we seek to calculate. To establish this connection, we introduce a method based on the phase space density $$\begin{aligned}
\varphi({\bm z},t) = \sum_{i=1}^N \delta({\bm z} - {\bm z}_i(t)),\end{aligned}$$ pioneered by Klimontovich [@klimontovich1967book]. Here, $\delta({\bm z})$ is the three-dimensional Dirac delta function. The average of the phase space density is related to $F_1$ as can be seen from $$\begin{aligned}
&\overline\varphi({\bm z},t) = \int d{\bm z}_1 \dots d{\bm z}_N  \sum_{i=1}^N \delta({\bm z} - {\bm z}_i) F_N({\bm z}_1, \dots, {\bm z}_N, t) \nonumber \\
& \qquad\qquad = N F_1({\bm z}, t),\end{aligned}$$ where we used Eq.. Fluctuations of the phase space density can formally be defined as $\delta \varphi = \varphi - \overline\varphi$, and their second moment is given by $$\begin{aligned}
& G_K( {\bm z}', {\bm z}'', t) \equiv \overline{\delta \varphi({\bm z}',t) \delta \varphi({\bm z}'',t)} \nonumber \\
& \qquad = N^2 G ( {\bm z}', {\bm z}'', t) + N F_1({\bm z}',t) \delta({\bm z}' - {\bm z}'').
\label{GK}\end{aligned}$$ Below, we refer to $G_K$ as the Klimontovich correlation function. Its utility is evident if one considers the spatial correlation function $C(R)$, defined in Eq. as $$\begin{aligned}
&C(R) = \lim_{t\rightarrow\infty} \frac{1}{V}\int d{\bm x} \overline{U^{\alpha}\left( {\bm x}, t \right) U^{\alpha}\left( {\bm x} + {\bm R}, t \right)}.\end{aligned}$$ 