Abstract: Maybe the first rigorous result on the propagation of chaos in kinetic models is that of Mark Kac, who first of all gave a strict definition of the concept, and then proved that it holds for a "caricature" of the Boltzmann equation.
I will discuss a generalization of his result and apply that to two models of schooling fish, which are also discussed in detail. One particular difference between these models and the one considered by Kac is that the stationary measures are not chaotic, but this does not prevent propagation of chaos for any finite time interval.
This is work in collaboration with Eric Carlen and Pierre Degond.
I will also briefly discuss a more abstract approach, to the propagation of chaos in random many particle systems, and new estimates on the rate of convergence. This approach is based on a representation of a the many particle system as an empirical measure, and goes back to Grünbaum's work on the Boltzmann equation.
This part of the talk is based on work in collaboration with Stéphane Mischler and Clément Mouhot. |