Structural states and conservation laws in a system of two-dimensional active swimmers

Ensembles of motile microswimmers display complex collective dynamics. Nonetheless, when confined to two dimensions, we show they can be expressed using a unifying formalism. A system of swimming particles such as algae or bacteria in a thin film can be described by a many-body Hamiltonian. When simulating a random arrangement of micro-swimmers, we find they evolve into sharp lines at a particular tilt. We call these states "escalators" as particles circulate along these canted conveyor belts. We argue that the conservation of the Hamiltonian and its symmetry germinate the self-assembly of the observed steady-state arrangements. The Hamiltonian is scale-invariant and depends strictly on the angles between the swimmers and their swimming orientation, thereby restricting their available phase space. Stability analysis predicts that an initial alignment at either a low or a high angle is unstable and that the ensemble of swimmers will break into the observed escalators.