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1.1. Lattice Gas

Consider a lattice gas in which the particles occupy the sites of a $d$ -dimensional lattice, with the constraint that each site cannot be occupied by more than one particle. Let $e_{i}$ equal a variable that takes values $\{0,1\}: 0$ when is the site is vacant and 1 when it is occupied. The interaction energy of each configuration is given by $$ \mathcal{H}=J \sum_{\langle i j\rangle} e_{i} e_{j} $$ Show that the grand canonical partition function of the lattice gas can be put in correspondence with the canonical partition function of the Ising model. Argue that the phase transition of the lattice gas, consists of the condensation of the particles, belongs to the same universality class of the Ising model.

I'm having trouble understanding why the phase transition would lead to the condensation of the particles.

To me, the phase transition occurs in a system where there are two, or more, competing forces, such as forces that try to minimise hamiltonian and forces that try to maximise entropy. In this case, the minimisation of energy would correspond to a repulsive force between the particles, whereas the entropic force tries to prevent this since on a lattice this would mean the particles will stay at the maximum possible distance from each other. Hence when the entropic forces at taken over by the energetic forces, the system will acquire a state of low density, but this would not corresponds to condensation but rather evaporation.

What am I missing in here?

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    $\begingroup$ $J$ is (presumably) negative, so there is attraction between neighboring particles. There is also a repulsive part to the interaction, but the latter is purely due to the fact that there can be at most one particle per site. $\endgroup$ Apr 19 at 16:43

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