2.1 Functional Integration

There are at least two ways to derive exact expressions for the excess free energy ${\rm\cal F}_{ex}[\rho]$.

• Integration in density space:
  
  Starting from some initial density $\rho_{i}(\vec{r})$ and integrating up to the final density $\rho(\vec{r})$ according to
  

  $$\rho_{\alpha} \equiv \rho(\vec{r};\alpha) = \rho_{i}(\vec{r... ... \right) = \rho_{i}(\vec{r})+ \alpha \Delta \rho(\vec{r})$$ (2.1)

  
  
  one finds, with
  

  $$\beta {\rm\cal{F}}_{ex}[\rho] = \beta {\rm\cal{F}}_{ex}[\rho... ...c{r} \; \Delta \rho (\vec{r}) c^{(1)}([\rho_{\alpha}];\vec{r})$$ (2.2)

  
  
  and a further integration over $c^{(2)}(\vec{r}_{1},\vec{r}_{2})$ the exact formula
  

  $$\beta {\rm\cal{F}}_{ex}[\rho] = \beta {\rm\cal{F}}_{ex}[\rho_{i}] -\int d\vec{r} \; \Delta \rho (\vec{r}) c^{(1)}([\rho_{i}];\vec{r})$$

  $$+ \int_{0}^{1} d\alpha (\alpha-1) \int d\vec{1}\int d\vec{2} \; \De... ... (\vec{1}) \Delta \rho (\vec{2}) c^{(2)}([\rho_{\alpha}];\vec{1},\vec{2})$$ (2.3)

  
  
  This expression serves as a starting point for density expansions of the functional ${\rm\cal F}[\rho]$. Introducing certain simplifications and approximating $c^{(2)}([\rho_{\alpha}];\vec{1},\vec{2})$ by a known function (such as Carnahan-Starling for hard spheres), one finds explicit expressions for ${\rm\cal {F}}_{ex}[\rho ]$.
  
  
• Integration in the Space of Pair Potentials:
  
  Starting point: system with the final density $\rho(\vec{r})$ but for a simple reference potential $u_{r}(\vec{i},\vec{j})$.
  
  Using the perturbative ansatz $u(\vec{i},\vec{j})=u_{r}(\vec{i},\vec{j})+u_{p}(\vec{i},\vec{j})$ and integrating via
  

  $$u_{\alpha}(\vec{i},\vec{j})= u_{r}(\vec{i},\vec{j})+\alpha (u_{p}(\vec{i},\vec{j}))$$ (2.4)

  
  
  one finds, noting the relation
  

  $$\frac{\delta {\rm\cal F}[\rho]}{\delta u(\vec{1},\vec{2})} =\frac{1}{2} \rho^{(2)}(\vec{1},\vec{2})$$ (2.5)

  
  
  the exact expression
  

  $$\beta {\rm\cal{F}}[\rho] = \beta {\rm\cal{F}}_{r}[\rho] +\fr... ...ho^{(2)}(u_{\alpha};\vec{1},\vec{2}) \; u_{p}(\vec{1},\vec{2})$$ (2.6)

  
  

In the theory of homogeneous fluids this formula served as a basis for perturbation theories such as Barker-Henderson and Weeks-Chandler-Andersen. In the case of inhomogeneous fluids the same idea may be used: start from a simpler pair potential and then ``switch on'' a perturbation - such as the long-ranged attractive part of the interaction.