# Ising model and fluids

As we have alreadt stated, the Ising model can also be used to describe systems different from magnets. The most important example is the correspondence that can be established between an Ising model and a fluid through a *lattice gas* model. We will now show how this equivalence can be defined; in order to do so, we will proceed in two steps: we will first show how an Ising model is equivalent to a lattice gas, and then show (qualitatively) that this lattice gas model is equivalent to the classical model for a fluid.
Before doing so, we briefly review the formalism used in classical statistical mechanics in order to describe fluids (see The canonical ensemble at work, for example).

Let us then consider a system of particles subjected to some generic potentials, so that its Hamiltonian can be written as:

^{[1]}:

^{[2]}). It is convenient to separate the contributes due to the kinetic and configurational terms:

*thermal wavelength*. Therefore:

^{[3]}:

## Ising model and lattice gas[edit | edit source]

The basic idea of the lattice gas model is to describe a fluid where the particles are located on the sites of a lattice instead of continuously occupying any position in space; it is a sort of "discretization" of the classical description of fluids. The correspondence with the Ising model is established relating the local density of a fluid with the local magnetization density of an Ising model.

Let us therefore consider a -dimensional lattice with coordination number . Each site of the lattice can be occupied by a particle, so if we call the occupation number of a site we will have either or , and the total number of particles will be:

^{[4]}. From now on we will neglect any potential that couples more than two particles; therefore, the first term of expression of becomes:

^{[5]}:

## Lattice gas and continuous fluids[edit | edit source]

We now want to show (although not really rigorously) that the lattice gas model can be derived from the "classical" model of a fluid. Consider the configurational sum:

*where*we can find a particle, but not

*which*specific particle is in a given cell; furthermore, the system is left unchanged if we interchange the particles. We can thus write:

^{[6]}is physically irrelevant, so and are equivalent

^{[7]}. Therefore, the grand partition functions of the continuum fluid and the lattice gas model are the same: we thus see that these two models are equivalent.

- ↑ This of course is valid for classical mechanics, but the correspondence we will establish holds also in quantum statistical mechanics.
- ↑ The only big difference with The canonical ensemble at work is that we are considering generic instead of equal to 3.
- ↑ A small remark: for finite and "reasonable" systems the grand free energy is not singular even if it involves an infinite sum over . The reason is that generally (in the "reasonable" cases we have just mentioned) the interaction potentials have a hard-core component that prevents the particles from overlapping: therefore, a finite system will be able to contain only a finite number of particles, so that the sum has in reality an upper limit and is not infinite.
- ↑ Of course, if two systems have the same partition function, their thermodynamics will coincide, so they are at all effects equivalent.
- ↑ From these definition we see that in reality the precise values of the coefficients , , that we have encountered are absolutely irrelevant.
- ↑ Remember that our system is at fixed temperature, so is constant.
- ↑ Equivalently, we can note that , so that we also have : the grand partition functions of the fluid and the lattice gas differ for a constant rescaling factor, similarly to what we have seen before.