the r.h.s. of Eq. is given in terms of $$\begin{aligned}
 &\mathcal{S}_{i,j}^{F} = 
F_1({\bm z}_j,t)\Big\{ \nabla_i^\alpha \left[ F_1({\bm z}_i,t) u_d^\alpha({\bm x}_i;{\bm z}_j)\right] \nonumber \\
&+  \mathbb{P}_i^{\alpha\beta} \frac{\partial}{\partial p_i^\beta}\left[ F_1({\bm z}_i, t) p_i^\gamma \mathbb{X}_i^{\alpha\mu\nu\gamma} \nabla_i^\mu u_d^\nu({\bm x}_i; {\bm z}_j) \right] \Big\},\end{aligned}$$ $$\begin{aligned}
 & \mathcal{S}_{i,j}^{G} = 
\nabla_i^\alpha \left[ G({\bm z}_i,{\bm z}_j,t) u_d^\alpha({\bm x}_i;{\bm z}_j)\right] \nonumber \\
&+  \mathbb{P}_i^{\alpha\beta} \frac{\partial}{\partial p_i^\beta}\left[ G({\bm z}_i,{\bm z}_j,t) p_i^\gamma \mathbb{X}_i^{\alpha\mu\nu\gamma} \nabla_i^\mu u_d^\nu({\bm x}_i; {\bm z}_j) \right], \end{aligned}$$ and $$\begin{aligned}
 & \mathcal{S}_{i,j}^{H} = 
N \int d{\bm z}' \Big\{
\nabla_i^\alpha \left[ H({\bm z}_i,{\bm z}_j,{\bm z}',t) u_d^\alpha({\bm x}_i;{\bm z}')\right] \nonumber \\
&+  \mathbb{P}_i^{\alpha\beta} \frac{\partial}{\partial p_i^\beta}\left[ H({\bm z}_i,{\bm z}_j,{\bm z}',t) p_i^\gamma \mathbb{X}_i^{\alpha\mu\nu\gamma} \nabla_i^\mu u_d^\nu({\bm x}_i; {\bm z}') \right] \Big\}.\end{aligned}$$ Eqs. and are the beginning of a BBGKY hierarchy of equations for partial distribution functions [@Balescu1975]. As such, they do not form a closed system as they also depend on the three-particle irreducible distribution function $H$.

Before discussing our choice of closure for this system of equations, let us briefly review the predictions of the mean-field approximation to Eqs. and , which consists of neglecting all correlation functions beyond $s=1$. The remaining equation determines the mean-field approximation to the one-particle correlation function $$\begin{aligned}
\partial_t F_1^{\text{MF}}({\bm z}, t) + {\mathcal L}[F_1^{\text{MF}}({\bm z}, t)]({\bm z})  