Passive, thermal particles may undergo phase-separation when attractive forces overcome the entropic cost of creating dense groups of particles. Active, self-propelled particles need no attractive forces to experience the same fate. When their propulsion speeds decreases sufficiently rapidly as the local density of particle increases, whether due to quorum-sensing like interactions or due to repulsive forces, they may undergo a so-called "Motility Induced Phase Separation" (MIPS).

In this talk I will both review this phenomenon and present a recent theory that accounts quantitatively for the phase equilibria observed in MIPS. Starting from a continuum description of scalar active matter, akin to a generalized Cahn-Hilliard equation, I will show how phase-separated profiles emerge from the extremization of an effective free energy. I will illustrate this approach on two well-known models: self-propelled particles interacting either through a density-dependent propulsion speed or via direct pairwise forces and show our theory to account quantitatively for their phase diagrams, providing a unified description of MIPS.