Thermodynamics of a hierarchical mixture of cubes

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 $2^j$, $j\in N_0$. 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 $E_j\equiv\lambda$ lead to a continuous phase transition, and prove a necessary and sufficient condition for the existence of a first-order phase transition.