The MEPHYSTO team is a follow up of the MEPHYSTO project-team. Since the former scientific leader, Antoine Gloria, left in September 2017, the scientific objectives have been modified.

The MEPHYSTO 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 focus on model derivation, study of stationary states and asymptotic behaviors, as well as links between different levels of description (e.g. micro and macro models) and numerical methods to simulate such models. Applications include nonlinear optics, thermodynamics and ferromagnetism.

The team investigates the existence of solitons and their link with the global dynamical behavior for nonlocal problems such as that of the Gross–Pitaevskii (GP) equation which arises in models of dipolar gases. These models, in general, also introduce nonzero boundary conditions which constitute an additional theoretical and numerical challenge. Numerous results are proved for local problems, and numerical simulations allow to verify and illustrate them, as well as making a link with physics. However, most fundamental questions are still open at the moment for nonlocal problems.

The nonlinear Schrödinger (NLS) equation finds applications in numerous fields of physics. We concentrate, in a continued collaboration with our colleagues from the physics department (PhLAM) of the Université de Lille (UdL), in the framework of the Laboratoire d’Excellence CEMPI, on its applications in nonlinear optics and cold atom physics. Issues of orbital stability and modulational instability are central here.

Another typical example of problems that the team wishes to address
concerns the Landau–Lifshitz (LL) equation,
which describes the dynamics of the spin
in ferromagnetic materials.
This equation is a fundamental model in the magnetic recording industry
and solitons in magnetic media are of particular interest
as a mechanism for data storage or information transfer .
It is a quasilinear PDE involving a function that takes values on the unit
sphere

The team investigates, from a microscopic viewpoint, the dynamical mechanism at play in the phenomenon of relaxation towards thermal equilibrium for large systems of interacting particles. For instance, a first step consists in giving a rigorous proof of the fact that a particle repeatedly scattered by random obstacles through a Hamiltonian scattering process will eventually reach thermal equilibrium, thereby completing previous work in this direction by the team. As a second step, similar models as the ones considered classically will be defined and analysed in the quantum mechanical setting, and more particularly in the setting of quantum optics.

Another challenging problem is to understand the interaction of large systems with the boundaries, which is responsible for most energy exchanges (forcing and dissipation), even though it is concentrated in very thin layers. The presence of boundary conditions to evolution equations sometimes lacks understanding from a physical and mathematical point of view. In order to legitimate the choice done at the macroscopic level of the mathematical definition of the boundary conditions, we investigate systems of atoms (precisely chains of oscillators) with different local microscopic defects. We apply our recent techniques to understand how anomalous (in particular fractional) diffusive systems interact with the boundaries. For instance, the powerful tool given by Wigner functions that we already used has been successfully applied to the derivation of anomalous behaviors in open systems (for instance in ). The next step consists in developing an extension of that tool to deal with bounded systems provided with fixed boundaries. We also intend to derive anomalous diffusion by adding long range interactions to diffusive models. There are very few rigorous results in this direction. Finally, we aim at obtaining from a microscopic description the fractional porous medium equation (FPME), a nonlinear variation of the fractional diffusion equation, involving the fractional Laplacian instead of the usual one. Its rigorous study carries out many mathematical difficulties in treating at the same time the nonlinearity and fractional diffusion. We want to make PDE theorists and probabilists work together, in order to take advantage of the analytical results which went far ahead and are more advanced than the statistical physics theory.

The team addresses both questions of precision and numerical cost of the schemes for the numerical integration of nonlinear evolution PDEs, such as the NLS equation. In particular, we aim at developing, studying and implementing numerical schemes with high order that are more efficient for these problems. We also want to contribute to the design and analysis of schemes with appropriate qualitative properties. These properties may as well be “asymptotic preserving” properties, energy-preserving properties, or convergence to an equilibrium properties. Other numerical goals of the team include the numerical simulation of standing waves of nonlinear nonlocal GP equations. We also keep on developing numerical methods to efficiently simulate and illustrate theoretical results on instability, in particular in the context of the modulational instability in optical fibers, where we study the influence of randomness in the physical parameters of the fibers.

The team also designs simulation methods to estimate the accuracy of the physical description via microscopic systems, by computing precisely the rate of convergence as the system size goes to infinity. One method under investigation is related to cloning algorithms, which were introduced very recently and turn out to be essential in molecular simulation.

