  A recently formulated description of homogeneous nucleation for Brownian
particles in the over-damped limit based on fluctuating hydrodynamics is used
to determine the nucleation pathway, characterized as the most likely path
(MLP), for the nucleation of a dense-concentration droplet of globular protein
from a dilute solution in a small, finite container. The calculations are
performed by directly discretizing the equations for the MLP and it is found
that they confirm previous results obtained for infinite systems: the process
of homogeneous nucleation begins with a long-wavelength, spatially-extended
concentration fluctuation that it condenses to form the pre-critical cluster.
This is followed by a classical growth processes. The calculations show that
the post-critical growth involves the formation of a depletion zone around the
cluster whereas no such depletion is observed in the pre-critical cluster. The
approach therefore captures dynamical effects not found in classical Density
Functional Theory studies while consistently describing the formation of the
pre-critical cluster.
