Cooling mechanism controls motility-induced phase separation in inertial active liquids

Motility-induced phase separation (MIPS) is a central collective phenomenon in active matter, theoretically established in the overdamped regime. We discover that the dynamical origin of MIPS is fundamentally altered by inertia, which induces a cooling mechanism absent in overdamped active matter. This conclusion is supported by an active variant of the direct simulation Monte Carlo method and by a kinetic theory for inertial self-propelled hard spheres derived from the microscopic dynamics. In contrast to the overdamped case, both analyses demonstrate that inertial MIPS does not rely on volume exclusion but on a cooling mechanism involving density, polarization, and temperature fields. This mechanism emerges from the competition between activity and a density dependent collision rate, arising from spatial correlations between colliding particles. These findings open a pathway to fundamentally connect inertial active matter with granular physics.