 VECS) in order to understand more deeply the corrosion phenomenon following the polarization state of the steel. In complement EEC investigations will be made under different temperatures and in electrolyte dynamic conditions (thanks to a rotating electrode) for hull steel samples at corrosion potential. The propeller material will also be characterized following speed influence of the solution and temperature. Simultaneously painted hull steel is also investigated in regards to temperature, polarization state and mechanical ageing. Transition state (ICCP turn on/off) and galvanic coupling between propeller and hull materials coated or not will be investigated to complete elementary investigations. Painted hull steel samples give equivalent result whatever experimental conditions. Indeed freshly painted hull steel samples show an almost pure capacitance behavior if we refer to the (Loveday et al.; 2004) [1] work’s suggesting a strong protective behavior of freshly painted sample. To model this behavior a simple Randle circuit,which only take into account the impedance of the paint, is used. The impact of mechanical defects will be presented to simulate paint degradation and investigate evolution of polarization law. In case of uncoated hull steel, EEC are more complex and change following the polarization and experimental conditions. In all cases, a deposit is formed but its nature and protective behavior change with polarization: a formation of calcareous deposit under cathodic polarization and corrosion products deposit under anodic polarization. That requires in consequence to include additional impedances in the EEC modeling: impedance of a deposit or diffusion impedance for example. Temperature variation and dynamic conditions influence also the kinetic of the different interfacial phenomena. Simultaneously some results of simulation obtain with the use of this first result give encouraging results with good correlations between data obtain in at laboratory scale and simulations. (C Rannou and JL Coulomd; 2008 ) : Optimization of the cathodic protection system of military ships with respect to the double constraint:cathodic protection and electromagnetic silencing ; HAL ; MARELEC 2006; Amesterdam; Netherlands; 2004 (A.Guibert ) : Diagnostique de corrosion et prédiction de signature électromagnétique de structures sous-marines sous protection cathodique ; Science de l’ingénieur ; Institut National Polytechnique de Grenoble-INPG ; HAL ; 2009 (Loveday et al.; 2004) : Evaluation of organic Coatings with Electrochemical Impedance Spectroscopy, part 2; Gamy Instruments; JCT Coating tech; 2004 Figure 1

We address in this work the problem of minimizing quantum entropies under local constraints. We suppose that macroscopic quantities, such as the particle density, current, and kinetic energy, are fixed at each point of [Formula: see text] and look for a density operator over [Formula: see text], minimizing an entropy functional. Such minimizers are referred to as local Gibbs states. This setting is in contrast with the classical problem of prescribing global constraints, where the total number of particles, total current, and total energy in the system are fixed. The question arises, for instance, in the derivation of fluid models from quantum dynamics. We prove, under fairly general conditions, that the entropy admits a unique constrained minimizer. Due to a lack of compactness, the main difficulty in the proof is to show that limits of minimizing sequences satisfy the local energy constraint. We tackle this issue by introducing a simpler auxiliary minimization problem and by using a monotonicity argument involving the entropy.

We consider the time-reversal of waves in time-dependent media. The constitutive parameters of a wave equation are assumed constant in time during intervals $(0,t-\tau)\cup(t+\tau,2t)$ for $\tau\ll...

This paper concerns the kinetic limit of the Dirac equation with a random electromagnetic field. We give a detailed mathematical analysis of the radiative transport limit for the phase space energy density of solutions to the Dirac equation. Our derivation is based on a martingale method and a perturbed test function expansion. This requires the electromagnetic field to be a Markovian space-time random field. The main mathematical tool in the derivation of the kinetic limit is the matrix-valued Wigner transform of the vector-valued Dirac solution. The major novelty compared with the scalar (Schrodinger) case is the proof of the weak convergence of cross modes to zero. The propagating modes are shown to converge in an appropriate probabilistic sense to their deterministic limit.

