Centre de Physique Théorique


March 2021

Wednesday 3 March 14:00-15:00, online-for link write to Annalisa Panati

Chaos in navigation satellites

Jérôme Daquin,Jerome Daquin, University of Namur, Dpt of Mathematics

The threat raised by space debris has vivified long-term studies of terrestrial
orbits. To apprehend the motion on long time scales, the Astrodynamics
community has translated, deployed and adapted many tools coming from
Dynamical System theory and Celestial Mechanics. During this seminar, I will
review and discuss some aspects of the long-term dynamics of terrestrial
orbits, especially in the range of medium altitude. In particular, I’ll reveal
a manifold structure in a 3 degrees-of-freedom Hamiltonian system physically
representative of the dynamics of navigation satellites. Long-time properties
of the system are explained in terms of the possible normally hyperbolic
manifold structure and the associated stable and unstable manifolds regulating
transport properties. Those transport properties could be appealing for end-
of-life strategies based on manifolds dynamics.

Wednesday 3 March 15:30-17:00, online-for link write to Annalisa Panati

Large deviations and dynamical phase transitions in stochastic chemical networks

Alexandre Lazarescu (Université de Louvain)

The standard (macroscopic) description of chemical reactions in a well-mixed dilute solution is a set of coupled first-order differential equations in time on the concentration of every species in solution. A natural way to describe the same system at a microscopic scale (i.e. for a priori finite numbers of chemical particles, rather than finite concentrations) is through the chemical master equation, which is a Markov jump process on probability distributions of chemical compositions (the number of particles of each species), which is consistent with the former macroscopic description, in the appropriate scaling limit.

Large deviation functions, sometimes called dynamical free energies, are the natural language to describe fluctuations around the typical behaviour of such stochastic systems in the limit of either a large size, or a long observation time. In this talk, I will present recent results on the structure of large deviations for stochastic chemical networks with mass-action dynamics. We will see how, in the limit of a large volume with finite concentrations, the stochastic evolution of the system can be described through a path integral involving a non-quadratic Hamiltonian which can be calculated explicitly. We will then look at probabilities of observing certain unlikely averages of concentrations and chemical currents over a long time, which can be recast as solutions of Hamilton’s equations of motion for said Hamiltonian. We will see how those distributions generically exhibit dynamical phase transitions if the macroscopic set of coupled differential equations has several stable attractors, and we will illustrate this fact on the Schlögl model, which is one of the simplest bistable chemical systems.

Wednesday 17 March 14:00-15:00, online-for link write to Annalisa Panati

Mean field models for fluctuation driven population dynamics

Matteo di Volo (LPTM, Université de Cergy-Pontoise)

Lorentzian distributions (LD) have been largely employed in statistical mechanics to obtain exact results for heterogeneous systems. However, due to the divergence of all the conventional moments and cumulants it is not possible to go beyond the ideal case, even for slight deformations of the LD due to noise sources. We will discuss a reduction method to analytically
investigate heterogenous systems with perturbed LD by considering an expansion of the characteristic function in terms of ’pseudo-cumulants’.
In particular, this approach allows to derive low dimensional mean field models for the macroscopic evolution of heterogeneous spiking networks of quadratic integrate-and-fire
neurons subject to extrinsinc and endogenous random fluctuations. Finally, the mean field model will be employed to describe collective oscillations measured in the Hippocampus with macroscopic recordings.

Wednesday 17 March 15:30-17:00, online-for link write to Annalisa Panati

L’équation non-linéaire de Schrödinger, ses généralisations, ses applications en optique et hydrodynamique

Andrea Armaroli, PhLAM, Université de Lille

Après un rappel des propriétés mathématiques de l’équation non-linéaire de Schrödinger (ENLS), je présenterai quelques applications dans la physique des ondes non-linéaires, notamment dans l’optique en fibre et dans l’hydrodynamique. Quelques résultats récents sur les généralisations de l’ENLS seront discutés. L’inclusion de termes d’ordre supérieure et la variation de ses paramètres pendant l’évolution représenteront le cœur de mon exposé : les approches théoriques et leur pertinence dans la physique seront discutés.

