论文标题
在用于相互作用粒子系统的随机批处理方法的平均场上极限上
On the mean field limit of the Random Batch Method for interacting particle systems
论文作者
论文摘要
我们以前的工作中提出的随机批处理方法[Jin等，J。Comput。 Phys。，400（1），2020]不仅是一种用于相互作用粒子系统及其平均场极限的数值方法，而且也可以将其视为粒子系统模型，其中粒子在离散时间与随机选择的小批量粒子相互作用。在本文中，我们研究了该模型的平均场限制，因为粒子的数量$ n \ to \ infty $。与相互作用的粒子系统的经典平均场限制不同，大数字定律扮演角色并将混乱传播到以后的时间，平均场限制现在不依赖于大数字定律，并且在每个离散时间都会施加混乱。尽管如此，我们不仅将证明这种平均场限制（及时离散）是合理的，而且还将表明，作为离散时间间隔$τ\ 0 $，我们的限制是解决非线性fokker-planck方程解决方案的方法，作为在WasserStein距离中原始相互作用粒子系统的平均场景限制而产生的。
The Random Batch Method proposed in our previous work [Jin et al., J. Comput. Phys., 400(1), 2020] is not only a numerical method for interacting particle systems and its mean-field limit, but also can be viewed as a model of particle system in which particles interact, at discrete time, with randomly selected mini-batch of particles. In this paper we investigate the mean-field limit of this model as the number of particles $N \to \infty$. Unlike the classical mean field limit for interacting particle systems where the law of large numbers plays the role and the chaos is propagated to later times, the mean field limit now does not rely on the law of large numbers and chaos is imposed at every discrete time. Despite this, we will not only justify this mean-field limit (discrete in time) but will also show that the limit, as the discrete time interval $τ\to 0$, approaches to the solution of a nonlinear Fokker-Planck equation arising as the mean-field limit of the original interacting particle system in Wasserstein distance.
