We derive a microscopic expression for the mechanical pressure $P$ in a system of spherical active Brownian particles at density $\rho$. Our exact result relates $P$, defined as the force per unit area on a bounding wall, to bulk correlation functions evaluated far away from the wall. It 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; and (iii) $P$ 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 $P$.

• Received 18 December 2014

DOI:https://doi.org/10.1103/PhysRevLett.114.198301