群集运动方程的渐近定速约化动力学
Asymptotic Fixed-Speed Reduced Dynamics for Kinetic Equations in Swarming
摘要：
我们进行渐近分析的一般粒子系统中产生的集体行为的限制大的自推进力和摩擦力。这些渐近性在极限中强加了固定的速度，从而将动力学简化为速度变量中的球体。极限模型通过相对于快速动态平均而获得。我们可以在应用中包括所有典型的效果：短程排斥，长程吸引和对齐。例如，我们可以严格地证明，Cucker-Smale模型在这个渐近极限中被简化为没有噪声的Vicsek模型。最后，一个正式的扩展的基础上减少动力学允许我们处理扩散的情况下。这种技术与研究带电粒子的磁约束时所用的陀螺平均法密切相关。主要的新的数学困难是在这个扩展过程中处理测量解决方案。
英文摘要：
We perform an asymptotic analysis of general particle systems arising in collective behavior in the limit of large self-propulsion and friction forces. These asymptotics impose a fixed speed in the limit, and thus a reduction of the dynamics to a sphere in the velocity variables. The limit models are obtained by averaging with respect to the fast dynamics. We can include all typical effects in the applications: short-range repulsion, long-range attraction, and alignment. For instance, we can rigorously show that the Cucker-Smale model is reduced to the Vicsek model without noise in this asymptotic limit. Finally, a formal expansion based on the reduced dynamics allows us to treat the case of diffusion. This technique follows closely the gyroaverage method used when studying the magnetic confinement of charged particles. The main new mathematical difficulty is to deal with measure solutions in this expansion procedure.
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