AI 中文总结
本文研究一维有限程归一化对齐系统中的碎片化现象，通过线性系统与格林核方法，提出精确的终端分离公式和递归预测算法，并给出碎片化判据与临界对齐强度。
AI 中文摘要
我们研究一维有限程归一化对齐系统中的碎片化现象，其中每个智能体将其速度松弛至固定相互作用半径内智能体的平均速度。由于通信图依赖于智能体的位置，随着智能体分离，边可能消失，从而导致多个渐近速度簇。我们首先考虑空间有序且速度同向有序的初始数据。在这种扩张机制下，我们证明速度有序性是前向不变的，所有成对分离距离是非递减的，且通信边只能被删除。因此，动力学经历有限次拓扑变化。在变化之间，速度动力学形成一个由随机游走拉普拉斯算子生成的线性系统。利用其群逆，我们推导出每个活动边的精确终端分离公式，并获得一个有限递归程序来确定终端通信图和每个簇的渐近速度。对于路径图，该理论变得显式：三对角格林核给出了有限时间碎片化的充分必要条件，以及区分碎片化与单群聚的临界对齐强度。我们还识别了一类非单调初始速度，其在第一次拓扑变化前进入扩张机制，并在随机初始速度下推导出终端分离的高斯统计。对于这类有序且安全进入的初始数据，我们的结果为早期数值研究中观察到的自发群体分裂提供了严格机制。任意初始构型的通用切换问题仍然开放。可复现的计算验证了路径阈值、首次碎片化时间预测以及有限事件递归，并与切换动力学的独立直接积分进行了对比。
英文摘要
Finite-range alignment can end either in a single flock or in several noninteracting clusters, yet convergence results rarely determine which outcome follows from a given finite-particle state. We study a one-dimensional normalized alignment model with a hard interaction cut-off and obtain exact predictions in an expansive regime. Velocity order is invariant, so pair separations are nondecreasing and the communication graph evolves through finitely many irreversible edge deletions. On each fixed graph, the group inverse of the random-walk Laplacian gives the total relative displacement remaining before relaxation. Comparing this displacement with the available interaction slack selects the next deletion and yields a finite recursion for the complete switching sequence, terminal partition, limiting cluster velocities, and internal geometry. For path configurations, an explicit Green kernel gives a necessary-and-sufficient fragmentation criterion and a sharp critical alignment rate, including asymptotic boundary contact at criticality. A spectral-geometric condition extends the theory to an open set of initially nonordered velocities, while a common self-weight extension covers both self-excluding and self-including local averages. Numerical computations reproduce the thresholds and multi-event cascades. The results provide an exact finite-size theory of fragmentation and terminal state selection for a class of finite-range interacting particle systems.
