  We investigate motility-induced phase separation of active Brownian
particles, which are modeled as purely repulsive spheres that move due to a
constant swim force with freely diffusing orientation. We develop on the basis
of power functional concepts an analytical theory for nonequilibrium phase
coexistence and interfacial structure. Theoretical predictions are validated
against Brownian dynamics computer simulations. We show that the internal
one-body force field has four nonequilibrium contributions: (i) isotropic drag
and (ii) interfacial drag forces against the forward motion, (iii) a
superadiabatic spherical pressure gradient and (iv) the quiet life gradient
force. The intrinsic spherical pressure is balanced by the swim pressure, which
arises from the polarization of the free interface. The quiet life force
opposes the adiabatic force, which is due to the inhomogeneous density
distribution. The balance of quiet life and adiabatic forces determines bulk
coexistence via equality of two bulk state functions, which are independent of
interfacial contributions. The internal force fields are kinematic functionals
which depend on density and current, but are independent of external and swim
forces, consistent with power functional theory. The phase transition
originates from nonequilibrium repulsion, with the agile gas being more
repulsive than the quiet liquid.
