Why does the 𝛿 (delta) term exist in the density-density correlation for simple liquids?

Question

Cross-posted from the PhysicsSE.

I am reading Theory of Simple Liquids by Hansen and McDonald, and they in chapter 3, they describe the density-density correlation for a simple liquid in the grand canonical ensemble. This is how they have defined it: $$H^{(2)}(r,r') = \langle [\rho (r) - \langle \rho (r) \rangle ][ \rho (r') - \langle \rho (r') \rangle ]\rangle \tag{1}$$ $$ = \rho ^{(2)}(r,r') - \rho ^{(1)}(r)\rho ^{(1)}(r') + \rho ^{(1)}(r) \delta (r-r').\tag{2}$$ This is my understanding of where the terms in the RHS comes from. I understand that the two-particle density $\rho^{(2)}$ term arises from: $$\langle \rho (r) \rho (r') \rangle\tag{3}$$ in the RHS.

The second term in the RHS comes from expanding the the product and taking like terms together.

$$-\rho ^{(1)}(r)\rho ^{(1)}(r') = -\langle \rho(r) \langle \rho (r') \rangle \rangle - \langle \rho(r') \langle \rho (r) \rangle \rangle + \rho ^{(1)}(r) \rho ^{(1)}(r'),\tag{4}$$ where $\rho^{(1)}(r)$ is the average single-particle density at $r$, for a homogeneous liquid.

But I still do not get why that $\delta$-function exists there. Is the $\delta$-function just there to say that if the two particles are in the same spot ($r=r'$), the density correlation is... maximized? How does the $\delta$ fall out of the averaging, mathematically?

I would appreciate any advice you have for me!

Answer 1

While deriving this formula in Ch. 3 of your source, the author references equation 2.5.13, \begin{equation} \rho_N^{(2)}\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=\left\langle\sum_{i=1}^N \sum_{j=1}^N{}^{'} \delta\left(\mathbf{r}-\mathbf{r}_i\right) \delta\left(\mathbf{r}^{\prime}-\mathbf{r}_j\right)\right\rangle.\tag{2.5.13} \end{equation} The author calls $\rho^{(2)}\left(\mathbf{r},\mathbf{r}'\right)$ an analogue of the above.

Notice the second sum has a prime indicating it is missing a set of terms where $i = j$. Those missing terms are

\begin{equation} \left\langle\sum_{i=1}^{N}\delta\left(\mathbf{r} - \mathbf{r}_{i}\right)\delta\left(\mathbf{r}' - \mathbf{r}_{i}\right)\right\rangle = \left\langle\sum_{i=1}^{N}\delta\left(\mathbf{r} - \mathbf{r}_{i}\right)\right\rangle\delta\left(\mathbf{r} - \mathbf{r}'\right) =\rho^{(1)}\left(\mathbf{r} \right)\delta\left(\mathbf{r} - \mathbf{r}'\right), \end{equation} where the final equality comes from Eq. (3.1.2) in the source. Anytime $\mathbf{r} \neq \mathbf{r}'$, the whole thing vanishes due to the two coupled $\delta$-functions.