Jansen, Sabine
(2020):
Thermodynamics of a Hierarchical Mixture of Cubes.
In: Journal of Statistical Physics, Vol. 179, No. 2: pp. 309340

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Abstract
We investigate a toy model for phase transitions in mixtures of incompressible droplets. The model consists of nonoverlapping hypercubes in Z(d) of sidelengths 2(j), j is an element of N0. Cubes belong to an admissible set 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 vanderWaals type equation of state, and invert the densityactivity relations. In addition we explore phase transitions for parameterdependent activities z j (mu) = exp(2(dj)mu  Ej). We prove a sufficient criterion for absence of phase transition, show that constant energies Ej equivalent to lambda lead to a continuous phase transition, and prove a necessary and sufficient condition for the existence of a firstorder phase transition.