Predicting MIPS coexistence densities from particle-level mechanics
In plain words
Self-propelled particles that separate into a crowd and a gas do not obey the equilibrium rules that fix the densities of the two phases. A complete replacement rule that predicts those densities from particle properties is not established.
Precise statement
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 $d = 3$.
What would settle it
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 in the literature
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.