# Mean field theories for weakly interacting systems

If a system is composed of weakly interacting particles we can use perturbative methods to compute the partition function of such systems. For example, consider a fluid composed of particles in a region of volume , interacting through a generic two-body potential that depends only on the relative distance between the particles:

^{[1]}:

## Virial and cluster expansion[edit | edit source]

The *virial expansion* is a systematic approach that can be used to incorporate the corrections due to the interactions between the particles, and as we will shortly see it can be obtained from a much general method, the *cluster expansion*.
The virial expansion consists in expanding the thermodynamic quantities of a system in powers of the density ; for example, the virial expansion for the pressure of a gas is:

*virial coefficients*, and in general they can depend on the temperature. The virial expansion is very useful because the coefficients can be experimentally measured (for example, in the case of the pressure they can be determined by properly fitting the isotherms of a system), and as we will see they can be related to microscopic properties of the interparticle interaction. Let us see for example the virial expansion of the Van der Waals equation. From Van der Waals equation we have:

*Boyle temperature*; in this case .

Now, let us see how the cluster expansion works and how we can obtain the virial expansion from it. Of course, we start from the general configurational partition function:

*Mayer function*:

*defined*the first virial coefficient. Of course, we should have set equal to a generic coefficient, but in the end we should have found exactly that this coefficient is : we have done this for the sake of simplicity. Therefore:

What we have seen now is how the cluster expansion works in general. Let us now apply it in order to find the virial expansion for real gases.
From what we have found, the configurational partition function of the system becomes:

^{[2]}:

## Computation of virial coefficients for some interaction potentials[edit | edit source]

Let us now see this method in action by explicitly computing some coefficients for particular interaction potentials.

### Hard sphere potential[edit | edit source]

As a first trial, we use a hard sphere potential similar to the one we have seen for the derivation of the Van der Waals equation:

### Square well potential[edit | edit source]

We now use a slight refinement of the previous potential:

### Lennard-Jones potential[edit | edit source]

This potential is a quite realistic representation of the interatomic interactions. It is defined as:

- ↑ We can always insert a multiplicative correction. We could have also written it as an additive correction, but the core of the subject doesn't change.
- ↑ Referring to what we have stated previously, if we set with a generic constant, then we should have found:
and proceeding like we have done now, in the end:and so we see that indeed .