This work is devoted to the asymptotic analysis of high frequency wave propagation in random media with long-range dependence. We are interested in two asymptotic regimes, that we investigate simultaneously: the paraxial approximation, where the wave is collimated and propagates along a privileged direction of propagation, and the white-noise limit, where random fluctuations in the background are well approximated in a statistical sense by a fractional white noise. The fractional nature of the fluctuations is reminiscent of the long-range correlations in the underlying random medium. A typical physical setting is laser beam propagation in turbulent atmosphere. Starting from the high frequency wave equation with fast non-Gaussian random oscillations in the velocity field, we derive the fractional It\^o-Schr\"odinger equation, that is a Schr\"odinger equation with potential equal to a fractional white noise. The proof involves a fine analysis of the backscattering and of the coupling between the propagating and evanescent modes. Because of the long-range dependence, classical diffusion-approximation theorems for equations with random coefficients do not apply, and we therefore use moment techniques to study the convergence., Comment: 58 pages, 5 figures

We consider in this work the problem of minimizing the von Neumann entropy under the constraints that the density of particles, the current, and the kinetic energy of the system is fixed at each point of space. The unique minimizer is a self-adjoint positive trace class operator, and our objective is to characterize its form. We will show that this minimizer is solution to a self-consistent nonlinear eigenvalue problem. One of the main difficulties in the proof is to parametrize the feasible set in order to derive the Euler–Lagrange equation, and we will proceed by constructing an appropriate form of perturbations of the minimizer. The question of deriving quantum statistical equilibria is at the heart of the quantum hydrodynamical models introduced by Degond and Ringhofer (J Statist Phys 112:(3–4), 587-628, 2003). An original feature of the problem is the local nature of constraints, i.e. they depend on position, while more classical models consider the total number of particles, the total current and the total energy in the system to be fixed.

In this paper, we consider the problem of minimizing quantum free energies under the constraint that the density of particles is fixed at each point of R d , for any d ≥ 1 . We are more particularly interested in the characterization of the minimizer, which is a self-adjoint nonnegative trace class operator, and will show that it is solution to a nonlinear self-consistent problem. This question of deriving quantum statistical equilibria is at the heart of the quantum hydrodynamical models introduced by Degond and Ringhofer in [4] . An original feature of the problem is the local nature of constraint, i.e. it depends on position, while more classical models consider the total number of particles in the system to be fixed. This raises difficulties in the derivation of the Euler-Lagrange equations and in the characterization of the minimizer, which are tackled in part by a careful parameterization of the feasible set.

This work is devoted to the stability/resolution analysis of several imaging functionals in complex environments. We consider both linear functionals in the wavefield as well as quadratic functionals based on wavefield correlations. Using simplified measurement settings and reduced functionals that retain the main features of functionals used in practice, we obtain optimal asymptotic estimates of the signal-to-noise ratios depending on the main physical parameters of the problem. We consider random media with possibly long-range dependence and with a correlation length that is less than or equal to the central wavelength of the source we aim to reconstruct. This corresponds to the wave propagation regimes of radiative transfer or homogenization.

This work concerns the analysis of wave propagation in random media. Our medium of interest is sea ice, which is a composite of a pure ice background and randomly located inclusions of brine and air. From a pulse emitted by a source above the sea ice layer, the main objective of this work is to derive a model for the backscattered signal measured at the source/detector location. The problem is difficult in that, in the practical configuration we consider, the wave impinges on the layer with a non-normal incidence. Since the sea ice is seen by the pulse as an effective (homogenized) medium, the energy is specularly reflected and the backscattered signal vanishes in a first order approximation. What is measured at the detector consists therefore of corrections to leading order terms, and we focus in this work on the homogenization corrector. We describe the propagation by a random Helmholtz equation, and derive an expression of the corrector in this layered framework. We moreover obtain a transport model for quadratic quantities in the random wavefield in a high frequency limit., 30 pages

International audience; This paper deals with the use of the Bayesian approach to inverse an underdetermined magnetostatic problem based on spherical harmonic expansion. Identification of the spherical harmonic coefficients is helped thanks to some a priori information. This information comes from a numerical model statistically studied to define an average-state vector and a covariance matrix. The whole approach is applied for the study of the magnetostatic field inside an electric vehicle, created by its power circuitry. It demonstrates the strength of merging a priori information and measured information in order to obtain an efficient identification of magnetic sources created by a complex set of conductors.

