# Ideal Mixture Approximation of Cluster Size Distributions at Low Density

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## Abstract

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.

## Keywords

Classical particle system Canonical ensemble Equilibrium statistical mechanics Dilute system Large deviations## Notes

### Acknowledgements

We gratefully acknowledge financial support by the DFG-Forschergruppe FOR718 “Analysis and stochastics in complex physical systems”.

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