Derivation of Higher-order correlation functions from definition

Question

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

Answer 1

The delta functions mean that you are fixing the values of any two of the $N$ vectors $r_k$. By permutation symmetry of the potential, you may take these always to be $r_1$ and $r_2$, and you get $N(N-1)$ times the same integral. Hence (using the definition of the expectation in terms of the canonical ensemble) your first expression for $\rho^{(2)}$ and your second expression say exactly the same.

This doesn't help you to evaluate the remaining integral, though. As $N$ is large, this is tough and can be done only approximately, using expansions such as those discussed in the statistical mechanics book by Reichl, which I'd recommend for further reading.

Comments

Thank you for the clarification. As far as I understand it for now, the Dirac delta function only light up for the right value of \((r_i,r_j)\)  and the expectation value \(\langle \cdot \rangle\)hides the normalized integral on all remaining \(r\). But at some point the "selected" values of the potential should come out from this expression ? In order to numerically compute this thing from a distribution of particles, don't we have to say at some point that the value of the potential doesn't really matter and can be $U_N = 0$ ? 

By similar construction, I would guess that the expression for \(\rho^{(3)}\) is 

\(\rho^{(3)}(r_1,r_2,r_3) =\langle \sum_i \sum_j'\sum_k'' \delta(r_i-r_1)\delta(r_j-r_2) \delta(r_k-r_3) \rangle\)

(not sure about the " convention, of $k$ not being equal to $i$ nor $j$ though ...) 

Anyway I'll give a look to the Reichl. For now, the Hansen McDonald (Theory of simple liquids) was far away beyond my capacities :) 

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The formulas to notice are

$$\langle f(r_1,...,r_n)\rangle= const \int dr_1...dr_n e^{-\beta U(r_1,...,r_n) } f(r_1,...,r_n)$$

and 

$$\int dr_i dr_k f(r_i,r_k)\delta(r_i-r_k)= \int dr_i f(r_i,r_i)$$ 

This turns the $n$-fold integral for the canonical expectation values into an $(n-2)$-fold integral without delta-terms.

The values of the potential do matter, of course, and determine the value of the pair correlation function.