We devise and study a random particle blob method for approximating the Vlasov-Poisson-Fokkker-Planck (VPFP) equations by a $N$-particle system subject to the Brownian motion in $\mathbb{R}^3$ space. More precisely, we show that maximal distance between the exact microscopic and the mean-field trajectories is bounded by $N^{-\frac{1}{3}+\varepsilon}$ ($\frac{1}{63}\leq\varepsilon 0$, which improves the cut-off in [10]. Our result thus leads to a derivation of VPFP equations from the microscopic $N$-particle system. In particular we prove the convergence rate between the empirical measure associated to the particle system and the solution of the VPFP equations. The technical novelty of this paper is that our estimates crucially rely on the randomness coming from the initial data and from the Brownian motion.

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关于 Vlasov-Poisson-Fokker-Planck 系统的平均场极限

我们设计并研究了一种随机粒子 blob 方法，用于通过在 $\mathbb{R}^3$ 空间中受布朗运动影响的 $N$-粒子系统来逼近 Vlasov-Poisson-Fokkker-Planck (VPFP) 方程。更准确地说，我们表明精确微观和平均场轨迹之间的最大距离以 $N^{-\frac{1}{3}+\varepsilon}$ ($\frac{1}{63}\ leq\varepsilon 0$，它改进了 [10] 中的截止。因此，我们的结果导致从微观 $N$-粒子系统推导出 VPFP 方程。特别是我们证明了与粒子系统和 VPFP 方程的解。本文的技术新颖之处在于我们的估计主要依赖于来自初始数据和布朗运动的随机性。