\nabla_1^\alpha \left[ F_1({\bm z}_1,t) \int d{\bm z}' G({\bm z}_2,{\bm z}',t) u_d^\alpha({\bm x}_1;{\bm z}')\right]
+ N \nabla_2^\alpha \left[ F_1({\bm z}_2,t) \int d{\bm z}' G({\bm z}_1,{\bm z}',t) u_d^\alpha({\bm x}_2;{\bm z}')\right]
\nonumber \\ 
& \qquad \qquad + N \mathbb{P}_1^{\alpha\beta} \frac{\partial}{\partial p_1^\beta}\left[  F_1({\bm z}_1,t) p_1^\gamma \mathbb{X}_1^{\alpha\mu\nu\gamma} \int d{\bm z}' G({\bm z}_2,{\bm z}',t)\nabla_1^\mu u_d^\nu({\bm x}_1;{\bm z}')\right]
\nonumber \\ 
& \qquad \qquad + N \mathbb{P}_2^{\alpha\beta} \frac{\partial}{\partial p_2^\beta}\left[  F_1({\bm z}_2,t) p_2^\gamma \mathbb{X}_2^{\alpha\mu\nu\gamma} \int d{\bm z}' G({\bm z}_1,{\bm z}',t)\nabla_2^\mu u_d^\nu({\bm x}_2;{\bm z}')\right] \nonumber \\
& \qquad \qquad =  - \mathcal{S}_{1,2}^{F} - \mathcal{S}_{2,1}^{F} -  \mathcal{S}_{1,2}^{G} - \mathcal{S}_{2,1}^{G} - \mathcal{S}_{1,2}^{H} - \mathcal{S}_{2,1}^{H},
\label{EqG}\end{aligned}$$

where we have introduced the operator $$\begin{aligned}
&{\mathcal L}[\Phi]({\bm z}) \nonumber \\
& \qquad =
v_s p^\alpha \nabla^\alpha \Phi({\bm z}) 
+ N \nabla^\alpha \big[ \Phi({\bm z}) \mathcal{U}^\alpha_{\text{MF}}({\bm x})\big] 
+ N \mathbb{P}^{\alpha\beta} \frac{\partial}{\partial p^\beta}\big[ \Phi({\bm z}) p^\gamma \mathbb{X}^{\alpha\mu\nu\gamma} \nabla^\mu \mathcal{U}^\nu_{\text{MF}}({\bm x})\big] 
+ \lambda \Phi({\bm z})
- \frac{\lambda}{4\pi} \int d{\bm p}\, \Phi({\bm z}),
\label{operatorL}\end{aligned}$$

acting on the variable $\bm z$ of an arbitrary function $\Phi=\Phi({\bm z}_1,\dots,{\bm z}_N)$, and defined the mean-field velocity field as $$\begin{aligned}
 \mathcal{U}^\alpha_{\text{MF}}({\bm x}) = \int d{\bm z}' F_1({\bm z}',t) u^\alpha_d({\bm x}; {\bm z}').
 \label{UMF}\end{aligned}$$ The rank-4 tensor $$\begin{aligned}
\mathbb{X}_i^{\alpha\mu\nu\gamma} = \mathbb{P}_i^{\alpha\beta} \left[ \frac{B+1}{2}\delta^{\mu\gamma}\delta^{\nu\beta} + \frac{B-1}{2}\delta^{\mu\beta}\delta^{\nu\gamma} \right],\end{aligned}$$ encodes the tensorial structure of Jeffrey’s equation , and 