Discussion of history, methdods and open problems in mean field limits.

• Event: Workshop on "Mean-field dynamics of many particle systems" (2016)

The rigorous derivation of the Vlasov equation from Newtonian mechanics of N Coulomb-interacting particles is still an open problem. In the talk I will present recent results, where an N-dependent cutoff is used to make the derivation possible. The cutoff is removed as the particle number goes to infinity. Our result holds for typical initial conditions, only. This is, however, not a technical assumption: one can in fact prove deviation from the Vlasov equation for special initial conditions for the system we consider.

• Event: Workshop on "Mean-field dynamics of many particle systems" (2016)

We will consider the many-body evolution of initially confined fermions in a joint mean-field and semiclassical scaling, focusing on the case of Coulomb interaction. We will show that, for initial states close to Slater determinants and under some conditions on the solution of the time-dependent Hartree-Fock equation, the many-body evolution converges towards the Hartree-Fock dynamics. This is a joint work with M. Porta, S. Rademacher and B. Schlein.

• Event: Workshop on "Mean-field dynamics of many particle systems" (2016)

Starting from the many-body Schroedinger equation for bosons, I will discuss the rigorous derivation of the Hartree equation for the condensate and the Bogoliubov equation for the excited particles. The effective equations allows us to construct an approximation for the many-body wave function in norm. This talk is based on joint works with Phan Thanh Nam.

• Event: Workshop on "Mean-field dynamics of many particle systems" (2016)

We consider a large systems of first order coupled equations. The system model the interaction ofdiffusive particles through a very rough force field, which can be the derivative of a bounded stream function. Through a new, modified law of large numbers, we are able to give quantitative estimates between any statistical marginal of the discrete solution and the mean field limit. We are also able to extend the method to cover the case of the 2d incompressible Navier-Stokes system in the vorticity formulation.

• Event: Workshop on "Mean-field dynamics of many particle systems" (2016)

We provide a rigorous derivation of the linear Boltzmann equation without cut-off starting from a system of particles interacting via a potential with infinite range as the number of particles N goes to infinity under the Boltzmann-Grad scaling. The main difficulty in this context is that, due to the infinite range of the potential, a non-integrable singularity appears in the angular collision kernel, making no longer valid the single-use of Lanford's strategy. On this talk, I will present how a combination of Lanford's strategy, of tools developed recently by Bodineau, Gallagher and Saint-Raymond to study the collision process and of new duality arguments to study the additional terms associated with the infinite range interaction (leading to some explicit weak estimates) overcomes this difficulty.

• Event: Workshop on "Mean-field dynamics of many particle systems" (2016)

Quantization of probability densities on the Euclidean space refers to the approximation of a probability measure that is absolutely continuous with respect to the Lebesgue measure by convex combination of Dirac measures. The quality of the approximation is measured in terms of a distance metrizing the weak convergence of probability measures, typically a Monge-Kantorovich (or Vasershtein) distance. The talk with describe a gradient flow approach to the quantization problem in the limit as the number of points goes to infinity. (Work in collaboration with E. Caglioti and M. Iacobelli).

• Event: Workshop on "Mean-field dynamics of many particle systems" (2016)