Mini-lecture 1: Entropic fluctuations in quantum two-time measurement framework

• Claude-Alain Pillet (Université de Toulon)
• Annalisa Panati (Université de Toulon)

Room: CY Advanced Studies (Auditorium) Non-equilibrium statistical mechanics has seen some impressive developments in the last three decades, since the ground-breaking formulation of the transient and steady entropic Fluctuation Relations (FR) in the early nineties. The extension of these results to the quantum setting has turned out to be surprisingly challenging and it is still an ongoing effort. Kurchan’s seminal work (2000) showed quantum formulation of the transient version of FR is possible by introducing two time measurement framework. In this mini course, we present some recent results where we attempt to introduce a quantum equivalent of steady entropic functional and compare it the transient version for open quantum system. We consider both the case of idealised direct measurement on the reservoirs and experimentally accessible indirect measurement through coupling with an ancilla. We analyse in particular stability with respect to the initial state. In order to deal with the thermodynamics limit and to have general results, we use methods of $C^*$- algebras and modular theory. (Joint work with T. Benoist, L. Bruneau, V. Jakšić)

Strictly-Correlated Electrons systems and their dissociation at infinity

• Rodrigue Lelotte (Ecole des Ponts)

The Strictly-Correlated Electrons (SCE) is a formulation of density functional theory obtained in the semi-classical limit of the Levy-Lieb functional, which is used for strongly-correlated systems. I will present this problem, which arises as a multimarginal optimal transport of a special kind, and present a recent result on the dissociation at infinity of such systems.

On the energy of the dilute Bose gas

• Søren Fournais (University of Copenhagen)

The ground state energy density $e(\rho)$ of a large system of interacting bosons in $3$ dimensions at density $\rho$ satisfies the formula $$ e(\rho) = 4 \pi \rho^2 a \Big( 1 + \frac{15}{128 \sqrt{\pi}} \sqrt{\rho a^3} \Big) + \text{higher order terms}, $$ in the dilute limit $\rho a^3 \ll 1$. Here $a$ is the scattering length of the interaction potential. This is the celebrated Lee-Huang-Yang formula for the energy density. In this talk, I will review the proof of the lower bound in this formula. I will also comment on the harder $2$-dimensional case and how the proof can be modified to accommodate this case.

The ground state of the Dirac-Fock model: a new approach

• Eric Séré (Université Paris Dauphine - PSL)

I will present a new approach to the ground state of the relativistic Dirac-Fock energy inspired of Lieb’s relaxed variational principle for Hartree-Fock. This approach seems simpler and more suitable for theoretical and numerical investigations than previous ones. Based on this approach, I will present existence results for the ground state of molecules and crystals. The existence proof for this ground state suggests an algorithm for its computation, that generalizes the ODA algorithm of Cances-Lebris. The existence result for crystals is joint work with Isabelle Catto, Long Meng and Eric Paturel. The theoretical study and implementation of the algorithm is work in progress with Maxime Chupin, Guillaume Legendre and Long Meng.

Mini-lecture 2: Entropic fluctuations in quantum two-time measurement framework

• Claude-Alain Pillet (Université de Toulon)
• Annalisa Panati (Université de Toulon)

Leading-order term expansion for the Teukolsky equation on subextremal Kerr black holes

• Pascal Millet (Ecole Polytechnique)

The study of wave propagation on black hole spacetimes has been an intense field of research in the past decades. This interest has been driven by the stability problem for black holes and by questions related to scattering theory. In the analysis of Maxwell's equations and the equations of linearized gravity, the focus often shifts to the study of the Teukolsky equation, which offers the advantage of being scalar in nature. I will present a result providing the large time leading-order term for initially localized and regular solutions and valid for the full subextremal range of black hole parameters. I will also discuss some aspects of the proof which relies on recent advances in spectral and microlocal analysis.

Recovery of qubit state after noisy leakage in high-dimensional space

• Janet Anders (University of Exeter)

Experiments often encode qubit states in physical systems that have many more physical dimensions. Unfortunately, environmental noise can cause the logical qubit to leak into these dimensions, compromising the qubit nature of the state. This causes unwanted artefacts, such as increased entropies. I will describe a new mathematical method to recover a meaningful qubit state from a known noisy high-dimensional state [1]. This method is valid for many physical situations where noise acts separately on two subspaces. As an example, we apply the method to the tomographically obtained states of a microwave cavity, which was used in a Maxwell demon experiment [2]. We find excellent recovery of the encoded state and a massive reduction in entropy. The new recovery method paves the way for quantum experiments and technologies to extract meaningful qubit information from a jungle of noise. [1] J. Anders, S. Sevitz, et al, in preparation. [2] N. Cottet, et al., “Observing a quantum Maxwell demon at work”, PNAS 114, 7561 (2017).

Gibbs measures for Hamiltonian PDEs: KMS property and completeness

• Zied Ammari (Université de Rennes)

During the past few decades, Gibbs measures have been used in nonlinear PDEs to establish various remarkable results related to almost sure well-posedness and flow properties. The main ingredients are Fourier analysis, the Hamiltonian structure and the measure invariance. In this talk I will report on some recent contributions obtained in collaboration with Shahnaz Farhat and Vedran Sohinger. In particular, the following aspects will be discussed: (i) The Kubo-Martin-Schwinger (KMS) property: I will define the KMS equilibrium states for Hamiltonian PDEs and show under certain hypotheses that there exists a unique KMS equilibrium state for such system given by the Gibbs measure. (ii) Completeness: I will underline a general principle proving that if a (Hamiltonian) PDE admits a stationary probability measure then the PDE admits almost surely global solutions.

