Overall objectives

The PARADYSE team gathers mathematicians from different communities with the same motivation: to provide a better understanding of dynamical phenomena involving particles. These phenomena are described by fundamental models arising from several fields of physics. We shall focus on model derivation, study of stationary states and asymptotic behaviors, as well as links between different levels of description (from microscopic to macroscopic) and numerical methods to simulate such models. Applications include non-linear optics, thermodynamics and ferromagnetism. Research in this direction has a long history, that we shall only partially describe in the sequel. We are confident that the fact that we come from different mathematical communities (PDE theory, mathematical physics, probability theory and numerical analysis), as well as the fact that we have strong and effective collaborations with physicists, will bring new and efficient scientific approaches to the problems we plan to tackle and will make our team strong and unique in the scientific landscape. Our goal is to obtain original and important results on a restricted yet ambitious set of problems that we develop in this document.

Last activity report : 2022

New results

New results

Some of the results presented below overlap several of the main research themes presented in section 3. However, results presented in paragraphs 6.16.6 are mainly concerned with research axis 3.1, whereas paragraphs 6.76.16 mostly concern axis 3.2. Paragraphs 6.176.19 concern numerics-oriented results, and are encompassed in axis 3.3.

Crank-Nicolson scheme for logarithmic nonlinear Schrödinger equations with non standard dispersion

In 9, we consider a nonlinear Schrödinger equation with discontinuous modulation and logarithmic non linearity. We regularize the nonlinearity at 0 to avoid numerical problems; the regularization parameter is ε . We analyze the consistence of the classical Crank-Nicolson scheme and provide precise error estimates depending on the time step τ and on ε1 .

Standing waves for nonlinear Schrödinger equations with non standard dispersion

In 14 and in the one dimensional case we study the existence of standing waves for a nonlinear Schrödinger equation whose dispersion is singular at x=0 . We overcome the difficulty that the problem is not invariant by space translations by introducing a suitable framework that takes into account the symetries of the problem.

Existence and decay of traveling waves for the non-local Gross–Pitaevskii equation

The non-local Gross–Pitaevskii equation is a model that appears naturally in several areas of quantum physics, for instance in the description of superfluids and in optics when dealing with thermo-optic materials because the thermal non-linearity is usually highly non-local. A. de Laire and S. López-Martínez considered a non-local family of Gross–Pitaevskii equations in dimension one, and they found in 15 general conditions on the interactions, for which there is existence of dark solitons for almost every subsonic speed. Moreover, they established properties of the solitons such as exponential decay at infinity and analyticity. This work improves on the results obtained in P. Mennuni’s PhD thesis.

Recent results for the Landau–Lifshitz equation

In 16, A. de Laire surveys recent results concerning the Landau–Lifshitz equation, a fundamental non-linear PDE with a strong geometric content, describing the dynamics of the magnetization in ferromagnetic materials. He revisits the Cauchy problem for the anisotropic Landau–Lifshitz equation, without dissipation, for smooth solutions, and also in the energy space in dimension one. He also examines two approximations of the Landau–Lifshitz equation given by the sine–Gordon equation and the cubic Schrödinger equation, arising in certain singular limits of strong easy-plane and easy-axis anisotropy, respectively. Concerning localized solutions, he reviews the orbital and asymptotic stability problems for a sum of solitons in dimension one, exploiting the variational nature of the solitons in the hydrodynamical framework. Finally, he surveys results concerning the existence, uniqueness and stability of self-similar solutions (expanders and shrinkers) for the isotropic LL equation with Gilbert term.

Minimizing travelling waves for the Gross–Pitaevskii equation

In 27, A. de Laire, P. Gravejat and D. Smets study the 2D Gross–Pitaevskii equation with periodic conditions in one direction, or equivalently on the product space ×𝕋L where L>0 and 𝕋L=/L. They focus on the variational problem consisting in minimizing the Ginzburg–Landau energy under a fixed momentum constraint. They prove that there exists a threshold value for L below which minimizers are the one-dimensional dark solitons, and above which no minimizer can be one-dimensional.

