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 ei 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 H = J

ij

ei ej .

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.