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The Ising model can be used to simulate other systems besides ferromagnets; examples include anti ferromagnets, binary alloys, and even fluids. The Ising model of a fluid is called a lattice gas. We imagine that space is divided into a lattice of sites, each of which can be either occupied by a gas molecule or unoccupied. The system has no kinetic energy, and the only potential energy comes from interactions of molecules on adjacent sites. Specifically, there is a contribution of $-u_0$to the energy for each pair of neighbouring sites that are both occupied.

Hamiltonian of this system is:

$H=-u_0\underset{鉄�i,j鉄�}{鈭�}{n}_i{n}_j$

Hamiltonian of in the Ising model is:

$H_I=-系\underset{鉄�i,j鉄�}{鈭�}{s}_i{s}_j-2{渭}_{0}B鈭�s$

Because there can only be one particle in a single lattice site, n_i =0,1, and it can have spin s₁ = -1 or s₁ = 1, we can write following relation for the occupation of a lattice site and the spin:

$s_i=2n_i-1n_i=\frac{s_i+1}{2}$

Plugging this expression into the first Hamiltonian, we get:

$$\begin{gathered} H=-u_0\underset{鉄�i,j鉄�}{鈭�}\frac{s_i+1}{2}路\frac{s_j+1}{2} \\ =\frac{-u_0}{4}\underset{鉄�i,j鉄�}{鈭�}\left(s_i+1\right)\left(s_j+1\right) \\ =\frac{-u_0}{4}\underset{鉄�i,j鉄�}{鈭�}\left(s_i{s}_j+s_i+s_j+1\right) \\ =\frac{-u_0}{4}\underset{鉄�i,j鉄�}{鈭�}{s}_i{s}_j-\frac{u_0路尉}{2}鈭�s_i+\text{const} \end{gathered}$$

$尉$is the number of nearest neighbours, and for a square lattice all sites have the same number of neighbours: $尉=4$

By comparison with the Hamiltonian of the Ising model, we can see that$系=u_0/4\text{or}{u}_0=4系$

Grand canonical Hamiltonian is:

$$\begin{gathered} H-渭N=\frac{-u_0}{4}\underset{鉄�i,j鉄�}{鈭�}{s}_i{s}_j-2u_0鈭�s_i+\text{const.}-渭鈭�s_i \\ H-渭N=\frac{-u_0}{4}\underset{鉄�i,j鉄�}{鈭�}{s}_i{s}_j-\left(渭+2u_0\right)鈭�s_i+\text{const} \end{gathered}$$

Where, $渭$is chemical potential

Now we can see that we will get the Hamiltonian of the Ising model if we Use substitutions: $u_0=4系and2{渭}_{0}B=渭+8系$

Rewriting the equation:

$H-渭N=H_1-\left(渭+2u_0\right)N_L$

Where N_L is the total number of lattice sites

Grand partition function is:

$Z_N=鈭�\exp [-尾(H-渭N\left)\right]$

Canonical partition function of the Ising model is:$Z_1=鈭�\exp \left(-尾H_1\right)$. So these two partition functions are related:

$$\begin{gathered} Z_N=鈭�\exp [-尾(H-渭N\left)\right] \\ {Z}_N=Z_1路\exp \left[尾\left(渭+2u_0\right)N_L\right] \end{gathered}$$

If the relation between occupation of the lattice site and the spin is:

$s_i=2n_i-1$

We can see that for n_i = 0 (empty site) we will have s₁= -1. So, magnetic state of s_i = -1 will correspond to the low density of lattice gas.