Pressure and Phase Equilibria in Interacting Active Brownian Spheres

We derive from first principles the mechanical pressure \(P\), defined as the force per unit area on a bounding wall, in a system of spherical, overdamped, active Brownian particles at density \(\rho\). Our exact result relates \(P\), in closed form, to bulk correlators and shows that (i) \(P(\rho)\) is a state function, independent of the particle-wall interaction; (ii) interactions contribute two terms to \(P\), one encoding the slow-down that drives motility-induced phase separation, and the other a direct contribution well known for passive systems; (iii) \(P(\rho)\) is equal in coexisting phases. We discuss the consequences of these results for the motility-induced phase separation of active Brownian particles, and show that the densities at coexistence do not satisfy a Maxwell construction on \(P\).