Swarming in three dimensions.

We consider a three-dimensional model of active Brownian particles interacting via a Morse-type potential. The system exhibits two modes of motion: translation and a coherent rotation in a torus-shape structure. We observe noise-induced transitions in both directions between the two states. These occur at different noise intensities, thus leading to a hysteresis curve. For certain parameter regions, the system switches persistently between the states such that the center of mass alternates between ballistic and diffusive motion. The coherent rotation leads to a pronounced mean angular momentum that changes its direction diffusively. We derive an analytic expression for the diffusion of the angular momentum of one particle in an external harmonic potential and show that it is always faster than the stochastic switching of the direction of motion in the two-dimensional case.