Exposés scientifiques des Mean-Field Days
Giacomo Albi (Munich)
Mean-field optimal control is an useful tool to describe an optimal action which can be exerted on a large system of interacting agents. In this talk, I will briefly review some theoretical results about mean-field optimal control problems, and second I will focus on their numerical solution. In particular, I will introduce a novel approximating hierarchy based on a Boltzmann approach, whose solution requires a moderate computational effort compared with standard direct approaches. I will show that this hierarchy of controls well approximate the solution of the mean-field optimal control problem. Different examples will show the effectiveness of the proposed strategies.
Kleber Carrapatoso (Montpellier)
Kac introduced in the 50's a many-particle stochastic system from which we expect to rigorously derive the homogeneous Boltzmann equation in the mean-field limit. In order to do so, he introduced the notion of chaos (and propagation of chaos) for the joint probability of particles. I will present results on chaotic and entropic chaotic probabilities on the phase space of Kac's model.
Michele Coghi (Bielefeld)
A system of interacting particles described by stochastic differential equations is considered. As opposed to the usual model, where the noise perturbations acting on different particles are independent, here the particles are subject to the same space-dependent noise, similar to the (noninteracting) particles of the theory of diffusion of passive scalars. We prove a result of propagation of chaos and show that the limit PDE is stochastic and of inviscid
type, as opposed to the case when independent noises drive the different particles. Moreover, we use this result to derive a mean field approximation of the stochastic Euler equations for the vorticity of an incompressible fluid.
Mitia Duerinckx (Bruxelles)
We investigate the mean-field limit for the gradient flow evolution of particle systems with pairwise singular interactions, as the number of particles tends to infinity. Based on a modulated energy method, we prove a new weak-strong stability principle for the limiting equation. This yields a quantitative proof of the mean-field result in dimensions 1 and 2 for a family of interactions for which this problem was still open.
Thomas Holding (Warwick)
We develop a new technique for establishing quantitative propagation of chaos for systems of interacting particles. Using this technique we prove propagation of chaos for diffusing particles whose interaction kernel is merely Holder continuous, even at long ranges. Moreover, we do not require specially prepared initial data. On the way, we establish a law of large numbers for SDEs that holds over a class of vector fields simultaneously. The proofs bring together ideas from empirical process theory and stochastic flows.
Nicolas Rougerie (Grenoble)
We study the ground state of N quantum particles trapped in a symmetric double-well potential, letting the distance between the two wells increase to infinity with the number
of particles. In this context, one should expect an interaction-driven transition between a delocalized state (particles are independent and all live in both wells) and a localized state (particles are correlated, half of them live in each well).
We start from the full many-body Schrödinger Hamiltonian and study a mean-field situation where interaction and kinetic energies are comparable. When tunneling is negligible against interaction energy, we prove a localization estimate showing that the particle number fluctuations in each well are strongly suppressed. The modes in which the particles condense are minimizers of nonlinear Schrödinger-type functionals.
Joint work with Dominique Spehner, arXiv:1612.05758