Two-particle distribution function ($f_2(z,z_{\star },t)$)

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.

The spatial marginal $n_2=\int f_2{d}^{3}v{d}^3{v}_{\star }$ of a number-normalized two-particle distribution function. It counts ordered distinct pairs, so its integral over both positions is $N(N-1)$. Exchange symmetry gives $n_2(\mathbf{x},{\mathbf{x}}_{\star })=n_2({\mathbf{x}}_{\star },\mathbf{x})$. In a homogeneous isotropic fluid it is $n^{2}g(|{\mathbf{x}}_{\star }-\mathbf{x}|)$, where $g$ is the radial distribution function.