Predicting the bubble cutoff and microphase threshold in active phase separation
In plain words
In ordinary phase separation small droplets dissolve and big ones grow until one large region of each phase remains. In active systems small gas bubbles inside the dense phase persist, and a theory that predicts their largest size and when they take over from bulk coexistence is missing.
Precise statement
Active model $B+$ predicts reverse Ostwald ripening, which can arrest coarsening and produce a steady population of gas bubbles inside the dense phase; large $2\mathrm{D}$ simulations of active Brownian particles (ABPs) show bubbles with a power-law size distribution up to a cutoff and, at large system size $L$ or density, microphase separation. Derive the active model $B+$ coefficients $(\lambda,\zeta)$ from ABP parameters and predict the bubble cutoff scale and the size or density threshold for microphase separation in $d=2\ \text{and}\ 3$.
What would settle it
A derivation of the active model $B+$ coefficients from particle parameters whose predicted bubble cutoff and microphase threshold match particle simulations over a wide range of $L$ in $d = 2\ \text{and}\ 3$.
Status in the literature
Unverified note
Large 2D ABP simulations (Shi et al., PRL 2020, https://doi.org/10.1103/PhysRevLett.125.168001) found algebraically distributed bubbles and, at large L or density, microphase separation with a finite bubble cutoff; a quantitative derivation of active model B+ coefficients from particle parameters, and the 3D case, remain open (2026).