Abstract. This paper is devoted to establish quantitative and qualitative estimates related to the notion of chaos as firstly formulated by M. Kac [41] in his study of meanfield limit for systems of N undistinguishable particles as N → ∞.First, we quantitatively liken three usual measures of Kac's chaos, some involving the all N variables, other involving a finite fixed number of variables. The cornerstone of the proof is a new representation of the Monge-Kantorovich-Wasserstein (MKW) distance for symmetric N -particle probability measures in terms of the distance between the law of the associated empirical measures on the one hand, and a new estimate on some MKW distance on probability measures spaces endowed with a suitable Hilbert norm taking advantage of the associated good algebraic structure.Next, we define the notion of entropy chaos and Fisher information chaos in a similar way as defined by Carlen et al [17]. We show that Fisher information chaos is stronger than entropy chaos, which in turn is stronger than Kac's chaos. More importantly, with the help of the HWI inequality of Otto-Villani, we establish a quantitative estimate between these quantities, which in particular asserts that Kac's chaos plus Fisher information bound implies entropy chaos.We then extend the above quantitative and qualitative results about chaos in the framework of probability measures with support on the Kac's spheres, revisiting [17] and giving a possible answer to [17, Open problem 11]. Additionally to the above mentioned tool, we use and prove an optimal rate local CLT in L

These notes are devoted to a summary on the mean-field limit of large ensembles of interacting particles with applications in swarming models. We first make a summary of the kinetic models derived as continuum versions of second order models for swarming. We focus on the question of passing from the discrete to the continuum model in the Dobrushin framework. We show how to use related techniques from fluid mechanics equations applied to first order models for swarming, also called the aggregation equation. We give qualitative bounds on the approximation of initial data by particles to obtain the mean-field limit for radial singular (at the origin) potentials up to the Newtonian singularity. We also show the propagation of chaos for more restricted set of singular potentials.1 (Young-Pil Choi)

We consider a stochastic system of N particles, usually called vortices in that setting, approximating the 2D Navier-Stokes equation written in vorticity. Assuming that the initial distribution of the position and circulation of the vortices has finite (partial) entropy and a finite moment of positive order, we show that the empirical measure of the particle system converges in law to the unique (under suitable a priori estimates) solution of the 2D Navier-Stokes equation. We actually prove a slightly stronger result: the propagation of chaos of the stochastic paths towards the solution of the expected nonlinear stochastic differential equation. Moreover, the convergence holds in a strong sense, usually called entropic (there is no loss of entropy in the limit). The result holds without restriction (except positivity) on the viscosity parameter.The main difficulty is the presence of the singular Biot-Savart kernel in the equation. To overcome this problem, we use the dissipation of entropy which provides some (uniform in N ) bound on the Fisher information of the particle system, and then use that bound extensively together with classical and new properties of Fisher information.