矢量结晶问题与集体行为
Vectorial crystallization problems and collective behavior
摘要：
我们提出并分析了一类矢量结晶问题，并将其应用于各向异性分子的结晶和鸟类群集和鱼类集群等集体行为。我们专注于二维系统的“定向”粒子：允许的配置表示的矢量经验措施与密度$\mathcal S^1$。我们赋予这样的配置与图形结构，其中的债券代表的“方便”粒子之间的相互作用，和建议的变分原理包括在最大限度地增加它们的数量。键的类别由硬球型成对势确定，这取决于颗粒之间的距离和连接两个颗粒的片段之间的角度及其取向，通过阈值标准。不同的基态出现调整的角度依赖的潜力，模仿鸭子游泳的行形成和预测，以及，对于一些特定的值的角度参数，所谓的{ \it钻石形成}在鱼集群。
英文摘要：
We propose and analyze a class of vectorial crystallization problems, with applications to crystallization of anisotropic molecules and collective behavior such as birds flocking and fish schooling. We focus on two-dimensional systems of "oriented" particles: Admissible configurations are represented by vectorial empirical measures with density in $\mathcal S^1$. We endow such configurations with a graph structure, where the bonds represent the "convenient" interactions between particles, and the proposed variational principle consists in maximizing their number. The class of bonds is determined by hard sphere type pairwise potentials, depending both on the distance between the particles and on the angles between the segment joining two particles and their orientations, through threshold criteria. Different ground states emerge by tuning the angular dependence in the potential, mimicking ducklings swimming in a row formation and predicting as well, for some specific values of the angular parameter, the so-called {\it diamond formation} in fish schooling.
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