# 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|)=−1/(μn) (ξ−1/n |i−j|) given by the superposition of a universal attractive interaction with strength −1/(μn) ξ<0, and a linear attractive potential 1/(μn{sup 2}) |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 ξ > 0. In the large temperature limit T ≫ T{sub 0}(ρ) = ρ{sup 2}/(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{sub c}{sup (vdW)}=1/(12μ) (3ξ−1)

- Authors:

- Department of Particle Physics, National Institute for Physics and Nuclear Engineering, 077125 Bucharest (Romania)
- Departamento de Física, FCEyN, Universidad de Buenos Aires and IFIBA, CONICET, Ciudad Universitaria, Pab. 1, 1428 Buenos Aires (Argentina)

- Publication Date:

- OSTI Identifier:
- 22403087

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 1; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CRITICAL TEMPERATURE; EQUATIONS OF STATE; EQUILIBRIUM; EXACT SOLUTIONS; INTERACTING BOSON MODEL; INTERACTIONS; ONE-DIMENSIONAL CALCULATIONS; PHASE TRANSFORMATIONS; POTENTIALS; THERMODYNAMICS

### Citation Formats

```
Pirjol, Dan, and Schat, Carlos.
```*Thermodynamics of a lattice gas with linear attractive potential*. United States: N. p., 2015.
Web. doi:10.1063/1.4904833.

```
Pirjol, Dan, & Schat, Carlos.
```*Thermodynamics of a lattice gas with linear attractive potential*. United States. doi:10.1063/1.4904833.

```
Pirjol, Dan, and Schat, Carlos. Thu .
"Thermodynamics of a lattice gas with linear attractive potential". United States.
doi:10.1063/1.4904833.
```

```
@article{osti_22403087,
```

title = {Thermodynamics of a lattice gas with linear attractive potential},

author = {Pirjol, Dan and Schat, Carlos},

abstractNote = {We study the equilibrium thermodynamics of a one-dimensional lattice gas with interaction V(|i−j|)=−1/(μn) (ξ−1/n |i−j|) given by the superposition of a universal attractive interaction with strength −1/(μn) ξ<0, and a linear attractive potential 1/(μn{sup 2}) |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 ξ > 0. In the large temperature limit T ≫ T{sub 0}(ρ) = ρ{sup 2}/(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{sub c}{sup (vdW)}=1/(12μ) (3ξ−1)},

doi = {10.1063/1.4904833},

journal = {Journal of Mathematical Physics},

number = 1,

volume = 56,

place = {United States},

year = {Thu Jan 15 00:00:00 EST 2015},

month = {Thu Jan 15 00:00:00 EST 2015}

}