复杂多体系统的动力学行为研究正在高速发展之中，与社会科学、工程控制以及生命科学等学科都密切相关。Kinetic Cucker-Smale（KCS）模型正是研究多体系统集体行为的非常重要和经典的模型之一。目前的研究主要集中在正则交互作用的情形，而在很多情形下，类似于库仑力等带奇性的交互作用也具有现实意义，能很好地与物理、生物等学科的需求相兼容。本项目针对随机环境下、带奇性交互作用的KCS模型所做的研究包括：1. 严格证明平均场极限，从而论证微观模型与介观模型的统一性；2. 证明KCS模型经典解在小初值条件下的存在性；3. 研究KCS模型小初值解的大时间行为，描述系统中集体行为的涌现。本课题是多体系统动力学行为研究的一个自然的深入和发展，对于多体系统中集体行为涌现的研究有重要的学术价值，也为样本数量特别巨大的系统的仿真模拟提供了有效的方法和理论基础。

Collective dynamics of multi-agent systems is ubiquitous in our nature. For instance, in biological system, the flocking of birds, the aggregation of bacteria, and swarming of fish are all collective behavior; while the herding of volatilities in financial markets and formation of dominant opinions in social systems are the collective behavior in our human society. Because of these closed connections with various scientific areas, the research on the collective dynamics nowadays becomes more and more popularized and a lot of phenomenological models were proposed for the collective motions of multi-agent systems. Among them, the Cucker-Smale (CS) model introduced by F. Cucker and S. Smale is a very seminal model and has been widely used in the engineering community due to the applications of unmanned aerial vehicles and sensor networks...In this project, we will focus on the mean field limit and large time behavior of Kinetic CS model with singular communication function in random environment. The singular setting is very natural, in fact, the Coulomb force is a singular force and has been widely applied in our science community. Moreover, the singular communication weight will automatically guarantee the avoidance of collision, which is very important in the application. Therefore, the study on the CS model with singular communication is very valuable and important. ..However, comparing to the kinetic CS model with regular communication, which has been extensively studied in the past ten years, only a few results have been done for the kinetic CS model with singular communication weight. Actually, only global existence of measure valued solution and local existence of classical solution were guaranteed so far. ..The main purposes of this project are three folds: 1. Apply the probabilistic approach to prove rigorously the mean field limit; 2. Prove the global well posed-ness of the kinetic CS model provided small initial data; 3. Prove the large time behavior of the classical solution provided small initial data and show the emergence of alignment of macroscopic velocity. ..This direction is a very natural and important development of the study on the kinetic CS model. The results provide a solid fundamental theory of kinetic CS model and also provide an efficient and valid method to simulate the dynamics in multi-agent system.