Self-induced polar order of active Brownian particles in a harmonic trap

Hydrodynamically interacting active particles in an external harmonic potential form a self-assembled fluid pump at large enough P\'eclet numbers. Here, we give a quantitative criterion for the formation of the pump and show that particle orientations align in the self-induced flow field in surprising analogy to ferromagnetic order where the active P\'eclet number plays the role of inverse temperature. The particle orientations follow a Boltzmann distribution $\Phi(\mathbf{p}) \sim \exp(A p_z)$ where the ordering mean field $A$ scales with active P\'eclet number and polar order parameter. The mean flow field in which the particles' swimming directions align corresponds to a regularized stokeslet with strength proportional to swimming speed. Analytic mean-field results are compared with results from Brownian dynamics simulations with hydrodynamic interactions included and are found to capture the self-induced alignment very well.