Differences in activities in colloidal particles are sufficient to drive
phase separation between active and passive (or less active) particles, even if
they have only excluded volume interactions. In this paper, we study the phase
separation kinetics and propose a theory of phase separation of colloidal
mixtures in the diffusive limit. Our model considers a mixture of diffusing
particles coupled to different thermostats, it thus has a non-equilibrium
nature due to the temperature differences. However, we show that indeed the
system recovers an effective equilibrium thermodynamics in the dilute limit. We
obtain phase diagrams showing the asymmetry in concentrations due to activity
differences. By using a more general approach, we show the equivalence of phase
separation kinetics with the well known Cahn-Hilliard theory. On the other
hand, higher order expansions in concentration indicate the emergence of
non-equilibrium effects leading to a breakdown of the equilibrium analogy. We
lay out the general theory in terms of accessible parameters which we
demonstrate by several applications. In this simple formalism, we capture a
positive surface tension for hard spheres}, and interesting scaling laws for
interfacial properties, droplet growth dynamics, and phase segregation
conditions. \rev{Several of our results are in agreement with existing
numerical simulations while we also propose testable predictions.Comment: Published version, 19 pages (main text+appendix), 4 figure