Thermodynamics of a lattice gas with linear attractive potential

We study the equilibrium thermodynamics of a one-dimensional lattice gas with interaction V ( i - j ) = -1/μn{ε - 1/n |i- j|} given by the superposition of a universal attractive interaction with strength -1/μn ε< 0, and a linear attractive potential 1/μn₂ |i-j|. The interaction is rescaled with the lattice size n, such that the thermodynamical limit n → ∞ is well-behaved. The thermodynamical properties of the system can be found exactly, both for a finite size lattice and in the thermodynamical limit n → ∞. The lattice gas can be mapped to a system of non-interacting bosons which are placed on known energy levels. The exact solution shows that the system has a liquid-gas phase transition for ε > 0. In the large temperature limit T » T₀(ρ) = ρ²/(4μ) with ρ the density, the system becomes spatially homogeneous, and the equation of state is given to a good approximation by a lattice version of the van der Waals equation, with critical temperature T_c ^(vdW) = 1/12μ(3ε - 1 ).