  Homogeneous nucleation, a textbook transition path for phase transitions, is
typically understood on thermodynamic grounds through the prism of classical
nucleation theory. However, recent studies have suggested the applicability of
classical nucleation theory to systems far from equilibrium. In this Article,
we formulate a purely mechanical perspective of homogeneous nucleation and
growth, elucidating the criteria for the properties of a critical nucleus
without appealing to equilibrium notions. Applying this theory to active fluids
undergoing motility-induced phase separation, we find that nucleation proceeds
in a qualitatively similar fashion to equilibrium systems, with concepts such
as the Gibbs-Thomson effect and nucleation barriers remaining valid. We further
demonstrate that the recovery of such concepts allows us to extend classical
theories of nucleation rates and coarsening dynamics to active systems upon
using the mechanically-derived definitions of the nucleation barrier and
surface tensions.Three distinct surface tensions -- the mechanical, capillary,
and Ostwald tensions -- play a central role in our theory. While these three
surface tensions are identical in equilibrium, our work highlights the
distinctive role of each tension in the stability of active interfaces and the
nucleation and growth of motility-induced phases.
