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Motility-induced phase separation (MIPS) is a purely non-equilibriumphenomenon in which self-propelled particles phase separate without anyattractive interactions. One surprising feature of MIPS is the emergence ofpolar, nematic, and higher order structures at the interfacial region, whoseunderlying physics remains poorly understood. Starting with a model of MIPS inwhich all many-body interactions are captured by an effective speed functionand an effective pressure function that depend solely on the local particledensity, I derive analytically an infinite set of integral formulae (IF) forthe ordering structures at the interface. I then demonstrate that half of theseIF 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 particledynamics simulation and find that all the IF remain valid to a great extent.

An Infinite Set of Integral Formulae for Polar, Nematic, and Higher Order Structures at the Interface of Motility-Induced Phase Separation