Macroscopic dynamics and bifurcations of active particle systems

活性粒子系统的宏观动力学和分叉

• 批准号: EP/M006883/1
• 负责人: Pierre Degond
• 金额: $ 48.55万
• 依托单位: Imperial College London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2015
• 资助国家: 英国
• 起止时间: 2015 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FM006883%2F1
• 关键词: Macroscopic; dynamics; bifurcations; active; particle

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.

生命世界呈现出许多协调主体的大型集合的例子，例如昆虫群、鸟群或人群，以及在更微观的尺度上，集群的细菌菌落或集体迁移的细胞。这些物质类似于组成惰性物质的粒子，但显着的区别是它们产生自己的运动。它们通常被称为活性粒子。像牛群和羊群一样，大多数活跃的粒子系统都表现出自组织的集体运动。自组织出现的机制仍然知之甚少。目前对这个问题的研究非常深入。在这项工作中，我们将自组织的出现视为系统从非协调状态到集体协调状态的分叉。分岔与物理学家所说的相变密切相关，即相变。 e.当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，水从液态变为气态。在自然界中，动物群体可能以类似的方式从随机运动状态（例如，当它们觅食时）转变为协调运动状态（当它们想要逃避捕食者的攻击时）。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量很大时对这些系统进行宏观描述，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，单独跟踪每个代理既不可能也不有效。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，并且计算效率更高。它们的严格推导涉及动力学理论的复杂数学工具，但它们产生了分析分岔的有效方法。与物质一样，活性粒子系统中也有许多不同类型的分叉。在本提案中，我们将重点关注两个具体但重要的示例。第一个是当系统状态改变其底层对称性时对称破缺分叉。第二个是由于干扰而产生的分叉，当有限尺寸的粒子达到彼此接触的密度（例如在密集的人群中）时，就会发生这种情况。为了测试我们研究结果的一般特征，我们还将通过观察通过态度协调相互作用的刚体系统来研究其他类型的分叉，并将集体精子细胞动力学作为应用。数学模型的性质根据它们所适应的系统状态而变化。当多个状态同时存在时，它们被突变的过渡界面分开。为了在数值上近似这种情况，将开发在过渡界面上一致精确的数值方法。它们将使我们能够通过将模型与两个选定应用（即集体精子细胞动力学和行人动力学）中的真实数据进行比较来验证模型。在这两个示例中，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策略的结果来展示模型的有用性。

项目成果

Phase Transitions in a kinetic flocking model of Cucker-Smale type

Cucker-Smale 型动力学植绒模型中的相变

• DOI: 10.48550/arxiv.1510.04009
• 发表时间: 2015
• 作者: Barbaro A

A new model for the emergence of blood capillary networks

毛细血管网络出现的新模型

• DOI: 10.3934/nhm.2021001
• 发表时间: 2021
• 期刊: Networks & Heterogeneous Media
• 影响因子: 1
• 作者: Aceves-Sanchez P

Large-Scale Dynamics of Self-propelled Particles Moving Through Obstacles: Model Derivation and Pattern Formation.

• DOI: 10.1007/s11538-020-00805-z
• 发表时间: 2020-09-25
• 期刊: Bulletin of mathematical biology
• 影响因子: 3.5
• 作者: Aceves-Sanchez P;Degond P;Keaveny EE;Manhart A;Merino-Aceituno S;Peurichard D
• 通讯作者: Peurichard D

Hydrodynamic limits for kinetic flocking models of Cucker-Smale type

Cucker-Smale 型动力学植绒模型的水动力极限

• DOI: 10.48550/arxiv.1901.11132
• 发表时间: 2019
• 作者: Aceves-Sánchez P

Pedestrian Models based on Rational Behaviour

基于理性行为的行人模型

• DOI: 10.48550/arxiv.1808.07426
• 发表时间: 2018
• 作者: Bailo R