in Eq. is then proportional to the total flow rate through a surface surrounding the dipole. The latter is zero due to incompressibility. Fluctuations around the homogeneous and isotropic state are then governed by Eq. with $F_1=1/(4\pi V)$, and $\mathcal{S}_{i,j}^{G}$ and $\mathcal{S}_{i,j}^{H}$ neglected $$\begin{aligned}
& \partial_t G({\bm z}_1,{\bm z}_2, t) + {\mathcal L}_{12} [G] + {\mathcal L}_{21} [G] \nonumber \\
& \qquad\qquad = \frac{3B}{\left(4\pi V\right)^2} \Big\{ p_1^\mu p_1^\nu \nabla_1^\mu u_d^\nu({\bm x}_1;{\bm z}_2) \nonumber \\
& \qquad\qquad\qquad\qquad\qquad+ p_2^\mu p_2^\nu \nabla_2^\mu u_d^\nu({\bm x}_2;{\bm z}_1) \Big\},
\label{EqGfinal}\end{aligned}$$ where $$\begin{aligned}
& {\mathcal L}_{ij} [G] = 
v_s p_i^\alpha \nabla_i^\alpha G({\bm z}_1,{\bm z}_2, t) \nonumber \\
&\qquad\qquad - \frac{3 n B}{4\pi} p_i^\mu p_i^\nu \int d{\bm z}' G({\bm z}_j,{\bm z}', t) \nabla_i^\mu u_d^\nu({\bm x}_i;{\bm z}') \nonumber \\
& \qquad\qquad + \lambda G({\bm z}_1,{\bm z}_2, t)
- \frac{\lambda}{4\pi} \int d{\bm p}_i\, G({\bm z}_1,{\bm z}_2, t).
\label{Lij}\end{aligned}$$ This equation has previously been derived and analysed for the case of shakers ($v_s=0$) [@Stenhammar2017]. Here, we solve it in the general case $v_s>0$.

Phase-space density fluctuations
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While the two-point distribution function $G$, given by Eq., contains statistical information about fluctuations in the system, it is not straightforward to relate it to the spatial and temporal correlation function $C(R,T)$, Eq., that 