  Motility-induced phase separation (MIPS) is a purely non-equilibrium
phenomenon in which self-propelled particles phase separate without any
attractive interactions. One surprising feature of MIPS is the emergence of
polar, nematic, and higher order structures at the interfacial region, whose
underlying physics remains poorly understood. Starting with a model of MIPS in
which all many-body interactions are captured by an effective speed function
and an effective pressure function that depend solely on the local particle
density, I derive analytically an infinite set of integral formulae (IF) for
the ordering structures at the interface. I then demonstrate that half of these
IF are in fact exact for a wide class of active Brownian particle systems.
Finally, I test the IF by applying them to numerical data from direct particle
dynamics simulation and find that all the IF remain valid to a great extent.
