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A complete replacement rule that predicts those densities from particle properties is not established.","posed_since":"2015","precise":"For active Brownian particles with purely repulsive interactions in $d = 2\\ \\text{and}\\ 3$, the mechanical pressure is a state function (no torques), so equal pressure is one coexistence condition, but the second condition replacing equal chemical potential is not given by equilibrium thermodynamics. Derive the binodal densities $\\rho_{\\mathrm{gas}}(\\mathrm{Pe})$ and $\\rho_{\\mathrm{liquid}}(\\mathrm{Pe})$ without fitted parameters and determine whether liquid-gas MIPS coexistence is stable or metastable with respect to an active crystal at high Pe in $d = 3$.","problem_ref":null,"references":"","settled_by":"A theory whose predicted binodals match large-scale particle simulations in $d = 2$ and 3 within statistical error, including the location of the crystal-fluid boundary.","status_note":"In 2015 the mechanical pressure was shown to lack an equation of state for generic active fluids but to be a state function for torque-free particles (Solon et al., Nature Physics, https://doi.org/10.1038/nphys3377); mechanical coexistence theories followed, and a 2021 simulation study found $3\\mathrm{D}$ MIPS metastable with respect to active crystallization.","title":"Predicting MIPS coexistence densities from particle-level 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