Quantum-classical motion of charged particles interacting with scalar fields

• Shahnaz Farhat (Jacobs University Bremen)

The goal of this talk is to investigate the dynamics of semi-relativistic or non-relativistic charged particles in interaction with a scalar meson field. Our main contribution is the derivation of the classical dynamics of a particle-field system as an effective equation of the quantum microscopic Nelson model, in the classical limit where the value of the Planck constant approaches zero ($\hbar \rightarrow 0$). Thus, we prove the validity of Bohr’s correspondence principle, that is to establish the transition from quantum to classical dynamics. We use a Wigner measure approach to study such transition. Then, as a consequence of this interplay between classical and quantum dynamics, we establish the global well-posedness of the classical particle-field interacting system, despite the low regularity of the related vector field, which prevents the use of a fixed point argument.

Delocalization of the Laplace eigenmodes on Anosov surfaces

• Stéphane Nonnenmacher (Université Paris-Saclay)

The eigenmodes of the Laplace-Beltrami operator on a smooth compact Riemannian manifold $(M,g)$ can exhibit various localization properties in the high frequency regime, which strongly depend on the properties of the geodesic flow. We will focus on situations where this flow is strongly chaotic (Anosov), e.g. if the sectional curvature of $(M,g)$ is negative. The Quantum Ergodicity theorem then states that almost all the eigenmodes become equidistributed on $M$, in the the high frequency limit. The Quantum Unique Ergodicity conjecture claims that this behaviour admits no exception. What can be said about possible "exceptional" eigenstates? In the case of compact surfaces with Anosov geodesic flow, we prove that all eigenmodes fully delocalize across $M$: for any open set $\Omega$ on $M$, the $L^2$ mass on $\Omega$ of any eigenstate is uniformly bounded from below. This is in contrast with, e.g., the case of eigenstates on the round sphere, which may be strongly concentrated along a closed geodesic. The proof uses various methods of semiclassical analysis, the structure of stable and unstable manifolds of the Anosov flow, and a Fractal Uncertainty Principle due to Bourgain-Dyatlov. Joint work with S.Dyatlov and L.Jin.

Disorder relevance for the random field Ising chain in the large interaction limit

• Giambattista Giacomin (Université Paris Cité)

I will present results that describe the domain-wall structure appearing in the large interaction limit for the one dimensional Ising model with centered random external field. These results match with the D. Fisher infinite disorder fixed point prediction (developed for this model in [D. Fisher, P. Le Doussand, C. Monthus, Phys. Rev. E 2001]) and represent a case in which a very strong form of disorder relevance is present. The aim is to present the results along with some ideas from the proof. Hopefully I will be able to discuss also the delicate (and widely open) issue of understanding the universality class to which Fisher’s idea should apply. The talk will be mostly based on joint work with Orphée Collin and Yueyun Hu.

Mini-lecture 3: Entropic fluctuations in quantum two-time measurement framework

• Annalisa Panati (Université de Toulon)
• Claude-Alain Pillet (Université de Toulon)

Boundary states of the magnetic Robin Laplacian

• Rayan Fahs (Université Toulouse III - Paul Sabatier)

In this talk, we will discuss the spectral analysis of the Robin Laplacian on a smooth bounded two-dimensional domain in the presence of a constant magnetic field. In the semi-classical limit, I will explain how to get a uniform description of the spectrum located between the Landau levels. The corresponding eigenfunctions, called edge states, are exponentially localized near the boundary. By means of a microlocal dimensional reduction, I will explain how to derive a very precise Weyl law, and also how to simultaneously refine old results about the low-lying eigenvalues in the Robin case and recent ones about edge states in the Dirichlet case.

Small eigenvalues and metastability for semiclassical Boltzmann operators

• Thomas Normand (Université de Nantes)

We consider some inhomogeneous linear Boltzmann equations in a low temperature regime and in the presence of an external force deriving from a potential. We provide a sharp description of the spectrum near 0 of the associated operators. It enables us to obtain some precise information on the long time behavior of the solutions with in particular some quantitative results of return to equilibrium and metastability. This type of work usually requires two steps. First, we establish some resolvent estimates thanks to hypocoercive methods. We can then use and adapt some constructions of ‘gaussian quasimodes’ (approximated eigenfunctions) involving some tools from semiclassical microlocal analysis which will provide the desired sharp localization of the small eigenvalues.

Phase transition in the Integrated Density of States of the Anderson model arising from a supersymmetric sigma model

• Margherita Disertori (University of Bonn)

Localization/delocalization transition in random Schrödinger operators cannot in general be seen from the behavior of the corresponding Integrated Density of States (IDS). Here we consider a random Schrödinger operator appearing in the study of certain reinforced random processes in connection with a supersymmetric sigma-model, and show that the IDS undergoes a phase transition between weak and strong disorder regime in dimension larger or equal to three. This transition follows from a phase transition in the corresponding random process and supersymmetric sigma-model. This is joint work with V. Rapenne, C. Rojas-Molina and X. Zeng.