  Simulation studies of the phase diagram of repulsive active Brownian
particles in three dimensions reveal that the region of motility-induced phase
separation between a high and low density phase is enclosed by a region of
gas-crystal phase separation. Near-critical loci and structural crossovers can
additionally be identified in analogy with simple fluids. Motivated by the
striking similarity to the behaviour of equilibrium fluids with short-ranged
pair-wise attractions, we show that a direct mapping to pair potentials in the
dilute limit implies interactions that are insufficiently attractive to
engender phase separation. Instead, this is driven by the emergence of
multi-body effects associated with particle caging that occurs at sufficiently
high number density. We quantify these effects via information-theoretical
measures of $n$-body effective interactions extracted from the configurational
structure.
