Thermodynamics of a Hierarchical Mixture of Cubes

Abstract

We investigate a toy model for phase transitions in mixtures of incompressible droplets. The model consists of non-overlapping hypercubes in {mathbb {Z}}^d of sidelengths 2^j, jin {mathbb {N}}_0. Cubes belong to an admissible set {mathbb {B}} such that if two cubes overlap, then one is contained in the other. Cubes of sidelength 2^j have activity z_j and density rho _j. We prove explicit formulas for the pressure and entropy, prove a van-der-Waals type equation of state, and invert the density-activity relations. In addition we explore phase transitions for parameter-dependent activities z_j(mu ) = exp ( 2^{dj} mu - E_j). We prove a sufficient criterion for absence of phase transition, show that constant energies E_jequiv lambda lead to a continuous phase transition, and prove a necessary and sufficient condition for the existence of a first-order phase transition.