Abstract. We study the structure of the constrained minimizers of the Gates–Lebowitz–Penrose free energy functional , non-local functional of a density field m(x), , a d-dimensional torus of side length L. At low temperatures, is not convex, and has two distinct global minimizers, corresponding to two equilibrium states. Here we constrain the average density to be a fixed value n between the densities in the two equilibrium states, but close to the low density equilibrium value. In this case, a 'droplet' of the high density phase may or may not form in a background of the low density phase, depending on the values n and L. We determine the critical density for droplet formation, and the nature of the droplet, as a function of n and L. The relation between the free energy and the large deviations functional for a particle model with long-range Kac potentials, proven in some cases, and expected to be true in general, then provides information on the structure of typical microscopic configurations of the Gibbs measure when the range of the Kac potential is large enough.

Mathematics Subject Classification: 49S05, 52A40, 82B26

Print publication: Issue 12 (December 2009)
Received 22 May 2009, in final form 6 October 2009
Published 30 October 2009