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

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