Chemical potential in active systems: predicting phase equilibrium from bulk equations of state?

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.