Collaborative Research: Hierarchical kinetic models for chemically and hydrodynamically coupled organisms

本项目发展了自推进粒子动力学理论的数学基础。用运动理论的方法来描述所有粒子的位置和速度的概率密度。数学分析改编自用于描述分子集合的理论，包括自我推进。由此产生的方程提供了一种方法，通过该方法可以计算一组流体动力学相互作用的自推进粒子的统计特性。使用这些方法，探索了自推进粒子动力学的三个特性——它们的流体动力学耦合，它们的诞生和死亡，以及它们与边界的相互作用。自推进粒子的动力学是一个与广泛的现实世界系统相关的物理模型，包括动物群，细菌游泳和多机器人组合。需要新的、更有效的数学工具来有效地计算粒子群的主要性质。该项目利用运动理论开发了这样的工具，其中的函数描述了每个粒子具有特定位置和速度的概率。推导了这些函数所服从的方程，并对其解进行了探讨和分析。利用适当的方程，研究人员研究了粒子如何在它们共同的流体环境中相互耦合，粒子如何湮灭和繁殖，以及粒子在固体边界附近的行为。在该项目中获得的见解的应用包括对细菌生物膜如何形成的潜在机制理解，水生生物如何优化其游泳，以及如何更好地控制一系列通信人造机器人，如自主海洋交通工具。

This project develops the mathematical foundations underlying the kinetic theory of self-propelled particles. A kinetic theory approach is used to describe the probability densities of the positions and velocities of all the particles. The mathematical analysis is adapted from theories used to describe ensembles of molecules, to include self-propulsion. The resulting equations provide a means by which statistical properties of a collection of hydrodynamically interacting, self-propelled particles can be computed. Using these methods, three properties of the dynamics of self-propelled particles are explored--their hydrodynamic coupling, their birth and death, and their interactions with boundaries.The dynamics of self-propelled particles is a physical model relevant to a wide range of real-world systems including animal swarming, bacterial swimming, and multi-robot ensembles. New, more efficient mathematical tools are needed to efficiently compute the main properties of swarms of particles. This project develops such tools using kinetic theory, in which functions describing the probabilities of each particle having a certain position and velocity. Equations that these functions obey are derived and their solutions explored and analyzed. Using the appropriate equations, the investigators study how particles couple to each other through their common fluid environment, how particles annihilate and reproduce, and how particles behave near solid boundaries. Applications of the insight gained during this project include potentially a mechanistic understanding of how bacterial biofilms form, how aquatic organisms optimize their swimming, and how a collection of communicating man-made robots, such as autonomous marine vehicles, can be better controlled.