---
title: "The Lattice Gas model"

---


A crude representation of a fluid is the lattice gas model. Here
particles can occupy the sites of a hypercubic lattice. The occupancy of
a site $i$ is specified by the variable $c_i=1$ (occupied) or $c_i=0$
vacant. The complete list of these occupancies $\{c\}$ specifies a
microstate. The average particle number density (fraction of occupied
sites) is given by

$$c=L^{-d}\sum_i c_i $$ 

where $L$ is the linear extent of the lattice and $d$ its dimensionality.

The Hamiltonian of the lattice gas model is

$${\cal H}_{LG}=-\epsilon\sum_{<i,j>}c_ic_j - \mu\sum_ic_i$$

where $\epsilon$ is an attraction energy between a pair of particles on
adjacent (nearest neighbouring) sites and $\mu$ is a field known as the
chemical potential, which couples to the particle density which is
assumed to fluctuate around a mean value controlled by the prescribed
chemical potential. This representation of the model in which the
overall density fluctuates is known as the *Grand Canonical ensemble.*

## Mapping between Ising model and lattice gas

The lattice gas model is interesting because whilst being a plausible
model for a fluid, it maps onto the Ising model. This extends the
applicability of the Ising model. To expose the mapping we write the
grand partition function of the lattice gas:

$$ \Xi=\sum_\textrm{ state}\exp-\beta{\cal H}_{LG}=\sum_{\{c\}}\exp\left[\beta \epsilon\sum_{<i,j>}c_ic_j +\beta\mu\sum_ic_i\right] $$
where the sum is an unrestricted sum over the occupancies of the lattice
sites. We now change variables to

$$c_i=(1+s_i)/2; ~~~~ J=\frac{\epsilon}{4} ~~~~
h=\frac{\epsilon q+2\mu}{4}$$ Hence

$${\cal H}_{LG}={\cal H}_\textrm{ I} + \textrm{ constant}$$ Since the last term
does not depend on the configuration, it feeds through as an additive
constant in the free energy; and since all observables feature as
derivatives of the free energy, the constant has no physical
implications.

## Phase diagram

Using these translation rules we can plot the phase diagram of the
lattice gas in the number density-temperature plane.

 ![Phase diagram of the lattice gas model in the density-temperature plane.](Figs/lattice_gas_phase_diagram.png){width=60% fig-align="center" #fig-lattgas}

In the $\mu-T$ plane there is a line of first order phase transitions terminating at a
critical point. The first order line means that if $T<T_c$ we smoothly
increase the chemical potential through the coexistence value of $\mu$,
the density of particles on our lattice $\rho=N/L^d$ jumps discontinuously from a
low to a high value.

$$\rho_\textrm{ gas}=\frac{1-m^\star}{2} \to \rho_\textrm{ liquid}=\frac{1+m^\star}{2}
$$
These values merge at $T_c$, the gas-liquid critical point. At higher
temperatures, the distinction between the phases disappears.

## Real Fluids

You may wish to compare this with the results of (say) van der Waals
equation (see recommended textbooks for the required phase diagram). The
main difference is that the lattice gas has so-called "particle-hole" symmetry,
$\rho\to 1-\rho$ (inherited from the up-down symmetry of the Ising model) which is not present for a real fluid. Accordingly, the
phase diagram in a real fluid looks like a lopsided version of the above
picture as shown in @fig-realfluid. See [here](https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.55.2160) for some real experimental data showing the asymmetry of the coexistence curve in liquid metals.

![Schematic of the liquid-gas phase diagram in the $\rho-T$ plane for a realistic fluid .](Figs/real_fluid_pd_schem.png){width=60% fig-align="center" #fig-realfluid}