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.
The activities during this period included development of theoretical models and mathematical techniques for analyzing (1) how particles interact through the fluid environment that couples their motion. (2) how a finite number of particles self-assemble. The probabilities of observing a specified number of clusters of specific size are estimates. (3) the time required for particles to first self-assemble into a large cluster. Our research provides a detailed mechanistic and statistical understanding of how particles self-assemble in realistic environments (finite sized systems and with hydrodynamic coupling). The results have implications for how biomolecules come together, how self-assembly may be controlled in materials synthesis, and how single-celled organisms evolve to form multicellular organisms. We also derived equations to describe how swimming particles interact with each other through a viscous fluid medium. We found thatshort-ranged molecular interactions can give rise to longer-ranged interactions if fluid flow coupling is included. Moreover, we find that fluid interactions destabilize coherent structures formed by otherwise uncoupled self-propelled particles. Numerous students participated in the proposed research, including two undergraduate students, one graduate student, and two postdoctoral scholars. The overall projects required learning and understanding fluid mechanics, computer simulations, and statistics. Each trainee was trained in at least two of the relevant skills.
