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

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.