  Active Brownian particles display self-propelled movement, which can be
modelled as arising from a one-body force. Although their interparticle
interactions are purely repulsive, for strong self propulsion the swimmers
phase separate into dilute and dense phases. We describe in detail a recent
theory (Phys. Rev. E 100, 052604 (2019); Phys. Rev. Lett. 128, 26802 (2019))
for such motility induced phase-separation. Starting from the continuity
equation and the force density balance, the description is based on four
superadiabatic contributions to the internal force density. Here the
superadiabatic forces are due to the flow in the system and they act on top of
the adiabatic forces that arise from the equilibrium free energy. Phase
coexistence is described by bulk state functions and agrees quantitatively with
Brownian dynamics simulation results from the literature. We describe in detail
all analytical steps to fully resolve the spatial and orientational dependence
of the one-body density and current. The decomposition into angular Fourier
series leads to coupling of total density, polarization and all higher modes.
We describe the power functional approach, including the kinematic dependence
of the superadiabatic force fields and the quiet life effect that pushes
particles from fast to slow regions, and hence induces the phase separation.
