Collective behavior in complex systems

集体行为是一种普遍存在的自然现象，在动物世界和许多无生命的物理系统中都存在。在生物学中，从细菌群落、昆虫群、鱼群和鸟群到复杂的人类种群互动，动物界的各个层面都可以观察到群体形成。集体行为也出现在许多物理系统中，如颗粒气体、纳米粒子和DNA巴基球分子的自组装，以及玻色-爱因斯坦凝聚体中的晶格。集体行为的出现通常是个体（或原子）遵循非常简单的规则的结果，没有任何外部协调。近年来，人们提出了许多群体出现和行为的模型。该计划的目的是研究紧急行为的基础，开发新的数学和计算机工具，并与其他科学家合作，将这些想法应用于化学，物理和生物问题。在生物学中，集体行为对个体或群体整体有一定的好处，比如鸟群可以躲避捕食者，蚂蚁可以收集食物，企鹅可以挤在一起取暖。这就提出了许多数学、生物学甚至哲学问题，例如：导致聚合的物理/生物机制是什么？蜂群的最佳形状（或大小）是什么？在多大程度上群居是一种进化适应？物理学中集体行为的一个例子是玻色-爱因斯坦凝聚体（BEC）。这是一种物质状态，由玻色和爱因斯坦在20世纪20年代提出理论，它存在于极低的温度下（接近绝对零度），由原子浓度组成。直到20世纪90年代，BEC的存在才在麻省理工学院和JILA（联合实验室天体物理研究所）被实验观察到，这两个小组因此获得了诺贝尔物理学奖（2001年）。在本实验中，我们观察到BEC自组织成刚性旋转的“漩涡晶体”，这是一种非常规则的漩涡粒子的六边形晶格构型，每个漩涡代表一个原子的浓度。这项提议的目标之一是应用最近在生物群体研究中发展起来的新方法来深入了解这些涡旋晶体的结构：它们的形状、稳定性和所谓的特卡琴科振动模式。最近，bec被发现有许多潜在的应用，如拉姆齐光谱。诱人的是，bec有望在计算领域取得突破；实验证明，bec可以潜在地用于分解大数，并用量子比特代替量子计算机执行逻辑运算。群体和集体行为的研究是一个非常丰富的主题，在几个数学，生物和物理分支学科的交叉点，目前正在经历一个爆炸性的发展。任何这些问题的进展都会对多个学科产生影响，形成持久的合作和跨学科的交叉授粉是及时而重要的。各种各样的问题使这个领域成为培养各级（从本科生到博士后）和不同背景的HQP的理想问题来源。

Collective behaviour is an ubiquitous natural phenomenon that pervades both the animal world and many inanimate physical systems. In biology, group formation is observed across all levels of the animal kingdom, from bacterial colonies, insect swarms, fish schools and flocks of birds, to complex human population interactions. Collective behaviour also arises in a multitude of physical systems such as granular gases, self-assembly of nanoparticles and DNA buckyball molecules, and crystal lattices in Bose-Einstein Condensates. The emergence of collective group behaviour is often a consequence of individuals (or atoms) following very simple rules, without any external coordination. In recent years, many models of group emergence and behaviour have been proposed. The purpose of this program is to study the fundamentals of emergent behaviour, develop new mathematical and computer tools, and to collaborate with other scientists to apply these ideas to problems in chemistry, physics and biology. In biology, collective behaviour confers certain benefits on individuals or group as a whole, such as predator avoidance for bird flocks, food gathering for ants and heat preservation in penguin huddles. This poses numerous mathematical, biological and even philosophical questions such as: What are the physical/biological mechanisms that cause aggregation? What is the optimal shape (or size) of the swarm? To what extent is swarming an evolutionary adaptation? An example of collective behaviour in physics are Bose-Einstein Condensates (BEC). This is a state of matter, theorized in 1920's by Bose and Einstein, that exists at extremely low temperatures (close to absolute zero), and which consists of concentrations of atoms. It was not until 1990's that BEC's existence was experimentally observed at MIT and JILA (Joint Institute for Laboratory Astrophysics), a work for which these groups were awarded the Nobel prize in Physics (2001). In this experiments, BEC were observed to self-organize into rigidly rotating "vortex crystals", which are very regular hexagonal lattice configurations of vortex-particles, with each vortex representing a concentration of atoms. One of the goals of this proposal is to apply new methods recently developed in studies of biological swarms to gain insight into the structure of these vortex crystals: their shape, stability, and the so-called Tkatchenko vibrational modes. Recently, BECs have been found to have numerous potential applications such as Ramsey spectroscopy. Tantalizingly, BECs hold promise for potential breakthroughs in computing; it was demonstrated experimentally that BECs can potentially be used to factor large numbers and to perform logic operations with qubits in lieu of a quantum computer. The study of swarms and collective behaviour is a very rich topic at the intersection of several mathematical, biological and physical sub-disciplines, and is currently undergoing an explosive development. A progress in any of these problems will have repercussions across multiple disciplines and the formation of lasting collaborations and the cross-pollination across disciplines is expected to be both timely and important. The variety of problems makes this area an ideal source of problems for training HQP at all levels (from undergrad to postdoc) and with different backgrounds.