Three-body correlation function in kinetic theory In Kinetic Theory, one studies the evolution of a system of $N$ particles interacting with each other. We use the notation $\boldsymbol{w}_{i}$ to describe the coordinates in phase-space of each particle. Liouville's Equation gives the incompressible evolution of the $N-$Body distribution function
$$f^{(N)} (\boldsymbol{w}_{1},...,\boldsymbol{w}_{N})$$
One can then define the reduced distribution function $f_{i}$ , for $1 \leq i \leq N$ as
$$f_{i} (\boldsymbol{w}_{1},...,\boldsymbol{w}_{i}) = \int d \boldsymbol{w}_{i+1} ... \boldsymbol{w}_{N} \; f^{(N)} (\boldsymbol{w}_{1},...,\boldsymbol{w}_{N}) $$
A system is said to be separable, if we have $f^{(N)} (\boldsymbol{w}_{1},...,\boldsymbol{w}_{N}) = \prod\limits_{i = 1}^{N} f_{1} (\boldsymbol{w}_{i})$.
In order to measure the $\textit{distance}$ to separability, one first introduces the two-body correlation function $g_{2} (\boldsymbol{w}_{1} , \boldsymbol{w}_{2})$ defined as
$$f_{2} (\boldsymbol{w}_{1},\boldsymbol{w}_{2}) = f_{1} (\boldsymbol{w}_{1}) \, f_{1} (\boldsymbol{w}_{2}) + g_{2} (\boldsymbol{w}_{1},\boldsymbol{w}_{2}) $$
This two-body correlation function plays a crucial role in the collision term appearing in the first equation of the BBGKY hierarchy.
My question is : What is the definition of the three-body correlation function $g_{3} (\boldsymbol{w}_{1},\boldsymbol{w}_{2},\boldsymbol{w}_{3})$ ?
My feeling would be that it is defined as
$$ \begin{aligned}
f_{3} (\boldsymbol{w}_{1},\boldsymbol{w}_{2},\boldsymbol{w}_{3}) = &f_{1} (\boldsymbol{w}_{1}) \, f_{1} (\boldsymbol{w}_{2}) \, f_{1} (\boldsymbol{w}_{3})
  \\& + \; f_{1}(\boldsymbol{w}_{1}) \, g_{2} (\boldsymbol{w}_{2},\boldsymbol{w}_{3}) + f_{1} (\boldsymbol{w}_{2}) \, g_{2} (\boldsymbol{w}_{1},\boldsymbol{w}_{3}) + f_{1} (\boldsymbol{w}_{3}) \, g_{2} (\boldsymbol{w}_{1},\boldsymbol{w}_{2}) \\& + \; g_{3} (\boldsymbol{w}_{1},\boldsymbol{w}_{2},\boldsymbol{w}_{3})
\end{aligned}$$
Is this definition correct ?
Then, would you have a nice expression for the general $n-$body correlation function $g_{n} (\boldsymbol{w}_{1},...,\boldsymbol{w}_{n})$ ?
 A: In order to generalize the nice writing of the $3-$body correlation function, we may define recursively the $n-$body correlation function $g_{n}$ as
$$
f_{n} (1,..,n) = \sum_{k=0}^{n} \frac{1}{k!(n\!-\!k)!} \sum_{\sigma \in \mathcal{S}_{n}} f_{1} (\sigma [1]) \, ... \, f_{1} (\sigma [k]) \, g_{n-k} (\sigma [k\!+\!1],...,\sigma [n]) \, , 
$$
where $\mathcal{S}_{n}$ is the set of all permutations of ${1,...,n}$. The prefactor in ${ 1/ (k! (n\!-\!k)!) }$ ensures the correct normalization of the function. Indeed, we rely on the fact the correlation functions $g_{k}$ are invariant under the permutations of its arguments. In the sum $\sum_{k=0}^{n}$, the case ${k=n}$ corresponds to the case of separability for which ${f_n = f_{1} ...f_{1}}$. As $k$ decreases, we are adding higher order correlation terms.
A: Yes, your three-body expression is consistent with the BBGKY hierarchy. Regarding $n$-body correlations, there may be a way to express the collision integrand in terms of $g$ and $f$ functions, but I do not know it and can not remember ever seeing one. That said, I'll heartily upvote any answer that manages to write it out.
Most treatments of the problem opt for a diagrammatic representation, which tends to communicate the ideas behind many-body collision integrals in a more compact (if initially difficult-seeming) notation. I found this note to be a digestible introduction to this approach. I can provide additional references in comments if this turns out to be what you are looking for.
