Two-particle number density ($n_2(\mathbf{x},{\mathbf{x}}_{\star },t)$)

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.