Lattice Gas

The goal of this section is to show that with a simple change of variables, the Ising model can be mapped into the lattice gas which is a simple model of density fluctuations and liquid-gas transformations.

The 1-dimensional lattice gas model is described by the following diagram:

The lattice divides space into cells $c_1, c_2, ..., c_N$. Each cell has an occupation number $n_j=0, 1$. The interaction between particles occupying the cells is modeled by assuming that the energy associated with a pair of occupied nearest neighbor cells is $-\epsilon n_j n_k$ and the total energy of the system is

$$E=-\mu \sum_j n_j -\epsilon \sum_j \sum_k n_j n_k,$$ (297)

where $j$ is the index of the cell and $\mu$ is the chemical potential of a particle.

The partition function of the lattice model is

$$\Xi = \sum_{n_1=0}^1 \sum_{n_2=0}^1 ... \sum_{n_N=0}^1 e^{\beta \mu \sum_{j=1}^N n_j + \beta \epsilon \sum_j \sum_k n_j n_k}.$$ (298)

In order to show the correspondence between the lattice gas and the Ising model, we make the variable transformation $n_j \equiv (S_j + 1)/2$ and we obtain

$$\Xi = \sum_{S_1=-1}^1 \sum_{S_2=-1}^1 ... \sum_{S_N=-1}^1 e^{\bet......u}{2}\sum_j(S_j +1) + \beta \frac{\epsilon}{4} \sum_j \sum_k (S_j +1)(S_k+1) },$$ (299)

Therefore the lattice model is isomorphic with the Ising model: ``Spin up'' in the Ising model corresponds to an occupied cell in the lattice model, ``spin down'' corresponds to an empty cell, the magnetic field in the Ising model corresponds (within constants) to the chemical potential in the lattice gas and the coupling constant in the Ising model is $\epsilon/4$ in the lattice gas.

The Ising model can also be mapped into many other problems in Chemistry and beyond, ranging from models of population dynamics to models of the brain.