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.
This NSF grant allowed us to study several aspects of particle interaction and how they organize, specifically within biological processes. We analyzed different relevant cases. On one hand we investigated the formation of locust swarms that are responsible for great devastation in Northern Africa, the Middle East, Asia, and Australia. A locust swarm can travel hundreds of kilometers per day, stripping vegetation, disrupting agriculture, and destroying crops. Our work was a mathematical attempt to study locust swarm formation as a way to help find better ways to predict, manage, and control locust outbreaks. Locusts exhibit two interconvertible behavioral phases, solitarious and gregarious. Crowding conditions biases conversion towards the gregarious form. We studied phase change at the collective level constructing a partial integrodifferential equation model that incorporates the interplay between phase change and spatial movement. We performed numerical and analytical work and used our model to study the dynamics of swarm formation. Our linear stability analysis shows the conditions for an outbreak, characterized by a large scale transition to the gregarious phase. We also predicted the proportion of the population that will gregarize, and of the time scale for this to occur. Another problem we tackled was that of aggregation and self-assembly of particles into clusters. This is a common phenomenon in chemistry, polymer science, material science, and molecular biology and has been extensively studied, both experimentally and theoretically. Most theories developed thus far have employed mass-action, mean-field kinetics that describe the dynamics of the mean concentration of clusters of a given size These previous theories have primarily focused on infinite systems with unbounded cluster sizes. However, in finite-sized systems such as those encountered in cell biology, molecularly constrained maximum cluster sizes and coagulation-fragmentation processes naturally arise. We derived and analyze stochastic models for self-assembly of finite-sized systems and systematically computed nontrivial stochastic quantities. We were able to find unexpected behaviors in coarsening, equilibrium cluster size distributions, and first assembly times that cannot be predicted from previous, classical theories. Our work was applied to monomer attachment-detachment, coagulatio-fragmentation and to homogeneous and heterogeneous particles.Finally, through this NSF proposal we were able to support several undergraduate and master degree students who have continued their careers and and are now seeking PhDs.
