Macroscopic dynamics and bifurcations of active particle systems

活生生的世界呈现了许多协调因素的大型集合的例子，例如昆虫群、鸟群或人群，以及在更微观的尺度上，蜂拥的细菌群体或集体迁移的细胞。这些试剂类似于组成惰性物质的粒子，但显着的不同之处在于它们会产生自己的运动。它们通常被称为活性粒子。像羊群和羊群一样，大多数活跃的粒子系统都表现出自组织的集体运动。人们对自组织产生的机制仍知之甚少。目前对这一问题的研究十分激烈。在这项工作中，我们将自组织的出现视为从系统的非协调状态到集体协调状态的分叉。分叉与物理学家所说的相变密切相关，即当系统的环境参数改变时，系统状态的突然变化。日常的例子是物质状态的变化，例如当水的温度超过沸腾温度时，水从液体变成蒸汽。在自然界中，动物群体可能会以类似的方式从随机运动状态(例如当它们在觅食时)转变为协调运动状态(当它们想要躲避捕食者的攻击时)。我们的目标是研究活性粒子系统的数学模型。我们的目标是在粒子数目较大时发展这些系统的宏观描述，并分析它们从无序运动到集体运动的分叉。事实上，当代理数量很大时，单独跟踪每个代理是不可能的，也不是有效的。宏观模型描述了统计平均值的演变，例如粒子的平均密度或速度，在计算上要高效得多。它们的严格推导涉及运动学理论的复杂数学工具，但它们提供了一种分析分叉的有效方法。就像物质一样，在活跃的粒子系统中有许多不同类型的分叉。在这项提案中，我们将重点介绍两个具体但重要的例子。第一种是当系统状态改变其基本对称性时的对称性破缺分叉。第二种是由于堵塞造成的分叉，当有限大小的颗粒达到密度时，它们都相互接触，例如在密集的人群中。为了测试我们发现的共性，我们还将调查其他类型的分叉，通过观察通过姿态协调相互作用的刚体系统，将集体精子细胞动力学作为一个应用。数学模型的性质根据它们适应的系统状态而有所不同。当几种状态同时存在时，它们被突变的跃迁界面分开。为了在数值上近似这种情况，将开发出在过渡界面上一致准确的数值方法。他们将允许我们通过将模型与选定的两个应用中的真实数据进行比较来验证模型，这两个应用是集体精子细胞动力学和行人动力学。在这两个例子中，我们将通过使用模型来预测旨在改变系统集体行为的各种行动战略的结果，从而展示模型的有用性。

The living world presents many examples of large assemblies of coordinated agents such as insect swarms, bird flocks or crowds and, at a more microscopic scale, swarming bacterial colonies or collectively migrating cells. These agents resemble particles composing inert matter but a striking difference is that they produce their own motion. They are generically referred to as active particles. Like herds and flocks, most active particle systems exhibit self-organized collective motion. The mechanisms by which self-organization emerges are still poorly understood. Current research on this question is intense. In this work, we view the emergence of self-organization as a bifurcation from a non-coordinated state of the system to a collectively coordinated one. Bifurcations are intimately related to what physicists call phase transitions, i. e. abrupt changes of the state of a system when its environmental parameters are changed. Everyday examples are changes of state of matter such as water changing from liquid to vapor when its temperature crosses the boiling temperature. In nature, animals groups may change from a random motion state (when they are foraging for food for instance) to a coordinated motion state (when they want to escape the attack of a predator) in a similar way. Our goal is to study mathematical models for active particle systems. We aim to develop macroscopic descriptions of these systems when the number of particles is large and to analyse their bifurcation from disordered to collective motion. Indeed, when the number of agents is large, it is neither possible nor efficient to follow each agent individually. Macroscopic models describe the evolution of statistical averages such as the mean density or velocity of the particles and are computationally much more efficient. Their rigorous derivation involves complex mathematical tools of kinetic theory but they give rise to an efficient way of analysing bifurcations. Like for matter, there are many different types of bifurcations in active particle systems. In this proposal, we will focus on two specific but important examples. The first one is symmetry-breaking bifurcations when a system state changes its underlying symmetry. The second one is bifurcation due to jamming, which occurs when finite sized particles reach the density where they are all in contact with each other as in dense crowds for instance. To test the general character of our findings, we will also investigate other kinds of bifurcations, by looking at systems of rigid bodies interacting through attitude coordination, having collective sperm-cell dynamics as an application in mind. The nature of mathematical models varies according to which state of the system they are adapted to. When several states are present simultaneously, they are separated by abrupt transition interfaces. To numerically approximate such situations, numerical methods that are uniformly accurate across the transition interface will be developed. They will allow us to validate the models by comparing them with real data in two selected applications, namely collective sperm-cell dynamics and pedestrian dynamics. In these two examples, we will showcase the usefulness of the models by using them to anticipate the outcome of various strategies of action aiming to change the collective behaviour of the system.