  Recently, we have derived an effective Cahn-Hilliard equation for the phase
separation dynamics of active Brownian particles by performing a weakly
non-linear analysis of the effective hydrodynamic equations for density and
polarization [Phys. Rev. Lett. 112, 218304 (2014)]. Here we develop and explore
this strategy in more detail and show explicitly how to get to such a
large-scale, mean-field description starting from the microscopic dynamics. The
effective free energy emerging from this approach has the form of a
conventional Ginzburg-Landau function. On the coarsest scale, our results thus
agree with the mapping of active phase separation onto that of passive fluids
with attractive interactions through a global effective free energy
(mobility-induced phase transition). Particular attention is paid to the
square-gradient term necessary for the dynamics. We finally discuss results
from numerical simulations corroborating the analytical results.