International audience; This work is devoted to Monte Carlo methods for radiative transfer equations with singular kernels, and is motivated by the study of wave propagation in random media with long-range dependence. As opposed to the short-range case where the collision cross section is integrable and leads to a non-zero mean free time, the cross section is not integrable in the long-range situation and yields a vanishing mean free time. For computational efficiency, a particular care is then required in the construction of the stochastic processes used in the Monte Carlo methods. For this, we adapt a method of Asmussen-Rosi´nskiRosi´nski and Cohen-Rosi´nskiRosi´nski based on a small jumps/large jumps decomposition of the generator. We compare this method with another approach based on alpha-stable processes, and show the superiority of the first one. We consider various algorithms for the simulation of the jump distribution, and underline the efficiency of an appropriate stochastic collocation method. Comparisons between integrable and singular kernel solutions are given.

International audience; Fast-field cycling (FFC) nuclear magnetic resonance applications require compensation of the ambient magnetic field. This compensation may be stated in terms of a multi-objective optimization problem involving the spherical harmonics decomposition of the field generated by a set ofcompensation coils. In this article, this optimization-based approach is detailed and applied to the particular case of a FFC relaxometer.

This work is concerned with the minimization of quantum entropies under local constraints of density, current, and energy. The problem arises in the work of Degond and Ringhofer about the derivation of quantum hydrodynamical models from first principles, which is an adaptation to the quantum setting of the moment closure strategy by entropy minimization encountered in kinetic theory. The main mathematical difficulty is the lack of compactness needed to recover the energy constraint. We circumvent this issue by a monotonicity argument involving energy, temperature and entropy, that is inspired by some thermodynamical considerations.

International audience; Corrosion or adapted cathodic protection is responsible for UEP (underwater electricpotential) generation that may cause ship detection. As a consequence, ship electricsignatures are measured on dedicated ranges at given depth. Nevertheless, to assessthe threat, the signature must be known at larger distances and for differentparameters (conductivity, water depth, number of conductive layers) depending onship operation zone. In previous work, a dipolar model has been proposed torepresent the electric field sources and exhibited good agreement with experiment. Itwas based on ellipsoidal harmonic decomposition and the use of image methods.The present paper deals with the generalization of the model to its multipolarcomponents. After presenting the model elaboration, the approach is tested onnumerical cases computed by CorOnS software, developed by the G2Elab incooperation with the DGA (Direction Générale de l’Armement).

We consider scattering of a pulse propagating through a three-dimensional random media and study the shape of the pulse in the parabolic approximation. We show that, similarly to the one-dimensional O’Doherty-Anstey theory, the pulse undergoes a deterministic broadening. Its amplitude decays only algebraically and not exponentially in time, due to the signal low/midrange frequency component. We also argue that the parabolic approximation captures the front evolution (but not the signal away from the front) correctly even in the fully threedimensional situation.

This work is devoted to non-linear stochastic Schrodinger equations with multiplicative fractional noise, where the stochastic integral is defined following the Riemann–Stieljes approach of Zahle. Under the assumptions that the initial condition is in the Sobolev space H q ( R n ) for a dimension n less than three and q an integer greater than or equal to zero, that the noise is a Q-fractional Brownian motion with Hurst index H ∈ ( 1 2 , 1 ) and spatial regularity H q + 4 ( R n ) , as well as appropriate hypotheses on the non-linearity, we obtain the local existence of a unique pathwise solution in C 0 ( 0 , T , H q ( R n ) ) ∩ C 0 , γ ( 0 , T , H q − 2 ( R n ) ) , for any γ ∈ [ 0 , H ) . Contrary to the parabolic case, standard fixed point techniques based on the mild formulation of the SPDE cannot be directly used because of the weak smoothing in time properties of the Schrodinger semigroup. We follow here a different route and our proof relies on a change of phase that removes the noise and leads to a Schrodinger equation with a magnetic potential that is not differentiable in time.

High frequency waves propagating in highly oscillatory media are often modeled by radiative transfer equations that describes the propagation of the energy density of the waves. When the medium is statistically homogeneous, averaging effects occur in such a way that in the limit of vanishing wavelength, the wave energy density solves a deterministic radiative transfer equation. In this paper, we are interested in the remaining stochasticity of the energy density. More precisely, we wish to understand how such stochasticity depends on the statistics of the random medium and on the initial phase-space structure of the propagating wave packets. The analysis of stochasticity is a formidable task involving complicated analytical calculations. In this paper, we consider the propagation of waves modeled by a scalar Schrodinger equation and limit the interaction of the waves with the underlying structure to second order. We calculate the scintillation function (second statistical moment of the Wigner transform) f...

