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 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.

In order to understand abnormally diffusive phenomena which are physically observed in nanotube technologies, one mathematical approach consists in starting from deterministic system of Newtonian particles, and then perturb this system with a stochastic component which provides enough ergodicity to the dynamics. It is already well known that these stochastic chains model correctly the behavior of the conductivity . In , (published in Communications in Mathematical Physics) M. Simon with her coauthors C. Bernardin, P. Gonçalves, M. Jara, T. Komorowski, S. Olla and M. Sasada have observed both behaviors, normal and anomalous diffusion, in the context of low dimensional asymmetric systems. They manage to describe the microscopic phenomena at play which are responsible for each one of these phenomena, and they go beyond the predictions that have recently been done in , . Moreover, in , the authors manage to treat rigorously, for the first time, the case of an anharmonic potential: more precisely, they consider a small quartic anharmonicity and show that the result obtained in the harmonic (linear) case persists up to some small critical value of the nonlinear perturbation.

A large variety of models has been introduced to describe the evolution of a multiphase medium, e.g. the joint evolution of liquid and solid phases. These complex physical phenomena often feature absorbing phase transitions. For instance, the porous medium equation (PME)

where $m>1$ is a constant and $\text{div}$ and $\nabla$ are the divergence and gradient operators in ${ℝ}^d$, describes the evolution of the density $\rho :{ℝ}^d\times {ℝ}_{+}\to [0,1]$ of an ideal gas flowing in a homogeneous medium. It is known that, starting from an initial density ${\rho }_0$ with compact support, the solution $\rho (x,t)$ is nonnegative and has compact support in the space variable for each positive $t$. Thus there are interfaces separating the regions where $\rho$ is positive from those where it is zero.

In one submitted paper in collaboration with O. Blondel, C. Cancès, and M. Sasada, we have derived the PME () from a degenerate and conservative dynamics in , for any integer $m>1$. More precisely we improved the results previously obtained in , since we allow the solutions to feature moving interfaces, namely the initial condition may vanish. This moving boundary was not well apprehended at the microscopic level. Its rigorous definition is indeed very delicate, and its behavior (such that its speed, or fluctuation), as well as the relationship between the microscopic and macroscopic boundaries, are challenging questions that we aim to tackle in a near future.

When $m<1$, equation () is called fast diffusion equation. In a recent collaborative work (submitted) with O. Blondel, C. Erignoux and M. Sasada , we derive such a fast diffusion equation in dimension one from an interacting particle system belonging to the class of conserved lattice gases with active-absorbing phase transition . The microscopic dynamics is very constrained: in a few words, a particle can jump to the right (resp. left) empty neighboring site if and only if it has a particle to its left (resp. right) neighboring site. This model is really complex: the state space is divided into transient states, absorbing states and ergodic states. Depending on the initial number of particles, the transient good configurations will lead to the ergodic component and the transient bad configurations will be absorbed to an inactive state. Because of the jump constraint, there are two distinct regimes for the macroscopic behavior. Either the macroscopic density is larger than $\frac{1}{2}$, in which case the system behaves diffusively, or the density is lower than $\frac{1}{2}$, in which case the system freezes rapidly.

The interfaces between these two phases propagate as particles from the supercritical phase ($\rho >\frac{1}{2}$) diffuse towards the subcritical phase ($\rho <\frac{1}{2}$). We expect that the macroscopic density profile evolves under the diffusive scaling according to the Stefan problem

The microscopic derivation of such Stefan problems is a well known difficult problem, only partially solved , . In we treat the liquid part of the problem (i.e. when the initial profiles ${\rho }_0$ are uniformly larger than the critical density $\frac{1}{2}$) and we provide a refined estimation of the time needed by the system to enter into the ergodic state. Then, we show that the macroscopic density profile evolves under the diffusive time scaling according to () with $m=-1$. The extension to more general initial profiles is our next goal.

S. De Bièvre and his co-authors introduced and studied a kinetic equation of the Vlasov-Wave type, which arises in the description of the behavior of a large number of particles interacting weakly with an environment, composed of an infinite collection of local vibrational degrees of freedom, modeled by wave equations. They use variational techniques to establish the existence of large families of stationary states for this system, and analyze their stability .

With S. Rota Nodari, S. De Bièvre considered the orbital stability of relative equilibria of Hamiltonian dynamical systems on Banach spaces, in the presence of a multi-dimensional invariance group for the dynamics . They proved a persistence result for such relative equilibria, presented a generalization of the Vakhitov-Kolokolov slope condition to this higher dimensional setting, and showed how it allows to prove the local coercivity of the Lyapunov function, which in turn implies orbital stability. The method was applied to study the orbital stability of relative equilibria of nonlinear Schrödinger and Manakov equations. It extends and clarifies the approach of Grillakis-Shatah-Strauss.

