相互作用粒子通过振动介质的长时间行为：N粒子系统与自然动力学方程动力学的比较
Long time behaviour of interacting particles through a vibrating medium: comparison between the N-particle systems and the natural kinetic equation dynamics
摘要：
我们感兴趣的是一个动力学方程，用来描述粒子与环境的相互作用。我们专注于长期行为。我们证明了空间密度的时间导数为0，并表现出分布函数的欧米茄极限集。然后，我们将这个结果应用于与有限数量的粒子相关联的经验密度，并证明所有粒子的速度都为0。它也使我们能够证明所有的位置的收敛性与增加的外部潜力，并得到一个精确的描述，他们的长期行为与非减少的外部潜力。这些结果使我们能够证明，在非常大的时间，分布函数是不是一个很好的近似的N粒子系统。从所有这些考虑中，我们得到了N粒子系统动力学的非常详细的描述。
英文摘要：
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.
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