  The pressure of suspensions of self-propelled objects is studied
theoretically and by simulation of spherical active Brownian particles (ABP).
We show that for certain geometries, the mechanical pressure as force/area of a
confined systems can equally be expressed by bulk properties, which implies the
existence of an nonequilibrium equation of state. Exploiting the virial
theorem, we derive expressions for the pressure of ABPs confined by solid walls
or exposed to periodic boundary conditions. In both cases, the pressure
comprises three contributions: the ideal-gas pressure due to white-noise random
forces, an activity-induce pressure (swim pressure), which can be expressed in
terms of a product of the bare and a mean effective propulsion velocity, and
the contribution by interparticle forces. We find that the pressure of
spherical ABPs in confined systems explicitly depends on the presence of the
confining walls and the particle-wall interactions, which has no correspondence
in systems with periodic boundary conditions. Our simulations of
three-dimensional APBs in systems with periodic boundary conditions reveal a
pressure-concentration dependence that becomes increasingly nonmonotonic with
increasing activity. Above a critical activity and ABP concentration, a phase
transition occurs, which is reflected in a rapid and steep change of the
pressure. We present and discuss the pressure for various activities and
analyse the contributions of the individual pressure components.
