论文标题
具有平均场相互作用的粒子系统：固定分布的大规模极限
A particle system with mean-field interaction: Large-scale limit of stationary distributions
论文作者
论文摘要
我们考虑一个由$ n $颗粒组成的系统，在实际线路上跳跃。系统状态是粒子位置的经验分布。每个粒子在某些时间点``向前跳跃''，其瞬时跳跃速率是由于粒子位置位置量化函数在当前状态内（经验分布）的下降函数所给出的。在某些条件下，对该模型的以前的工作将系统随机动态的收敛性（作为$ n \ infty $）与确定性平均模型（MFM）的收敛性，这是对整数分化方程的解决方案。先前工作的另一条线确定了流动波的MFM的存在，以及MFM轨迹对行进波的吸引力。本文的主要结果是：（a）我们证明，作为$ n \至\ infty $，（重新定义）状态的固定分布集中在（重新中心的）行驶波上； （b）我们在$ n $时刻的固定分配中获得（重新定义）状态的统一； （c）我们证明了收敛到MFM的结果，这比以前的工作要大得多。结果（b）和（c）用作（a）证明的``成分''，但也具有独立的利益。
We consider a system consisting of $n$ particles, moving forward in jumps on the real line. System state is the empirical distribution of particle locations. Each particle ``jumps forward'' at some time points, with the instantaneous rate of jumps given by a decreasing function of the particle's location quantile within the current state (empirical distribution). Previous work on this model established, under certain conditions, the convergence, as $n\to\infty$, of the system random dynamics to that of a deterministic mean-field model (MFM), which is a solution to an integro-differential equation. Another line of previous work established the existence of MFMs that are traveling waves, as well as the attraction of MFM trajectories to traveling waves. The main results of this paper are: (a) We prove that, as $n\to\infty$, the stationary distributions of (re-centered) states concentrate on a (re-centered) traveling wave; (b) We obtain a uniform across $n$ moment bound on the stationary distributions of (re-centered) states; (c) We prove a convergence-to-MFM result, which is substantially more general than that in previous work. Results (b) and (c) serve as ``ingredients'' of the proof of (a), but also are of independent interest.
