{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"b44335f4e745739b464dfd39010105ed7672f3a5eb28224185c07a6565db97ce","created":"2026-10-03T07:17:56Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"847d7b4a9ea982f9db26ad4967cb0c47269e68766c3135cacbf4fbec6081a83a","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":[],"external_id":"bio.active-matter.mips-coexistence-rules","kind":"well-posed","literature_status":"partially-resolved","n":"1","parents":[],"plain":"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.","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 mechanics","topic_ref":"6a008dc19304a289e116a6d6e6371cc6de0ef3d1a904e572d5454f039b2ac217"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"9ce32d14d437c7b5801e44bd6fe871d63e3cddf16371ef8a7d80457413d3df8a","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"c8aca7af0af0c6c8919a428115be8123f62b148efe8664469a55be5c83f53ae2","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"031EKyfndCHF3m0B-mYdXBV5WD31T4TOf_z2MbPPMp45DeaS9YlhsBpiwGuYYxoHHHLwlNXFZUxFPJGduoG-Bw"},"schema":"pubphys.envelope/1"},"record_hash":"9ce32d14d437c7b5801e44bd6fe871d63e3cddf16371ef8a7d80457413d3df8a","leaf_index":722}