Modulational instability in random fibers and stochastic Schrödinger equations

The team achieved an analysis of modulational instability in optical fibers with a normal dispersion perturbed with a coloured noise in 11. The effect of coloured noise on the modulational instability was investigated in order to assess whether it can produce a larger modulational instability than periodic modulations or homogeneous fibers with anomalous dispersion. They found that generally this is not the case. In 19, randomly dispersion-managed fibers are on the contrary shown to be able to produce such large instabilities. This research was carried out with physicists from the PhLAM laboratory in Lille.

Large deviations principle for the SSEP with weak boundary interactions

Efficiently characterizing non-equilibrium stationary states (NESS) has been in recent years a central question in statistical physics. The Macroscopic Fluctuations Theory 33 developped by Bertini et al. has laid out a strong mathematical framework to understand NESS, however fully deriving and characterizing large deviations principles for NESS remains a challenging endeavour. In 20, C. Erignoux and his collaborators proved that a static large deviations principle holds for the NESS of the classical Symmetric Simple Exclusion Process (SSEP) in weak interaction with particles reservoirs. This result echoes a previous result by Derrida, Lebowitz and Speer 36, where the SSEP with strong boundary interactions was considered. In 20, it was also shown that the rate function can be characterized both by a variational formula involving the corresponding dynamical large deviations principle, and by the solution to a non-linear differential equation. The obtained differential equation is the same as in 36, with different boundary conditions corresponding to the different scales of boundary interaction.

Mapping hydrodynamics for the facilitated exclusion and zero-range processes

In 24, we derive the hydrodynamic limit for two degenerate lattice gases, the facilitated exclusion process (FEP) and the facilitated zero-range process (FZRP), both in the symmetric and the asymmetric case. For both processes, the hydrodynamic limit in the symmetric case takes the form of a diffusive Stefan problem, whereas the asymmetric case is characterized by a hyperbolic Stefan problem. Although the FZRP is attractive, a property that we extensively use to derive its hydrodynamic limits in both cases, the FEP is not. To derive the hydrodynamic limit for the latter, we exploit that of the zero-range process, together with a classical mapping between exclusion and zero-range processes, both at the microscopic and macroscopic level. Due to the degeneracy of both processes, the asymmetric case is a new result, but our work also provides a simpler proof than the one that was previously proposed for the FEP in the symmetric case in 35.

Equilibrium perturbations for stochastic interacting systems

In 28, we consider the equilibrium perturbations for two stochastic systems: the d -dimensional generalized exclusion process and the one-dimensional chain of anharmonic oscillators. We add a perturbation of order Nα to the equilibrium profile, and speed up the process by N1+κ for parameters 0<κα . Under some additional constraints on κ and α , we show the perturbed quantities evolve according to the Burgers equation in the exclusion process, and to two decoupled Burgers equations in the anharmonic chain, both in the smooth regime.

Moderate deviations for the current and tagged particle in symmetric simple exclusion processes

In 29, we prove moderate deviation principles for the tagged particle position and current in one dimensional symmetric simple exclusion processes. There is at most one particle per site. A particle jumps to one of its two neighbors at rate 1/2 , and the jump is suppressed if there is already one at the target site. We distinguish one particular particle which is called the tagged particle. We first establish a variational formula for the moderate deviation rate functions of the tagged particle positions based on moderate deviation principles from hydrodynamic limit proved by Gao and Quastel 38 Then we construct a minimizer of the variational formula and obtain explicit expressions for the moderate deviation rate functions.

The voter model with a slow membrane

In 30, we introduce the voter model on the infinite lattice with a slow membrane and investigate its hydrodynamic behavior. The model is defined as follows: a voter adopts one of its neighbors’ opinion at rate one except for neighbors crossing the hyperplane {x:x1=1/2} , where the rate is αNβ . Above, α>0,β0 are two parameters and N is the scaling parameter. The hydrodynamic equation turns out to be heat equation with various boundary conditions depending on the value of β . The proof is based on duality method.

Long-time behavior of SSEP with slow boundary

In 17, we consider the symmetric simple exclusion process with slow boundary first introduced in 31. We prove a law of large number for the empirical measure of the process under a longer time scaling instead of the usual diffusive time scaling.

