无截止Kac方程的掠碰撞和粒子近似的渐近性
Asymptotic of grazing collisions and particle approximation for the Kac equation without cutoff
摘要：
本文的研究对象是无截止的Kac方程。我们首先证明了在掠碰撞的渐近过程中，Kac方程可以近似为Fokker-Planck方程。收敛是一致的时间，我们给出了一个明确的收敛速度。接下来，我们用一个小的扩散项来代替小的碰撞，以近似Kac方程的解，并研究由此产生的误差。最后，我们建立了一个系统的随机粒子进行碰撞和扩散，我们可以很容易地模拟，它近似的Kac方程的解没有截止。我们给出了一些估计的收敛速度。
英文摘要：
The subject of this article is the Kac equation without cutoff. We first show that in the asymptotic of grazing collisions, the Kac equation can be approximated by a Fokker-Planck equation. The convergence is uniform in time and we give an explicit rate of convergence. Next, we replace the small collisions by a small diffusion term in order to approximate the solution of the Kac equation and study the resulting error. We finally build a system of stochastic particles undergoing collisions and diffusion, that we can easily simulate, which approximates the solution of the Kac equation without cutoff. We give some estimates on the rate of convergence.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
