Dry Active Matter on a Sphere

当成群的鸟向南迁徙时，想象一下越来越大的鸟群在一个不断缩小的球体上一起移动，直到一个巨大的鸟群覆盖整个星球。如果他们到达极点，他们的运动会发生什么？我们建议建立和分析一个被限制在球体表面上运动的活性粒子的模型。与平面中所有粒子可以相互排列并以相同的速度相互平行移动不同，球体上的运动要特殊得多。这是著名的毛球定理的直接结果，该定理指出：“你不可能在不产生卷曲的情况下梳理毛球平面。”翻译成主动运动的语言，整个鸟群不可能在球体表面以均匀的速度运动。因此，人们预计会观察到一组丰富而复杂的运动模式，这些模式从曲率和粒子排列之间复杂的相互作用中出现。我们的目标是识别、理解和分类这些运动模式。

When flocks of birds migrate southwards imagine larger and larger swarms moving together on a shrinking globe, until one gigantic flock covers the entire planet. What becomes of their motion if they reach the pole? We propose to build and analyse a model of active particles confined to move on the surface of a sphere. Unlike in the plane where all particles can align with each and travel with the same velocity parallel to each other, motion on the sphere is far more peculiar. This is a direct consequence of the famous Hairy ball theorem that states that "you can't comb a hairy ball flat without creating a cowlick". Translated into the language of active motion, it is not possible for the entire flock to travel with a uniform velocity on the surface of a sphere. As a result, one expects to observe a rich and complex set of motion patterns emerging from the intricate interplay between the curvature and particles' alignment. It is our aim to identify, understand and classify those motion patterns.