z}_2, {\bm z}_3, t\right) = F_1\left( {\bm z}_1, t\right) F_1\left( {\bm z}_2, t \right) F_1\left( {\bm z}_3, t \right) \nonumber \\
+ G\left( {\bm z}_1, {\bm z}_2, t\right) F_1\left( {\bm z}_3, t \right) + G\left( {\bm z}_1, {\bm z}_3, t\right) F_1\left( {\bm z}_2, t \right) \nonumber \\
+ G\left( {\bm z}_2, {\bm z}_3, t\right) F_1\left( {\bm z}_1, t \right) + H\left( {\bm z}_1, {\bm z}_2, {\bm z}_3, t\right),
\label{F3}\end{aligned}$$ where $G$ and $H$ are the irreducible (connected) correlation functions [@Balescu1975]. The time evolution of $F_s$ can be deduced from the Master equation by integrating it over $\left\{{\bm z}_{s+1},\dots,{\bm z}_N\right\}$. Integrating by parts and using Eqs. and , we obtain the following equations for the one- and two-particle irreducible correlation functions

$$\begin{aligned}
& \partial_t F_1({\bm z}, t) + {\mathcal L}[F_1({\bm z}, t)]({\bm z}) \nonumber \\
& \qquad\qquad = -N \nabla^\alpha \int d{\bm z}' G({\bm z},{\bm z}',t) u^\alpha_d ({\bm x}; {\bm z}')  
- N \mathbb{P}^{\alpha\beta} \frac{\partial}{\partial p^\beta} \int d{\bm z}' G({\bm z},{\bm z}',t)
p^\gamma \mathbb{X}^{\alpha\mu\nu\gamma} \nabla^\mu u^\nu_d ({\bm x}; {\bm z}'), 
\label{EqF1}
\\ 
\nonumber \\
& \partial_t G({\bm z}_1,{\bm z}_2, t) + {\mathcal L}[G({\bm z}_1,{\bm z}_2, t)]({\bm z}_1) + {\mathcal L}[G({\bm z}_1,{\bm z}_2, t)]({\bm z}_2) \nonumber \\
& \qquad \qquad + N 