  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.

Soft Condensed Matter;Biological Physics;Fluid Dynamics