  In equilibrium, the collective behaviour of particles interacting via steep,
short-ranged potentials is well captured by the virial expansion of the free
energy at low density. Here, we extend this approach beyond equilibrium to the
case of active matter with self-propelled particles. Given that active systems
do not admit any free-energy description in general, our aim is to build the
dynamics of the coarse-grained density from first principles without any
equilibrium assumption. Starting from microscopic equations of motion, we
obtain the hierarchy of density correlations, which we close with an ansatz for
the two-point density valid in the dilute regime at small activity. This
closure yields the nonlinear dynamics of the one-point density, with
hydrodynamic coefficients depending explicitly on microscopic interactions, by
analogy with the equilibrium virial expansion. This dynamics admits a spinodal
instability for purely repulsive interactions, a signature of motility-induced
phase separation. Therefore, although our approach should be restricted to
dilute, weakly-active systems a priori, it actually captures the features of a
broader class of active matter.
