  We derive a microscopic expression for a quantity $\mu$ that plays the role
of chemical potential of Active Brownian Particles (ABPs) in a steady state in
the absence of vortices. We show that $\mu$ consists of (i) an intrinsic
chemical potential similar to passive systems, which depends on density and
self-propulsion speed, but not on the external potential, (ii) the external
potential, and (iii) a newly derived one-body swim potential due to the
activity of the particles. Our simulations on active Brownian particles show
good agreement with our Fokker-Planck calculations, and confirm that $\mu(z)$
is spatially constant for several inhomogeneous active fluids in their steady
states in a planar geometry. Finally, we show that phase coexistence of ABPs
with a planar interface satisfies not only mechanical but also diffusive
equilibrium. The coexistence can be well-described by equating the bulk
chemical potential and bulk pressure obtained from bulk simulations for systems
with low activity but requires explicit evaluation of the interfacial
contributions at high activity.
