# Mean field theories for fluids

Ideal gases are exceedingly idealised systems and are not suited to describe the behaviour of real systems: they always obey the same state equation and never undergo phase transitions (for example they never condense). We must therefore step a little further: using the "philosophy" of mean field theories we can make the description of fluids a little bit more realistic. As we will see this will also lead to the derivation of the Van der Waals equation, which better describes the behaviour of real fluids (even if, as we will shortly see, it still has some problems).

In general, in a real gas all the atoms or molecules interact through a certain potential ${\displaystyle \Phi (\lbrace {\vec {r}}_{i}\rbrace )}$ that will depend on the positions of all the particles. For a system of ${\displaystyle N}$ particles the configurational contribution to the partition function will therefore be:

${\displaystyle Q_{N}=\int \prod _{i=1}^{N}d{\vec {r}}_{i}e^{-\beta \left(\sum _{i=1}^{N}\varphi _{\text{ext}}(\lbrace {\vec {r}}_{i}\rbrace )+\Phi (\lbrace {\vec {r}}_{i}\rbrace )\right)}}$
where ${\displaystyle \varphi _{\text{ext}}}$ is an external potential and in general:
${\displaystyle \Phi (\lbrace {\vec {r}}_{i}\rbrace )=\sum _{i\neq j}U_{2}({\vec {r}}_{i},{\vec {r}}_{j})+\sum _{i\neq j\neq k}U_{3}({\vec {r}}_{i},{\vec {r}}_{j},{\vec {r}}_{k})+\cdots }$
(where ${\displaystyle U_{n}}$ can be a generic ${\displaystyle n}$-body interaction potential). Generally ${\displaystyle \varphi _{\text{ext}}}$ does not pose great problems while it is ${\displaystyle \Phi }$ that makes ${\displaystyle Q_{N}}$ impossible to compute exactly, forcing us to resort to approximations. In the framework of mean field theories we substitute the interaction potential ${\displaystyle \Phi }$ with an effective single-particle potential${\displaystyle \varphi ({\vec {r}}_{i})}$ that acts on every particle in the same way: ${\displaystyle \Phi (\lbrace {\vec {r}}_{i}\rbrace )\approx \sum _{i}\varphi ({\vec {r}}_{i})}$. Therefore, neglecting the external term for the sake of simplicity, mean field theories allow us to compute ${\displaystyle Q_{N}}$ as:
${\displaystyle Q_{N}=\left(\int d{\vec {r}}e^{-\beta \varphi ({\vec {r}})}\right)^{N}}$
Of course, every particular mean field theory will provide a different form of ${\displaystyle \varphi }$, which will lead to different results.