标题： 关于Vlasov-Poisson系统的平均场极限 显示英文标题

标题： On the mean-field limit for the Vlasov-Poisson system

摘要： 我们提出了一个概率证明，证明了三维空间中一类具有库仑相互作用力的N粒子系统的平均场极限和混沌传播，该相互作用力的形式为$f^N(q)=\pm\frac{q}{|q|^3}$，并且在$|q|>N^{-\frac{5}{12}+\sigma}$处有依赖于$N$的截断，其中$\sigma>0$可以任意小选择。 这种截断尺寸远小于最近邻的距离。 特别是，对于典型的初始数据，我们展示了牛顿轨迹收敛到 Vlasov-Poisson 系统的特征线。 证明基于精确微观动力学与近似平均场动力学之间最大距离的Gronwall估计。 因此，我们的结果导致从微观$N$粒子动力学（力项可任意接近物理相关的库仑力）导出 Vlasov-Poisson 方程。 显示英文摘要

摘要： We present a probabilistic proof of the mean-field limit and propagation of chaos of a classical N-particle system in three dimensions with Coulomb interaction force of the form $f^N(q)=\pm\frac{q}{|q|^3}$ and $N$-dependent cut-off at $|q|>N^{-\frac{5}{12}+\sigma}$ where $\sigma>0$ can be chosen arbitrarily small. This cut-off size is much smaller than the typical distance to the nearest neighbour. In particular, for typical initial data, we show convergence of the Newtonian trajectories to the characteristics of the Vlasov-Poisson system. The proof is based on a Gronwall estimate for the maximal distance between the exact microscopic dynamics and the approximate mean-field dynamics. Thus our result leads to a derivation of the Vlasov-Poisson equation from the microscopic $N$-particle dynamics with force term arbitrary close to the physically relevant Coulomb force.