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

Ideal mixture approximation of cluster size distributions at low density