Orateur
Description
We consider a toy model of self-propelled particles (velocities constrained on the sphere) with alignment interaction (each individual tends to align with its close neighbors) and angular noise (to explore different directions).
At the mesoscopic scale (in the limit of a large number of particles), it takes the form of a kinetic equation coupling free transport (at constant velocity) and a local alignment-diffusion operator on the sphere (the velocity variable). The homogeneous version is well understood with a phase transition phenomenon: below a certain noise (or density) threshold, the velocity distribution converges towards isotropic equilibrium, while above this threshold, stable equilibria form a family of profiles that are more concentrated in velocity (von Mises distributions).
I will begin with an overview of the macroscopic behaviors that we expect to observe (or that we do not really understand) in the spatially inhomogeneous model. For example, we can formally derive fluid-type equations for densities above the critical threshold.
I will then present some results on the behavior (in short and long time) of the spatially inhomogeneous equation below the threshold, using hypoellipticity and hypocoercivity techniques, which require specific adjustments due to the fact that the velocity variable is constrained on the unit sphere. We obtain local (nonlinear) and quantitative stability of the isotropic equilibrium, with an exponential convergence rate when the space variable is on a flat torus. These results come from a long-standing collaboration with Émeric Bouin (Ceremade, Université Paris Dauphine - PSL).