Conserved Quantities and Structural States in Active Systems

Both biological organisms such as bacteria or spermatozoa as well as lab-generated self-propelled particles can be described as microswimmers - force-free particles interacting with one another through the flow created by their motion. By solving the flow generated by a swimmer using the multipole expansion, we represent a microswimmer with a set of basic flow modes. When the swimmers are bound to a two-dimensional fluid, the leading modes are very long-ranged, decaying algebraically as $1/r$. We show that an ensemble of swimmers can be described by a geometric Hamiltonian formalism. Using Noether's theorem, we link the symmetries of the Hamiltonian to corresponding structural conservation laws, which can lead to ordering or to collective motion.
We analyze a few setups via theory and simulations, starting from two swimmers - showing that the angle between them is conserved, scaling up to several swimmers and eventually to a large collection of particles. We study perturbations to a line of oriented swimmers (a "street" of swimmers) with either (a) a single swimmer deviating from the centerline resulting in a propagating wavefront with predictable velocity, or (b) wave deformation resulting in breakage into sub-streets that maintain their shape for an extended period.