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 \(2^j\), \(j\in {\mathbb {N}}_0\). 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 \(E_j\equiv \lambda \) lead to a continuous phase transition, and prove a necessary and sufficient condition for the existence of a first-order phase transition.

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I thank Serena Cenatiempo, Diana Conache, Dimitrios Tsagkarogiannis, and Steffen Winter for stimulating discussions, and David Brydges for pointing out the possible usefulness of renormalization for treating mixtures of objects of different sizes.

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Correspondence to Sabine Jansen.

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Communicated by Alessandro Giuliani.

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Jansen, S. Thermodynamics of a Hierarchical Mixture of Cubes. J Stat Phys 179, 309–340 (2020).

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  • Incompressible droplets
  • Condensation
  • Excluded volume
  • Polymer partition function
  • Hierarchical model

Mathematics Subject Classification

  • 82B20
  • 82B26