The team has almost completed the process of creation of a new project-team named Paradyse (for PARticles And DYnamical SystEms), between Inria and the Laboratoire Paul Painlevé of the Université de Lille in 2019.

In 2019, the Mephysto team has been granted an Action de Développement Technologique (ADT) by Inria. This allowed the team to hire Alexandre Roget as an engineer for 2 years. The goal of this ADT is to develop software using mathematical techniques developed in the team, to be used by theoretical and experimental physics communities.

In 2019, the team also had individual successes that can be highlighted. Amongst others, M. Simon submitted an ERC Starting Grant project which was ranked A, and S. De Bièvre became Associate Editor of the Journal of Mathematical Physics.

The nonlocal 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 nonlinearity is usually highly nonlocal. A. de Laire and P. Mennuni have considered a nonlocal family of Gross–Pitaevskii equations in dimension one, and they have provided in conditions on the nonlocal interaction such that there is a branch of traveling waves solutions with nonvanishing conditions at infinity. Moreover, they showed that the branch is orbitally stable. In this manner, this result generalizes known properties for the contact interaction given by a Dirac delta function. Their proof relies on the minimization of the energy at fixed momentum.

As a follow-up of the previous result, P. Mennuni and G. Dujardin carried out numerical simulations of traveling waves for some nonlocal nonlinear Gross-Pitaevskii equations with nonzero conditions at infinity in dimensions 1 and 2. Using a numerical analogue of the minimization of the energy at fixed momentum, they used gradient methods with nonuniform fast Fourier transforms (to deal with the nonlocal terms numerically) to carry out significant numerical simulations to illustrate numerically the theoretical results and to discuss the hypotheses numerically. These results can be found in P. Mennuni's PhD manuscript .

It is well-known that the dynamics of biaxial ferromagnets with a strong easy-axis anisotropy is essentially governed by the cubic Schrödinger equation. A. de Laire and P. Gravejat provided in a rigorous justification to this observation. More precisely, they showed the convergence of the solutions to the Landau-Lifshitz equation for biaxial ferromagnets towards the solutions to the cubic Schrödinger equation in the regime of an easy-axis anisotropy. This result holds for solutions to the Landau–Lifshitz equation in high order Sobolev spaces. By introducing high order energy quantities with good symmetrization properties, they derived the convergence from the consistency of the Landau–Lifshitz equation with the Sine-Gordon equation by using well-tailored energy estimates.

In this regime, they additionally classified the one-dimensional solitons of the Landau-Lifshitz equation and quantified their convergence towards the solitons of the one-dimensional cubic Schrödinger equation.

These two macroscopic behaviors belong to the class of moving interfaces problems, which are particularly hard to derive from the microscopic point of view.

In physics, the rotor chain has been investigated as an example of a system with two conserved quantities (angular momentum and energy), for which the thermal conductivity is finite (and therefore energy diffuses). Numerics shows an unexpected behaviour of the chain when the latter is connected at the boundaries to two thermostats, and a mechanical force imposes an average angular momentum at one boundary: the stationary temperature profile coincides with the values of the thermostats, but in the middle of the chain it raises to a much higher value. This behaviour is related to the presence of two conserved quantities and is sometimes referred to as Joule effect. Since the rotor model is too difficult to be treated analytically, T. Komorowski, S. Olla and M. Simon investigate in the harmonic chain of oscillators, perturbed with a stochastic noise, which makes the heat transport diffusive, namely: the noise destroys the conservation law of the total momentum, but keeps the other two conservation laws (energy and stretch) intact. The boundaries of the chain are connected to two Langevin thermostats and an external force acting on one boundary puts the system in a non-equilibrium stationary state. The authors rigorously derive the Joule effect for a particular value of the noise intensity.

In , S. De Bièvre and his co-authors introduce a new measure of the nonclassicality of the quantum states of an optical field, the so-called “ordering sensitivity” of the state, that measures the fluctuations of its Wigner function. This work is prolonged in two subsequent papers. In , S. De Bièvre and the same co-authors investigate a new class of quantum states they call the “Thermal Difference States” that can be generated by parametric down conversion. They investigate in particular the degree to which such states are nonclassical. In , S. De Bièvre and his postdoc A. Hertz, re-interpret the ordering sensitivity in terms of another physical property of the quantum states of an optical field, namely their quadrature coherence scale. It is shown in particular that a large such coherence scale is responsible for very fast environmental decoherence of the state.

