I'm trying to understand the definition of the n-th order correlation function. My aim is to translate the math into a numerical implementation in order to compute the correlation function $g^{(n)}$ for a distribution of spherical particles with $n=\lbrace3,4,5,\ldots\rbrace$.

I'm going to present you what I have understood for now, in order for you to get an better insight of what I'm looking for. (To reduce the verbiage of the following, I over-abbreviated sometimes $\mathbf r_1, \mathbf r_2$ into $r$, sorry if it kills the readability)

I'm used to the radial distribution (or pair distribution function) that we call $g(r)$ (which should be called rigorously $g^{(2)}(r)$). I want to derive the expression of $g^{(2)}(r)$ from the definition of $g^{(n)}$ (because I hope that if I'm able to do it for $n=2$, I will be able to do it for any $n$ !)

So let's start with the definition of $g^{(n)}$ one can find in wikipedia or any stat mech book :

$$\rho^{(n)}(\mathbf{r}_{1}\, \ldots, \, \mathbf{r}_{n}) = \rho^{n}g^{(n)}(\mathbf{r}_{1}\, \ldots, \, \mathbf{r}_{n}) \, ,$$
with $\rho$ the particle number density and $\rho^{(n)}$ the probability of (all the permutations of) elementary configuration $(\mathbf{r}_{1}\, \ldots, \, \mathbf{r}_{n})$ :
$$ \rho^{(n)}(\mathbf{r}_{1}\, \ldots, \, \mathbf{r}_{n}) = \frac{N!}{(N-n)!} \frac{1}{Z_N} \int \cdots \int \mathrm{e}^{-\beta U_N} \, \mathrm{d} \mathbf{r}_{n+1} \cdots \mathrm{d} \mathbf{r}_N \, .$$
with $U_N(\mathbf{r}_{1}\, \ldots, \, \mathbf{r}_{N})$ the potential energy of the configuration and $Z_N$ the configurational integral, taken over all possible combinations of particle positions.

Rewriting the previous with $n=2$ leads to :

$$ \rho^{(2)}(\mathbf{r}_{1}\, \mathbf{r}_{2}) = N(N-1) \frac{1}{Z_N} \int \cdots \int \mathrm{e}^{-\beta U_N} \, \mathrm{d} \mathbf{r}_{3} \cdots \mathrm{d} \mathbf{r}_N \, .$$

From that starting point, I should be able to derive the expression for $g^{(2)}(\mathbf{r}_{1}\, \mathbf{r}_{2})$ but I have no clue about the origin of the Dirac Delta function that appears in the definition of $g^{(2)}(\mathbf{r}_{1}\, \mathbf{r}_{2})$ or $\rho^{(2)}(\mathbf{r}_{1}\, \mathbf{r}_{2})$

$$ \rho^{(2)}(\mathbf{r}_{1}\, \mathbf{r}_{2}) = \left\langle \sum_i\sum_j' \delta(\mathbf r_1 - \mathbf r_i) \delta(\mathbf r_2 - \mathbf r_j) \right\rangle $$
I think there is something appearing from the potential energy that was defined before, but I'm not able to understand exactly the origin.

Any help will be highly appreciated :) Thank you in advance !

(*Note* : related questions like "Use and understanding of higher-order correlation functions" is absolutely not helpful, and the reference on the wikipedia page leads to a paper that has nothing to deal with correlations functions IM(H)O...)

This post imported from StackExchange Physics at 2014-06-08 08:17 (UCT), posted by SE-user Pascail