Mechanical pressure and momentum conservation in dry active matter

We relate the breakdown of equations of states (EOS) for the mechanical pressure of generic dry active systems to the lack of momentum conservation in such systems. We show how net sources and sinks of momentum arise generically close to confining walls. These typically depend on the interactions of the container with the particles, which makes the mechanical pressure a container-dependent quantity. We show that an EOS is recovered if the dynamics of the propulsive forces of the particles are decoupled from other degrees of freedom and lead to an apolar bulk steady-state. This recovery of an EOS stems from the mean steady-state active force density being the divergence of the flux of 'active impulse', an observable which measures the mean momentum particles will receive from the substrate in the future.