论文标题
矢量结晶问题和集体行为
Vectorial crystallization problems and collective behavior
论文作者
论文摘要
我们提出和分析了一类矢量结晶问题，并应用了各向异性分子的结晶和集体行为，例如鸟类蜂拥而至和鱼类教育。 我们专注于“定向”粒子的二维系统：可允许的配置由vectorial经验度量表示，密度为$ \ Mathcal s^1 $。我们将这种配置赋予了图结构，其中键代表粒子之间的“方便”相互作用，而所提出的变分原理在于最大化其数量。 键类是由硬球型成对电势决定的，这取决于粒子之间的距离和连接两个粒子及其方向的段之间的距离，通过阈值标准。 不同的基态通过调节电势中的角度依赖性，模仿小鸭在连续地形成并预测角参数的某些特定值，即在鱼类学校中所谓的{\ it钻石形成}。
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.
