论文标题
统计动力学中的平均场限制
Mean-Field Limits in Statistical Dynamics
论文作者
论文摘要
这些讲座的注释旨在将读者引入统计动力学中平均场限制的一些最新数学工具和结果。作为热身，第1节审查了经典力学中平均场限制的方法，遵循W.Braun，K。Hepp和R.L. Dobrushin的思想，基于相位空间经验措施，Klimontovich Solutions和Monge-Kantorovich-Wassine的概念的概念。第2节讨论了量子动力学中克里马托维奇解决方案概念的类似物，并解释了这种概念如何在Pickl的方法中出现，以处理与库仑型奇异性在起源的相互作用势的情况。最后，讲座3解释了如何在量子$ n $ - 粒子动力学上共同采用平均场和经典限制，从而导致弗拉索夫方程。这些讲座基于与C. Mouhot和T. Paul的一系列联合作品。
These lectures notes are aimed at introducing the reader to some recent mathematical tools and results for the mean-field limit in statistical dynamics. As a warm-up, lecture 1 reviews the approach to the mean-field limit in classical mechanics following the ideas of W. Braun, K. Hepp and R.L. Dobrushin, based on the notions of phase space empirical measures, Klimontovich solutions and Monge-Kantorovich-Wasserstein distances between probability measures. Lecture 2 discusses an analogue of the notion of Klimontovich solution in quantum dynamics, and explains how this notion appears in Pickl's method to handle the case of interaction potentials with a Coulomb type singularity at the origin. Finally, lecture 3 explains how the mean-field and the classical limits can be taken jointly on quantum $N$-particle dynamics, leading to the Vlasov equation. These lectures are based on a series of joint works with C. Mouhot and T. Paul.