S. De Bièvre and his collaborators introduced a new distance-based measure for the nonclassicality of the states of a bosonic field, which outperforms the existing such measures in several ways . They defined for that purpose the operator ordering sensitivity of the state which evaluates the sensitivity to operator ordering of the Renyi entropy of its quasi-probabilities and which measures the oscillations in its Wigner function. Through a sharp control on the operator ordering sensitivity of classical states they obtained a precise geometric image of their location in the density matrix space allowing them to introduce a distance-based measure of nonclassicality. They analyze the link between this nonclassicality measure and a recently introduced quantum macroscopicity measure, showing how the two notions are distinct.

A. de Laire and S. Gutierrez established in a global well-posedness result for the Landau–Lifshitz equation with Gilbert damping, provided that the BMO semi-norm of the initial data is small. As a consequence, they deduced the existence of self-similar solutions in any dimension. Moreover, in the one-dimensional case, they characterized the self-similar solutions when the initial data is given by some (${\S }^2$-valued) step function and established their stability. They also showed the existence of multiple solutions if the damping is strong enough.

It is well-known that the dynamics of biaxial ferromagnets with a strong easy-plane anisotropy is essentially governed by the Sine-Gordon 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 Sine-Gordon equation in the regime of a strong easy-plane anisotropy. This result holds for solutions to the Landau–Lifshitz equation in high order Sobolev spaces. They also provided an alternative proof for local well-posedness in this setting by introducing high order energy quantities with better symmetrization properties. Then they derived the convergence from the consistency of the Landau–Lifshitz equation with the Sine-Gordon equation by using well-tailored energy estimates. As a by-product, they also obtained a further derivation of the free wave regime of the Landau–Lifshitz equation.

In telecommunication models the quality of the transferred data is assessed through the entropy of the channel, a theoretical quantity that is usually not computable in practice. W. Hachem, A. Hardy and S. Shamai prove in that one can relate this quantity for a large class of models involving several antennas (MIMO) to the equilibrium measure of a matrix valued Markov chain associated with the model, and so does its asymptotic behavior when the signal-noise-ratio parameter becomes large. By means of ergodicity results, this yields estimates for these quantities that are implementable faster than the naive estimators.

The Sine-beta process is a universal object appearing in the study of large Hermitian random matrices and statistical systems in a logarithmic interaction, such as low dimensional Coulomb gases. However, the only description available yet relied on a rather complicated and non-physical system of coupled stochastic differential equations. In , D. Dereudre, A. Hardy, T. Leblé and M. Maïda obtain a statistical physic interpretation of the Sine-beta process as probability measure on infinite configurations of points described by means of the DLR formalism. This allows to obtain more information on the Sine-beta process: for instance, it is rigid, it is tolerant, and the number of particles in a compact box has gaussian fluctuations as the box becomes large.

In signal processing, an important challenge is to be able to separate signals from ambient noises. In time-frequency analysis, this problem reduces to identify what is the spectrogram of a white noise to derive statistical tests in order to decide if some partial signal is noise or not. P. Fandrin recently put forward that the understanding of the zeros of the spectrograms would be already an important step by analyticity of the spectrograms. R. Bardenet and A. Hardy observed in that there is a canonical way to identify the zeros of the usual white noise transforms associated to classical spectrograms and zeroes of Gaussian analytic functions associated with classical orthogonal polynomials in the background. In particular the zeros satisfy some invariance properties leading to computable correlation functions. In specific cases, one can identify some transforms whose zeros form a determinantal point process, in which case all the statistics of interests can be computed explicitly and this allows an exact numerical treatment.

In relation to the 7th Smale problem, which is about finding polynomial time algorithm to produce well spread configuration of points on the sphere in a quantified manner, C. Beltran and A. Hardy proved in that the Coulomb gas on the sphere at a temperature proportional to the inverse number of points in a configuration reaches the numerical precision required by this problem. We however did not discuss yet the algorithmic procedure, which is currently in investigation by A. Hardy and M. Simon.

Determinantal point processes can be of important use in applications as soon as one is interested in producing configurations of well spread points on an arbitrary space. A class of determinantal point processes on the real line that has been extensively studied recently are the so-called polynomial ensembles. A. Hardy gathered in several results concerning these models in relation to the recurrent coefficients associated with the orthogonal polynomials hidden in the background.

The convergence of the Coulomb gas, which is a statistical gas of charged particles in an electrostatic interaction, towards its limiting distribution as the number of particles goes to infinity is a result which is part of the folklore of potential theory. The speed at which this convergence arise, which can be assessed through concentration of measure estimates in, say, the Wasserstein-Kantorovich metric, are however new results obtained by D. Chafaï, A. Hardy and M. Maïda in . One of the main ingredient was to develop transport inequalities associated with the Coulomb interaction.