  We consider a continuum model for motility-induced phase separation (MIPS) of
active Brownian particles [J. Chem. Phys. 142, 224149 (2015)]. Using a recently
introduced perturbative analysis [Phys. Rev. E 98, 020604(R) (2018)], we show
that this continuum model reduces to the classic Cahn-Hilliard (CH) model near
the onset of MIPS. This makes MIPS another example of the so-called active
phase separation. We further introduce a generalization of the perturbative
analysis to the next higher order. This results in a generic higher order
extension of the CH model for active phase separation. Our analysis establishes
the mathematical link between the basic mean-field MIPS model on the one hand,
and the leading order and extended CH models on the other hand. Comparing
numerical simulations of the three models, we find that the leading order CH
model agrees nearly perfectly with the full continuum model near the onset of
MIPS. We also give estimates of the control parameter beyond which the higher
order corrections become relevant and compare the extended CH model to recent
phenomenological models.
