关于群集的数学：对齐动力学中的涌现行为
On the Mathematics of Swarming: Emergent Behavior in Alignment Dynamics
摘要：
我们概述了最近的发展，在对齐流体动力学的研究，驱动的一般类对称通信内核。感兴趣的一个主要问题是表征这种系统的紧急行为，我们量化的频谱间隙的加权拉普拉斯算子与对齐操作。我们对能量涨落的谱分析涵盖了长程和短程核，并且不需要热平衡（压力没有闭合）。特别是，在基于度量的短程内核的典型情况下，频谱间隙允许径向内核的离散傅里叶系数表示的下限，这使我们能够量化非真空解决方案的新兴群集行为。只要解保持光滑，这些大时间行为结果就适用。众所周知，全球光滑的解决方案存在于一个和两个空间维度，受到亚临界初始数据。我们解决了任意维数的问题，得到了保证多维整体光滑解存在的非平凡初始阈值条件。
英文摘要：
We overview recent developments in the study of alignment hydrodynamics, driven by a general class of symmetric communication kernels. A main question of interest is to characterize the emergent behavior of such systems, which we quantify in terms of the spectral gap of a weighted Laplacian associated with the alignment operator. Our spectral analysis of energy fluctuation covers both long-range and short-range kernels and does not require thermal equilibrium (no closure for the pressure). In particular, in the prototypical case of metric-based short-range kernels, the spectral gap admits a lower-bound expressed in terms of the discrete Fourier coefficients of the radial kernel, which enables us to quantify an emerging flocking behavior for non-vacuous solutions. These large-time behavior results apply as long as the solutions remain smooth. It is known that global smooth solutions exist in one and two spatial dimensions, subject to sub-critical initial data. We settle the question for arbitrary dimension, obtaining non-trivial initial threshold conditions which guarantee existence of multiD global smooth solutions.
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