Dynamics in multi-component systems: mathematical and physical aspects

19 oct. 2017, 13:30 → 20 oct. 2017, 17:05 Europe/Paris

Batiment M2, salle de réunion (Laboratoire Paul Painlevé)

Stephan De Bièvre (Laboratoire Paul Painlevé), Thomas Rey (Laboratoire Paul Painlevé)

Aim and scope

The goal of this workshop is to bring together physicists and mathematicians studying the dynamics of systems with a very large or infinite number of degrees of freedom, be it experimentally, numerically or analytically. More specifically, the following topics will be addressed:

• Ensembles of relativistic electrons in interaction in accelerators and Free-Electron Lasers

• Nonlinear waves in optics

• Stationary states of the nonlinear Schrödinger equation: soliton turbulence, integrable turbulence

• Kinetic equations: derivations and range of validity; stationary solutions and their stability.

Program

10 speakers are confirmed:

• Dmitry Agafontsev, (Moscow, Russia): Integrable turbulence and formation of rogue waves: new results;
• Enrico Allaria, (Trieste, Italy): Electron beam qualities required for FELs and seeded FEL;
• Nathalie Ayi, (Paris, France): High-field limit of a stochastic BGK model;
• Julien Barré, (Orléans, France): Bifurcations de l'équation de Vlasov;
• Clément Evain (Lille, France): Microbunching Instability in Storage Rings;
• Thierry Goudon (Nice, France): Modeling of Magneto-Optical Traps; from Vlasov-Poisson equations to the Incompressible Euler system, the case with finite mass;
• Maxime Hauray, (Marseille, France): Propagation of chaos for the 3D homogenous Landau equation with moderalty soft potential;
• Maxime Herda, (Paris, France): Massless electron limit of the magnetized Vlasov-Poisson-Fokker-Planck equation;
• Jani Lukkarinen, Helsinki (Finland): Thermalization and prethermalization in anharmonic oscillator chains;
• Clément Mouhot, (Cambridge, United Kingdom)

jeudi 19 octobre

13:30 Welcome

1 Clément Mouhot (University of Cambridge) - TBA

TBA

2 Maxime Hauray (Université d'Aix-Marseille) - Propagation of chaos for the 3D homogenous Landau equation with moderalty soft potential

I will present results obtained in collaboration with Nicolas Fournier, on the propagation of chaos for the Landau equation. The difficulty here is the presence of a singularity in the interaction kernel that appears in equation. For mild singularities, we obtain quantitative results of convergence using a weak-strong stability result for the Landau equation, and a perturbation of it, that allows to apply it also to empirical measures associated to particles system approximating the Landau equation. For stronger singularity, we obtain a qualitative result of convergence, relying on the techniques introduced previously with Stéphane Mischler for the case of vortex, but with several improvement in order to control the possibly degenerate Landau diffusion.

15:25 Cofee break

3 Enrico Allaria (Sincrotrone Trieste) - Electron beam qualities required for FELs and seeded FEL

Modern high gain x-ray free electron lasers rely on the FEL instability that requires high brightness electron beam to develop. This has lead to the need of an accurate control of the electron beam dynamic in the linear accelerator in order to suppress other kind of instabilities that may deteriorate the electron beam quality. In the case of seeded FELs, that allow the generation of fully coherent radiation pulses, it is also required that the electron beam properties do not change over the length of the final FEL pulse. After a general overview, recent experimental results on FEL and electron beam control will be presented.

4 Julien Barré (Université d'Orléans) - Bifurcations de l'équation de Vlasov

Les équations de Vlasov ont une dynamique très riche. En particulier, elles possèdent une infinité d'états stationnaires, dont la stabilité est l'objet de plusieurs résultats mathématiques récents. Je m'intéresserai ici à des états stationnaires légèrement instables, typiques d'une situation de bifurcation. Je présenterai le cas connu des états stationnaires homogènes, pour lesquels une certaine universalité est conjecturée. J'expliquerai les différences avec le cas moins étudié des états stationnaires inhomogènes, pertinents par exemple en astrophysique, pour lesquels nous conjecturons un nouveau type de bifurcation. C'est un travail en commun avec David Métivier (U. de Nice) et Yoshiyuki Yamaguchi (U. de Kyoto).

