\dots d{\bm z}_N F_N\left({\bm z}_1, \dots, {\bm z}_N, t \right) = 1.
\label{normalisation}\end{aligned}$$ Its time dynamics is governed by the Master equation [@Balescu1975] $$\begin{aligned}
\partial_{t} F_{N} & + \sum_{i=1}^{N} \Big[ \nabla_{i}^{\alpha}(\dot{x}_{i}^{\alpha} F_{N}) + \partial_{i}^{\alpha}(\dot{p}_{i}^{\alpha} F_{N}) \Big] \nonumber \\
& = - N \lambda F_{N} +  \frac{\lambda}{4\pi} \sum_{i=1}^{N} \int \mathrm{d} {\bm p}_{i} F_{N},
\label{Liouville}\end{aligned}$$ where we introduced $\partial_{i}^{\alpha} = \mathbb{P}_{i}^{\alpha\beta} \partial/\partial p^\beta_i$. The l.h.s. of Eq. describes the probability fluxes to and from a particular point in the phase space due to the deterministic particle dynamics given by Eqs. and , while the r.h.s. gives the changes of the probability due to random tumbling from and into that phase space point [@Subramanian2009; @Koch2011]. Next, we introduce the $s$-particle correlation functions defined as $$\begin{aligned}
&F_{s}\left({\bm z}_1,\dots, {\bm z}_s, t \right) = \frac{N!}{(N-s)!N^{s}} \nonumber \\
&\qquad\qquad \times \int d{\bm z}_{s+1} \dots d{\bm z}_N F_N\left({\bm z}_1, \dots, {\bm z}_N, t \right), 
\label{Fs}\end{aligned}$$ Below, we will only be interested in the first partial correlation functions $F_1$, $F_2$, and $F_3$, that we further express as $$\begin{aligned}
F_2\left( {\bm z}_1, {\bm z}_2, t\right) = F_1\left( {\bm z}_1, t\right) F_1\left( {\bm z}_2, t \right) +  G\left( {\bm z}_1, {\bm z}_2, t\right),
\label{F2}\end{aligned}$$ and $$\begin{aligned}
F_3\left( {\bm z}_1, {\bm 