Wednesday 24 March 14:00-15:00, online-for link write to Annalisa Panati

On the ensemble controllability of quantum systems via adiabatic methods

Nicolas Augier (INRIA Sophia-Antipolis)

The principal issue that will be developed in this talk is how to control

a parameter-dependent family of quantum systems with a common control input,

that is, the ensemble controllability problem. Thanks to the study one-parametric

families of Hamiltonians and their generic singularities when the system is driven

by two real inputs, we will give an explicit adiabatic control strategy for the

ensemble controllability problem when geometric conditions on the spectrum of

the Hamiltonian are satisfied, in particular, the existence of conical or

semi-conical intersections of eigenvalues.
Then, in order to understand which controllability properties can be extended

to the case where the system is driven by a single real input, we will study the

compatibility of the adiabatic approximation with the rotating wave approximation.

Wednesday 24 March 15:30-16:30, online-for link write to Annalisa Panati

Random Schrödinger Operators arising in the study of aperiodic media

Constanza Rojas-Molina (CY Cergy Paris Université)

In this talk we review some recent results on random Schrödinger operators,
a standard framework for the study of the absence of wave propagation in random media, a phenomenon known
as Anderson localization. We focus on the case where the random medium is given by a random potential with
aperiodic underlying structure, which breaks the ergodicity of the model. We will see how information on the localization properties
for these non-ergodic models can be applied to
study the spectral and dynamical properties of Delone operators. The latter are deterministic models associated to
aperiodic structures (quasi-crystals). This is joint work with Peter Müller (LMU München)

Wednesday 31 March 14:00-15:00, online-for link write to Annalisa Panati

Dynamiques sauvages holomorphes

Sébastien Sbiebler (IMJ-PRG, Sorbonne Université)

Le phénomène de Newhouse est un des grands mystères des systèmes
dynamiques différentiables. Dans les années 60, Smale désirait décrire
le comportement d’un système dynamique typique. Pour ce faire, il
conjectura la densité des difféomorphismes uniformément hyperboliques
dans l’espace des C^r-difféomorphismes f d’une variété compacte M. Dans
les années 70, un phénomène découvert par l’étudiant de Smale, Newhouse,
se révéla être une obstruction à cette conjecture trop optimiste. Pour
tout 2 ≤ r ≤ ∞, il a montré l’existence d’un ouvert U de l’espace des
C^r-difféomorphismes d’une surface M, tel que tout f dans un
sous-ensemble topologiquement générique de U possède une infinité de
points périodiques attractifs. Aussi la mesure de probabilité invariante
de chaque orbite attractive est très différente de celles des autres, et
le comportement statistique de tels systèmes ne peut donc pas être
décrit de façon satisfaisante avec un nombre fini de mesures.

Dans cet exposé, je definirai précisément le phénomène de Newhouse et je
montrerai comment il peut être étendu au cas de dynamiques holomorphes
de C^2. Je présenterai ensuite un résultat récent en commun avec Pierre
Berger (CNRS, Sorbonne Université, IMJ-PRG) dont la preuve est basée sur
le phénomène de Newhouse. Nous montrons qu’il existe des automorphismes
polynomiaux de C^2 ayant une composante de Fatou errante. L’ensemble de
Fatou est l’ouvert maximal où la dynamique est localement équicontinue.
Une composante de Fatou en est une composante connexe, et elle est dite
errante lorsqu’elle n’est pas prépériodique. Ceci contraste avec un
important théorème prouvé par Sullivan dans les années 80 montrant qu’il
n’y a pas de telles composantes errantes pour des dynamiques
rationnelles en une variable complexe. Nous étudions aussi le
comportement statistique des orbites des points dans la composante
errante et nous montrons que celui-ci est très compliqué à décrire, plus
précisément historique avec une émergence élevée.