Lattice gas

Particle (occupation) model on a lattice with 0–1 variables, equivalent to the Ising model in a field and used to model gas–liquid coexistence and adsorption.

Lattice gas

A lattice gas is a statistical-mechanical model of particles on a lattice, where each site is either empty or occupied. It can be formulated as a lattice spin system with spin space $\{0,1\}$ and is closely related to the Ising model .

Let $\Lambda\subset\mathbb{Z}^d$ be finite. A configuration is $n=(n_x)_{x\in\Lambda}$ with $n_x\in\{0,1\}$ , where $n_x=1$ means “occupied.” A common nearest-neighbor attractive lattice gas Hamiltonian is

H_\Lambda(n) ={} -J\sum_{\langle x,y\rangle:\, x,y\in\Lambda} n_x n_y -\mu\sum_{x\in\Lambda} n_x, \qquad J\ge 0,

where $\mu$ is the chemical potential (an analogue of an external field ) and $J\ge 0$ makes occupied neighbors energetically favorable. More general lattice gases can be defined using an interaction potential beyond nearest neighbors.

At inverse temperature $\beta$ , the finite-volume Gibbs distribution is the finite-volume Gibbs measure

\mu_{\Lambda,\beta}(n)\propto \exp\!\big(-\beta H_\Lambda(n)\big),

normalized by the partition function . The associated pressure is typically defined from $\frac{1}{|\Lambda|}\log Z_\Lambda$ (and its infinite-volume limit by thermodynamic limit of the pressure ).

Key properties

Density as the main observable. The natural order parameter is the particle density
$\rho_\Lambda=\frac{1}{|\Lambda|}\sum_{x\in\Lambda} n_x,$
or its infinite-volume expectation. Phase coexistence appears as multiple possible limiting densities at the same $(\beta,\mu)$ .
Equivalence to Ising in a field. Define Ising spins $\sigma_x\in\{-1,+1\}$ by $\sigma_x=2n_x-1$ . For nearest-neighbor interactions on $\mathbb{Z}^d$ (degree $2d$ ), one obtains (up to an additive constant)
$H_\Lambda(n) ={} -\frac{J}{4}\sum_{\langle x,y\rangle}\sigma_x\sigma_y -\left(\frac{\mu}{2}+\frac{Jd}{2}\right)\sum_{x\in\Lambda}\sigma_x +\text{const}.$
Thus an attractive lattice gas is equivalent to a ferromagnetic Ising model with an effective field, making lattice gases a convenient physical reinterpretation of Ising phase transitions .
Gas–liquid transition and coexistence line. In the Ising correspondence, the “coexistence” regime (two densities) corresponds to the regime with multiple infinite-volume Gibbs states. The symmetry point (Ising zero field) corresponds to a particular chemical potential.
Boundary conditions and phase separation. Different boundary conditions can select low-density (“gas”) or high-density (“liquid”) phases in finite volume, and interfaces/domain walls appear in mixed boundary conditions.

Physical interpretation

The lattice gas is a coarse-grained model of a fluid on discrete sites, useful for:

gas–liquid coexistence and criticality,
adsorption in porous materials (occupied vs empty sites),
phase separation and interfaces.

Because of its exact mapping to the Ising model, it provides a direct bridge between magnetic language (spins, fields, magnetization) and fluid language (occupations, chemical potential, density).