  We discuss different definitions of pressure for a system of active spherical
particles driven by a non-thermal coloured noise. We show that mechanical,
kinetic and free-energy based approaches lead to the same result up to first
order in the non-equilibrium expansion parameter. The first prescription is
based on a generalisation of the kinetic mesoscopic virial equation and
expresses the pressure exerted on the walls in terms of the average of the
virial of the inter-particle forces. In the second approach, the pressure and
the surface tension are identified with the volume and area derivatives,
respectively, of the partition function associated with the known stationary
non-equilibrium distribution of the model. The third method is a mechanical
approach and is related to the work necessary to deform the system. The
pressure is obtained by comparing the expression of the work in terms of local
stress and strain with the corresponding expression in terms of microscopic
distribution. This is determined from the force balance encoded in the
Born-Green-Yvon equation. Such a method has the advantage of giving a formula
for the local pressure tensor and the surface tension even in inhomogeneous
situations. By direct inspection, we show that the three procedures lead to the
same values of the pressure, and give support to the idea that the partition
function, obtained via the unified coloured noise approximation, is more than a
formal property of the system, but determines the stationary non-equilibrium
thermodynamics of the model.