Hydrodynamics for one-dimensional ASEP in contact with a class of reservoirs

In 41, we study the hydrodynamic behaviour of the asymmetric simple exclusion process (ASEP) on the lattice of size  n , in contact with a type of slow boundary reservoirs. A scalar conservation law with boundary-trace conditions is obtained as the hydrodynamic limit in the Euler space-time scale.

A Microscopic Derivation of Coupled SPDE’s with a KPZ Flavor

In 10, we consider an interacting particle system driven by a Hamiltonian dynamics and perturbed by a conservative stochastic noise so that the full system conserves two quantities: energy and volume. The Hamiltonian part is regulated by a scaling parameter vanishing in the limit. We study the form of the fluctuations of these quantities at equilibrium and derive coupled stochastic partial differential equations with a KPZ flavor.

Mathematical modeling for ecology

The team had an important contribution to multi-scale ecosystem modeling. O. Goubet and his collaborators computed in 12 the large population limit of a stochastic process that models the evolution of a complex forest ecosystem to an evolution convection-diffusion equation that is more suitable for concrete computations. Then, they proved on the limit equation that the existence of exchange of population between forest patches slows down the extinction of species.

Quantum optics and quantum information

Given two orthonormal bases in a d-dimensional Hilbert space, one may associate to each state its Kirkwood–Dirac (KD) quasi-probability distribution. KD-non-classical states – those for which the KD-distribution takes on negative and/or non-real values – have been shown to provide a quantum advantage in quantum metrology and information, raising the question of their identification. Under suitable conditions of incompatibility between the two bases, S. De Bièvre provided sharp lower bounds on the support uncertainty of states that guarantee their KD-non-classicality in 4 and 22. In particular, when the bases are completely incompatible, a new notion introduced in this work, states whose support uncertainty is not equal to its minimal value d+1 are necessarily KD-non-classical. The implications of these general results for various commonly used bases, including the mutually unbiased ones, and their perturbations, are detailed.

In quantum optics, the notion of classical state is different than in the discrete value systems described above. In that case one requires the Glauber-Sudarshan P-function to be positive. Characterizing the classical states is a longstanding problem in this context as well. In 25, S. De Bièvre and collaborators establish an interferometric protocol allowing to determine a recently introduced nonclassicality measure, known as the quadrature coherence scale (QCS) 7. A detailed study of the QCS of photon-added/subtracted states is provided in 26.

Linearly implicit high-order numerical methods for evolution problems

G. Dujardin and his collaborator derived in 13 a new class of numerical methods for the time integration of evolution equations set as Cauchy problems of ODEs or PDEs, in the research direction detailed in Section 3.3. The systematic design of these methods mixes the Runge–Kutta collocation formalism with collocation techniques, in such a way that the methods are linearly implicit and have high order. The fact that these methods are implicit allows to avoid CFL conditions when the large systems to integrate come from the space discretization of evolution PDEs. Moreover, these methods are expected to be efficient since they only require to solve one linear system of equations at each time step, and efficient techniques from the literature can be used to do so.

Numerical simulation of multispecies Bose–Einstein condensates

In 23, G. Dujardin, A. Nahas and I. Lacroix-Violet proposed a new numerical method for the simulation of multicomponent Bose–Einstein condensates in dimension 2. They implemented their method and demonstrated its efficiency compared to existing methods from the literature, in several physically relevant regimes (vortex nucleation, vortex sheets, giant holes, etc were obtained numerically). They verified numerically several theoretical results known for the minimizers in strong confinment regimes. They also supported numerically theoretical conjectures in other physically relevant contexts. In addition, they developped post-processing algorithms for the automatic detection of vortex structures (simple vortices, vortex sheets, etc), as well as for the numerical computation of indices.

Discrete quantum harmonic oscillator and Kravchuk transform

We consider in 21 a particular discretization of the harmonic oscillator which admits an orthogonal basis of eigenfunctions called Kravchuk functions possessing appealing properties from the numerical point of view. We analytically prove the almost second-order convergence of these discrete functions towards Hermite functions, uniformly for large numbers of modes. We then describe an efficient way to simulate these eigenfunctions and the corresponding transformation. We finally show some numerical experiments corroborating our different results.

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