Thermodynamics of a Hierarchical Mixture of Cubes

Abstract

AbstractWe investigate a toy model for phase transitions in mixtures of incompressible droplets. The model consists of non-overlapping hypercubes in $${\mathbb {Z}}^d$$ Z d of sidelengths $$2^j$$ 2 j , $$j\in {\mathbb {N}}_0$$ j ∈ N 0 . Cubes belong to an admissible set $${\mathbb {B}}$$ B such that if two cubes overlap, then one is contained in the other. Cubes of sidelength $$2^j$$ 2 j have activity $$z_j$$ z j and density $$\rho _j$$ ρ 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 $$z_j(\mu ) = \exp ( 2^{dj} \mu - E_j)$$ z j ( μ ) = exp ( 2 dj μ - E j ) . We prove a sufficient criterion for absence of phase transition, show that constant energies $$E_j\equiv \lambda $$ E j ≡ λ lead to a continuous phase transition, and prove a necessary and sufficient condition for the existence of a first-order phase transition.