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

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