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Jansen, Sabine (2020): Thermodynamics of a Hierarchical Mixture of Cubes. In: Journal of Statistical Physics, Vol. 179, No. 2: pp. 309-340

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We investigate a toy model for phase transitions in mixtures of incompressible droplets. The model consists of non-overlapping hypercubes in Z(d) of sidelengths 2(j), j is an element of N-0. 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 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 equivalent to lambda lead to a continuous phase transition, and prove a necessary and sufficient condition for the existence of a first-order phase transition.

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