 Two-particle distribution function

For $N$ identical particles with symmetric, unit-normalized phase-space distribution function $D_N$, the number-normalized two-particle marginal is $f_2(z_1,z_2)=N(N-1)\int D_{N}d{z}_3\cdots d{z}_N$. Its integral over $z_2$ is $(N-1)f(z_1)$ and its full integral is $N(N-1)$. The factorial convention counts ordered distinct pairs and absorbs the combinatorial coefficient in the first BBGKY hierarchy equation. It does not imply statistical independence of the pair.