The Liouville equation of motion is written in terms of an $N$ particle distribution $f_N$.
\begin{equation}
\frac{\partial f_N}{\partial t}=\{H,f_N\}
\end{equation}
Where $\{\cdot ,\cdot \}$ is the Poisson bracket and $f_N=f_N(q_1,\dots ,q_N,p_1,\dots ,p_N)$. Let us now define an $n$ particle probability distribution function $f_n$ with $n< N$.
\begin{equation}
f _n(q_1,\dots,q_n,p_1,\dots ,p_n,t)=\frac{N!}{(N-n)!}\int \prod ^N_{i=n+1}d q^id p_if_{N+1}( q_1,\dots ,q_{N+1},p_1 ,\dots,p_{N+1},t)
\end{equation}
Now $f_n$ satisfies,
\begin{equation}
\frac{\partial f _n}{\partial t}=\{H_n,f_n\}+\sum ^n_{i=1}\int dq_{n+1}dp_{n+1}\frac{\partial U(q_i-q_{n+1})}{\partial q_i}\frac{\partial f _{n+1}}{\partial p_i}\ \ \ \ \ \ \ \ \ \ \ \ \ \ (*)
\end{equation}
With the $n$-body Hamiltonian $H_n$,
\begin{equation}
H_n=\sum ^n_{i=1}\bigg(\frac{p_i^2}{2m}+U(q_i)\bigg)+\sum _{i<j\leq n}U(q_{ij})
\end{equation}
And $q_{ij}=q_i-q_j$. Here $(*)$ is the BBGKY hierarchy. I am reading out of the following notes.

Despite reading the linked notes and wikipedia pages etc I am struggling to understand how the BBGKY hierarchy is related to the Liouville equation. In particular taking $n=N$ does not (to my naive understanding) regenerate the Liouville equation. Why do we not require $f_{N+1}$ in the Liouville equation by the logic of the $n$ particle distribution function? Lastly is the Boltzmann equation defined for $f_N$ or $f_1$ (or is it irrelevant, the equation holding in any case?).

Any help on the BBGKY formalism is appreciated so much!

This post imported from StackExchange Physics at 2015-08-27 17:51 (UTC), posted by SE-user RedPen