Hydrodynamic Hamiltonians and Structural States of Active Systems

We show that a two-dimensional system of co-oriented active particles interacting only hydrodynamically can be expressed using a Hamiltonian formalism. The Hamiltonian depends strictly on the positions of the particles, thereby restricting their available phase-space. The conservation of the Hamiltonian, coupled with its inherent symmetries, fosters the self-organization of the observed steady-state configurations. These emergent formations depend on the force distribution acting on the particles. We explore several cases of force distributions on the particles, verifying our conclusions by simulations of active particles with quenched orientation, interacting only hydrodynamically and by steric forces. Particles subject to two symmetric opposing forces, termed stresslets, create zigzag formations — sharp lines at a particular tilt along which particles circulate. Particles with two anti-symmetric opposing forces, termed rotlets, create a rotating hexagonal lattice. Higher-order force distributions lead to the aggregation of an ensemble of particles.