  We consider microscopic models of active particles whose velocities,
rotational diffusivities, and tumbling rates depend on the gradient of a local
field, which is either externally imposed or depends on all particle positions.
Despite the fundamental differences between active and passive dynamics at the
microscopic scale, we show that a large class of such tactic active systems
admit fluctuating hydrodynamics equivalent to those of interacting Brownian
colloids in equilibrium. We exploit this mapping to show how taxis may lead to
the lamellar and micellar phases observed for soft repulsive colloids. In the
context of chemotaxis, we show how the competition between chemoattractant and
chemorepellent may lead to a bona-fide equilibrium liquid-gas phase separation
in which a loss of thermodynamic stability of the fluid signals the onset of a
chemotactic collapse.
