起伏山丘上的角马群：任意曲面上的集群行为
Wildebeest Herds on Rolling Hills: Flocking on Arbitrary Curved Surfaces
AI总结：
本研究基于Toner-Tu集群理论，提出采用有限元方法求解任意曲面上的活性物质集群控制方程，验证了方法有效性并拓展了可研究的地形范围，为生物相关几何下的集体运动研究奠定基础。
摘要：
活性主体的集体行为——无论是角马群还是由分子马达驱动的微观肌动蛋白丝——都是生物物理与软物质物理领域一个令人兴奋的前沿方向。近三十年前，Toner和Tu提出了鸟群或兽群集体行动的连续介质理论，推动了现代活性物质领域的兴起。将连续介质活性物质理论应用于生命现象时面临的一个挑战是生物环境的复杂几何结构。宏观和微观的兽群都在非对称曲面上运动，比如起伏的草原，或是细胞、胚胎的表层，这会让问题在解析上难以求解。在本研究中，我们提出了Toner-Tu集群理论的一种新表述，采用有限元方法在任意曲面上求解控制方程。首先，我们在含散射障碍物的槽道流中，以及圆柱面、球面上测试了所建立的理论体系及其数值实现，并将结果与解析解进行对比。随后我们拓展到任意曲率的曲面，突破了此前可求解问题的限制，探索多种地形下的兽群行为。我们的方法能够研究解析方法无法揭示的瞬态过程和动力学解，还可以灵活引入新的几何结构与边界条件，并高效扫描参数空间。展望未来，本研究为Toner-Tu理论与生物相关几何结构下的集体运动数据搭建了对话基础，这些数据涵盖迁徙动物群的无人机 footage 到细胞内微观细胞骨架流动的成像记录。
英文摘要：
The collective behavior of active agents, whether herds of wildebeest or microscopic actin filaments propelled by molecular motors, is an exciting frontier in biological and soft matter physics. Almost three decades ago, Toner and Tu developed a continuum theory of the collective action of flocks, or herds, that helped launch the modern field of active matter. One challenge faced when applying continuum active matter theories to living phenomena is the complex geometric structure of biological environments. Both macroscopic and microscopic herds move on asymmetric curved surfaces, like undulating grass plains or the surface layers of cells or embryos, which can render problems analytically intractable. In this work, we present a formulation of the Toner-Tu flocking theory that uses the finite element method to solve the governing equations on arbitrary curved surfaces. First, we test the developed formalism and its numerical implementation in channel flow with scattering obstacles and on cylindrical and spherical surfaces, comparing our results to analytical solutions. We then progress to surfaces with arbitrary curvature, moving beyond previously accessible problems to explore herding behavior on a variety of landscapes. Our approach allows the investigation of transients and dynamic solutions not revealed by analytic methods. It also enables versatile incorporation of new geometries and boundary conditions and efficient sweeps of parameter space. Looking forward, the work presented here lays the groundwork for a dialogue between Toner-Tu theory and data on collective motion in biologically-relevant geometries, from drone footage of migrating animal herds to movies of microscopic cytoskeletal flows within cells.
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