发表机构
Plum Science Co., Ltd.; Hokkaido University; Graduate School of Engineering, The University of Osaka（Plum科学有限公司; 北海道大学; 大阪大学工学研究科）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究以足球追踪数据为基础，探究球员围绕球的集体运动是否可用类引力有效场描述，得出了球员径向漂移的经验定律及漂移场模式。
AI 中文摘要
在足球运动中，球员的集体运动围绕球展开。我们探究这种多主体响应是否可由类似引力的有效场描述，同时强调所测量的量是球员的径向漂移速度，如同过阻尼系统中的情况，而非牛顿加速度。利用三场匿名比赛的公开追踪数据，我们对相同球员径向漂移值进行两种平均：距离相关的径向漂移速度D(r)，通过在固定的球员-球距离处取平均得到；以及球位置漂移速度场K(x,y)，通过对球位于给定球场单元内的观测结果取相同向内速度的平均得到。排除开球、其他受限比赛阶段、离球最近的球员以及间距r<2m的情况后，仍存在持续的向内漂移，其全局平均值为0.70±0.05 m/s。距离依赖性无法用r的反幂很好描述：从球向外移动时，D(r)在r≈12m处降至一个浅最小值，上升至正的中程平台，随后在最大间距处降至接近零。完整曲线由一个简洁的经验漂移定律总结。漂移场K(x,y)在球场大部分区域大致为正，但其三场平均值存在一个浅的中央凹陷，比非球门区平均值低约25%，且在中央球门附近抑制作用更强，此处平均漂移可变为略负。纳入离球最近的球员或排除守门员后，径向定律和漂移场模式基本保持不变。因此，足球追踪数据可用于提取主动多体运动的简单有效定律。
英文摘要
In association football, collective player motion is organized around the ball. We ask whether this many-agent motion can be described by an effective field analogous to gravitational attraction, emphasizing that the measured quantity is a radial drift velocity, as in overdamped dynamics, rather than a Newtonian acceleration. Using public tracking data from ten matches, we characterize this radial drift velocity through the distance-dependent mean D(r) and the ball-position-dependent field K(x,y). After excluding restarts and other constrained phases of play, both goalkeepers, the player nearest to the ball, and small player-ball separations, a persistent inward drift velocity remains, with a global mean of about 0.9 m/s. The distance dependence is not described by an inverse power of r: D(r) decreases at short range and then forms a broad plateau that persists to the largest measured separations. The mean K(x,y) is positive over most of the pitch, with a central depression about 15 percent below the non-goalmouth average and stronger suppression near the goalmouths, where the mean drift velocity becomes slightly negative, corresponding to weak effective repulsion. The role-resolved D(r) shows a pronounced defender minimum near 12 m and a pronounced attacker maximum near 60 m, whereas only defenders produce effective repulsion near the goals in K(x,y). These results show that football tracking data can reveal simple effective laws of active many-body motion while retaining clear signatures of player role and pitch geometry.
Comments16 pages, 15 figures, 6 tables