In the research direction exposed in Section , G. Dujardin and his coauthors proposed and studied in several discrete versions of homogeneous and inhomogeneous one-dimensional Fokker-Planck equations. They proved in particular, for these discretizations of velocity and space, the exponential convergence to the equilibrium of the solutions, for time-continuous equations as well as for time-discrete equations. Their method uses new types of discrete Poincaré inequalities for a “two-direction” discretization of the derivative in velocity. For the inhomogeneous problem, they adapted for the very first time hypocoercive methods to the discrete level.

G. Dujardin and his coauthors have revisited and extended relaxation methods for nonlinear Schrödinger equations (NLS). The classical relaxation method for NLS is an energy preserving method and a mass preserving method. Moreover, it is only linearly implicit. A first proof of the second order accuracy was achieved in . Moreover, the method was extended to enable to treat noncubic nonlinearities, nonlocal nonlinearities, as well as rotation terms. The resulting methods are still energy preserving and mass preserving. Moreover, they are shown to have second order accuracy numerically. These new methods are compared with fully implicit, mass and energy preserving methods of Crank and Nicolson.

A. de Laire is a member of the ANR ODA project.

Title: Dispersive and random waves

ANR reference: ANR-18-CE40-0020-01

Coordinator: Nikolay Tzvetkov, Université de Cergy-Pontoise

A. Hardy is a member of the ANR BoB project.

Title: Inférence bayésienne à ressources limitées - données massives et modèles coûteux

Programme ANR: (DS0705) 2016

ANR reference: ANR-16-CE23-0003

Coordinator: R. Bardenet, CNRS & Université de Lille

Duration: October 2016 - October 2020

M. Simon has been a member of the ANR EDNHS project.

Title: Diffusion de l'énergie dans des système hamiltoniens bruités

Type: Défi de tous les savoirs (DS10) 2014

ANR reference: ANR-14-CE25-0011

Coordinator: C. Bernardin, Université de Nice

Duration: October 2014 - October 2019

M. Simon is a collaborator of the ERC Starting Grant HyLEF project.

Title: Hydrodynamic Limits and Equilibrium Fluctuations: universality from stochastic systems

Duration: May 2017 - April 2022

Coordinator: P. Gonçalves, Instituto Superior Técnico, Lisbon, Portugal

A. de Laire co-organized the “Journée des Doctorants en Mathématiques du Nord-Pas-de-Calais”.

S. De Bièvre is Associate Editor of the Journal of Mathematical Physics since January 2019.

M. Simon is reviewer for the main international peer-reviewed journals in probability and statistical physics. In 2019, G. Dujardin served as a reviewer for the numerical analysis journals ESAIM:M2AN, Numerische Mathematik and IMA Journal of Numerical Analysis.

M. Simon has been invited speaker at several conferences, among them:

*Interactions PDEs/Probability: particle systems, hyperbolic conservation laws*, CIRM, Marseille

*1st SFB International Workshop Taming Complexity in Partial Differential Systems*, University of Vienna, Austria

Licence: A. de Laire, Mathématiques fondamentales 1, 54 TD, L1, Université de Lille

Licence: G. Dujardin, Calcul différentiel et intégral, 60h, BA2, Université Libre de Bruxelles, Belgium.

Master: A. de Laire, “Etude de Problèmes Elliptiques”, 60h, M1, Université de Lille

Master: A. Hardy, “Probabilité, modèles et applications” 60h, M1, Université de Lille

Master: A. Hardy, “Séries temporelles”, 30h, M1, Université de Lille

Master: M. Simon, “Introduction à la physique statistique”, 56h, M2, Université de Lille

Master: M. Simon, “Markov Chains and Applications”, Université de Lille and École Centrale Lille

In addition, A. de Laire is in charge of the Master 2 of Applied Mathematics at Université de Lille.

PhD: P. Mennuni, “Ondes progressives de l'équation de Gross–Pitaevskii non locale : analyse et simulations”, Université de Lille, defended on the November 4, 2019; advisors: S. De Bièvre, A. de Laire, G. Dujardin.

M. Simon participated in the programs “Chercheurs itinérants” and “Fête de la Science”, and gave several lectures aimed at high-school students.