  We consider an interacting particle system in continuous configuration space.
The pair interaction has an attractive part. We show that, at low density, the
system behaves approximately like an ideal mixture of clusters (droplets): we
prove rigorous bounds (a) for the constrained free energy associated with a
given cluster size distribution, considered as an order parameter, (b) for the
free energy, obtained by minimising over the order parameter, and (c) for the
minimising cluster size distributions. It is known that, under suitable
assumptions, the ideal mixture has a transition from a gas phase to a condensed
phase as the density is varied; our bounds hold both in the gas phase and in
the coexistence region of the ideal mixture. The present paper improves our
earlier results by taking into account the mixing entropy.
