  Recently, there has been much interest in activity-induced phase separations
in concentrated suspensions of "active Brownian particles" (ABPs),
self-propelled spherical particles whose direction of motion relaxes through
thermal rotational diffusion. To date, almost all these studies have been
restricted to 2 dimensions. In this work we study activity-induced phase
separation in 3D and compare the results with previous and new 2D simulations.
To this end, we performed state-of-the-art Brownian dynamics simulations of up
to 40 million ABPs -- such very large system sizes are unavoidable to evade
finite size effects in 3D. Our results confirm the picture established for 2D
systems in which an activity-induced phase separation occurs, with strong
analogies to equilibrium gas-liquid spinodal decomposition, in spite of the
purely non-equilibrium nature of the driving force behind the phase separation.
However, we also find important differences between the 2D and 3D cases.
Firstly, the shape and position of the phase boundaries is markedly different
for the two cases. Secondly, for the 3D coarsening kinetics we find that the
domain size grows in time according to the classical diffusive $t^{1/3}$ law,
in contrast to the nonstandard subdiffusive exponent observed in 2D.
