The velocity of the fluid at a position $\bm x$ is given by the superposition of the velocity fields generated by all swimmers $$\begin{aligned}
&U^{\alpha}\left( {\bm x}, t \right) = \sum_{i=1}^N u_d^\alpha({\bm x}; {\bm z}_i(t)) \nonumber \\
&\qquad\qquad = \int d{\bm z}' \varphi({\bm z}',t) u_d^\alpha({\bm x}; {\bm z}'). \end{aligned}$$ Separating the phase space density into its average and fluctuations, $\varphi =\overline\varphi +  \delta \varphi$, the spatial correlation function becomes $$\begin{aligned}
&C(R) = \lim_{t\rightarrow\infty} \frac{1}{V}\int d{\bm x} \int d{\bm z}'d{\bm z}''  u_d^\alpha({\bm x}; {\bm z}') u_d^\alpha({\bm x}+{\bm R}; {\bm z}'')  \nonumber \\
& \qquad\qquad\qquad \times  \Bigg[ \left(\frac{n}{4\pi}\right)^2 + G_K( {\bm z}', {\bm z}'', t)\Bigg].\end{aligned}$$ The integral with the constant term vanishes, demonstrating that $G_K$ fully determines the spatial correlation function.

Time evolution of the Klimontovich correlation function can readily be derived from Eqs. and , yielding $$\begin{aligned}
& \partial_t G_K({\bm z}_1,{\bm z}_2, t) + {\mathcal L}_{12} [G_K] + {\mathcal L}_{21} [G_K] \nonumber \\
& \qquad\qquad = 
2 \lambda \frac{n}{4\pi} \delta({\bm x}_1 - {\bm x}_2 ) \left[ \delta({\bm p}_1 - {\bm p}_2 ) - \frac{1}{4\pi} \right],
\label{EqGK}\end{aligned}$$ where $ {\mathcal L}_{ij}$ is defined in Eq., and we used $F_1 = 1/(4\pi V)$ in the homogeneous and isotropic state. To solve Eq., we 