Long time behaviour of interacting particles through a vibrating medium: comparison between the N-particle systems and the natural kinetic equation dynamics

We are interested in a kinetic equation intended to describe the interactions of particles with their environment. We focus on the long time behaviour. We prove that the time derivative of the spatial density goes to 0 and exhibit the omega limit set for the distribution function. We then apply this result to the empirical density associated to a finite number of particles and prove that the speeds of all of them go to 0. It also allows us to prove the convergence of all the positions with an increasing external potential and to get a precise description of their long time behaviour with a non decreasing external potential. Those results allow us to prove that in very large time, the distribution function is not a good approximation of the N-particle system. From all those considerations, we get a very detailed description of the dynamic of the N-particle system.