Thermodynamics of a lattice gas with linear attractive potential

Abstract

We study the equilibrium thermodynamics of a one-dimensional lattice gas with interaction V(i−j)=−1μn{ξ−1ni−j} given by the superposition of a universal attractive interaction with strength −1μnξ<0, and a linear attractive potential 1μn2i−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 ξ > 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 Tc(vdW)=112μ(3ξ−1).