19:20 Social dinner

vendredi 20 octobre

5 Jani Lukkarinen (University of Helsinki) - Thermalization and prethermalization in anharmonic oscillator chains

In a joint work with Christian Mendl and Jianfeng Lu [Phys. Rev. E 94 (2016) 062104], we consider a particle chain with an onsite anharmonicity, known to exhibit normal heat conduction. We make a direct comparison between the relevant spatially homogeneous, but time-dependent, Boltzmann-Peierls equation and the average Wigner function computed from numerical simulations of the chain, and we demonstrate quantitative agreement after an initial transient time interval. In particular, besides energy conservation, we observe additional quasi-conservation of the phonon density, as predict by the kinetic equation. On super-kinetic time scales, density quasi-conservation is lost while energy remains conserved, and we find evidence for eventual relaxation of the density to its canonical ensemble value. However, the final relaxation can be extremely slow, similar to "prethermalization" observed in certain quantum systems, and it begs for an explanation going beyond the framework of standard kinetic theory.

6 Dmitry Agafontsev, (University of Moscow) - Integrable turbulence and formation of rogue waves: new results

We study numerically the nonlinear stage of modulational instability (MI) of cnoidal waves, in the framework of the focusing one-dimensional Nonlinear Schrodinger (NLS) equation. Cnoidal waves are the exact periodic solutions of the NLS equation and can be represented as a lattice of overlapping solitons. MI of these lattices lead to development of "integrable turbulence". We study the major characteristics of the turbulence and demonstrate their dependence on the degree of "overlapping" between the solitons within the cnoidal wave. Our analysis shows that, in the asymptotic state, when the overlapping is weak, the interactions of the system reduce to two-soliton collisions with two-fold increase in amplitude, while for strong overlapping the probability of rogue waves occurrence is not significantly higher than that for a linear system.

10:55 Coffee Break

7 Nathalie Ayi (Université Pierre et Marie Curie) - High-field limit of a stochastic BGK model

After discussing the problematic of hydrodynamical limits in the context of gas dynamics for the Boltzmann equation, we took an interest in the more general context of BGK models which are some kinetic equations. More precisely, we study a stochastic BGK model with a high field scaling as an approximation of a hyperbolic conservation law with stochastic forcing. After establishing the existence of a solution for any fixed parameter $\varepsilon$, we prove the convergence to a kinetic equation where a modified Maxwellian appears under an additional assumption on these solutions. We deduce from it the existence of a weak solution to the studied scalar conservation law with stochastic forcing.

12:10 Lunch

8 Maxime Herda (UPMC) - Massless electron limit of the magnetized Vlasov-Poisson-Fokker-Planck equation

In this talk, I will present results obtained in collaboration with L. Miguel Rodrigues. We consider a plasma of electrons in an inhomogeneous background of ions. We are interested in the dynamics of the light particles which is modeled by the Vlasov-Poisson-Fokker-Planck equation. In the appropriate scaling where characteristic time scales are those of the ions, an important dimensionless parameter appears, the mass ratio between an electron and an ion. Our focus is on deriving an asymptotic model when the mass ratio tends to $0$. In this regime, strong anisotropic phenomena occurs; while adiabatic equilibrium along magnetic field lines is asymptotically reached our limit model captures a non trivial guiding-center dynamics in the perpendicular directions. We do check that in any case the obtained asymptotic model defines a well-posed dynamical system and when self consistent electric fields are neglected we provide a rigorous mathematical justification of the formally derived systems. In this last step we provide a complete control on solutions by developing anisotropic hypocoercive estimates.

9 Clément Évain (Université de Lille) - Microbunching Instability in Storage Rings

"In storage rings - where relativistic electron bunches circulate in a pseudo-circular orbit - a spatio-temporal instability appears when the number of electrons exceeds a threshold value. This so-called microbunching instability is characterized by the spontaneous apparition of micro-structures at the millimeter scale in the longitudinal profile of the electron bunch and form a complex dynamics. A general overview of this instability will be presented. Experimental, numerical (in particular based on Vlasov-Fokker-Planck equation) and theoretical aspects will be discussed.

10 Thierry Goudon (Inria Sophia Antipolis) - TBA

Magneto-Optical Traps (MOT) are experimental devices used to trap cold atoms. The mathematical modeling of such devices involves the Vlasov-Poisson(-Fokker-Planck) system with an external potential. We can identify physically relevant regimes which allow us to replace this equation by a model of macroscopic nature. The analysis relies on the derivation of the Incompressible Euler system from the Vlasov equation in the quasi-neutral regime by Y. Brenier and N. Masmoudi. However, by contrast to these cases studied on the torus or with infinite charge, here, the strong external field governs the shape of the domain on which the limit equation is posed. The discussion of these phenomena has unexpected connections with the analysis of the obstacle problem. This is a joint work with J. Barré (Univ. Orléans), D. Chiron (Univ. Côte d'Azur) and N. Masmoudi (CIMS-NYU).

16:05 Final word and goodbye