This work is concerned with the semi-classical analysis of mixed state solutions to a Schrodinger–Position equation perturbed by a random potential with weak amplitude and fast oscillations in time and space. We show that the Wigner transform of the density matrix converges weakly and in probability to solutions of a Vlasov–Poisson–Boltzmann equation with a linear collision kernel.Aconsequence of this result is that a smooth non-linearity such as the Poisson potential (repulsive or attractive) does not change the statistical stability property of the Wigner transform observed in linear problems.We obtain, in addition, that the local density and current are self-averaging, which is of importance for some imaging problems in random media. The proof brings together the martingale method for stochastic equations with compactness techniques for non-linear PDEs in a semi-classical regime. It relies partly on the derivation of an energy estimate that is straightforward in a deterministic setting but requires the use of a martingale formulation and well-chosen perturbed test functions in the random context.

This paper reviews recent work in two complementary aspects of uncertainty quantification of linear kinetic models. First, we review the modeling of uncertainties in linear kinetic equations as high frequency limits of models for wave propagations in heterogeneous media. Second, we analyze the propagation of stochasticity from the constitutive coefficients to the solutions of linear kinetic equations. Such uncertainty quantifications find many important applications, e.g. in physics-based modeling of errors in inverse problems measurement.

International audience; This work is devoted to the analysis of the quantum Liouville-BGK equation. This equation arises in the work of Degond and Ringhofer on the derivation of quantum hydrodynamical models from first principles. Their theory consists in transposing to the quantum setting the closure strategy by entropy minimization used for kinetic equations. The starting point is the quantum Liouville-BGK equation, where the collision term is defined via a so-called quantum local equilibrium, defined as a minimizer of the quantum free energy under a local density constraint. We then address three related problems: we prove new results about the regularity of these quantum equilibria; we prove that the quantum Liouville-BGK equation admits a classical solution; and we investigate the long-time behavior of the solutions. The core of the proofs is based on a fine analysis of the properties of the minimizers of the free energy.

We consider the imaging of objects buried in unknown heterogeneous media. The medium is probed by using classical (e.g. acoustic or electromagnetic) waves. When heterogeneities in the medium become too strong, inversion methodologies based on a microscopic description of wave propagation (e.g. a wave equation or Maxwell's equations) become strongly dependent on the unknown details of the heterogeneous medium. In some situations, it is preferable to use a macroscopic model for a quantity that is quadratic in the wave fields. Here, such macroscopic models take the form of radiative transfer equations also referred to as transport equations. They can model either the energy density of the propagating wave fields or more generally the correlation of two wave fields propagating in possibly different media. In particular, we consider the correlation of the two fields propagating in the heterogeneous medium when the inclusion is absent and present, respectively. We present theoretical and numerical results showing that reconstructions based on this correlation are more accurate than reconstructions based on measurements of the energy density.

This paper concerns the asymptotic structure of the scintillation function in the simplified setting of wave propagation modeled by an Ito–Schrodinger equation. We show that the size of the scintil...

International audience; We address the following inverse problem in quantum statistical physics: does the quantum free energy (von Neumann entropy + kinetic energy) admit a unique minimizer among the density operators having a given local density $n(x)$? We give a positive answer to that question, in dimension one. This enables to define rigourously the notion of local quantum equilibrium, or quantum Maxwellian, which is at the basis of recently derived quantum hydrodynamic models and quantum drift-diffusion models. We also characterize this unique minimizer, which takes the form of a global thermodynamic equilibrium (canonical ensemble) with a quantum chemical potential.

We analyze in this work the regularity properties of the density operator solution to the quantum Liouville equation. As was recently done for the Strichartz inequalities, we extend to systems of orthonormal functions the local smoothing estimates satisfied by the solutions to the Schrodinger equation. We show in particular that the local density associated to the solution to the free, linear, quantum Liouville equation admits locally fractional derivatives of given order provided the data belong to some Schatten spaces.

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