## Appendix A Fourier Transform of the Mass Fluxes Fluctuation Correlations  
Here, we derive the Fourier transform of the fluctuating flux correlation (23), or  
\[ \begin{split}&\langle\delta\hat{\mathbf{j}}(\mathbf{k}_{1},\omega_{1}) \delta\hat{\mathbf{j}}(\mathbf{k}_{2},\omega_{2})\rangle=\\ &\frac{1}{(2\pi)^{4}}\int_{-\infty}^{\infty}d\mathbf{r}_{1}\int_ {-\infty}^{\infty}dt_{1}\int_{-\infty}^{\infty}d\mathbf{r}_{2}\int_{-\infty}^ {\infty}dt_{2}\langle\delta\hat{\mathbf{j}}(\mathbf{r}_{1},t_{1})\delta\hat{ \mathbf{j}}(\mathbf{r}_{2},t_{2})\rangle e^{i(\omega_{1}t_{1}-\mathbf{k}_{1} \cdot\mathbf{r}_{1})}e^{i(\omega_{2}t_{2}-\mathbf{k}_{2}\cdot\mathbf{r}_{2})}. \end{split}  \tag{11}\]  
Hence, all that is needed is to transform two delta functions \(\delta(t_{1}-t_{2})\) and \(\delta(\mathbf{r}_{1}-\mathbf{r}_{2})\) over the two times (\(t_{1}\) and \(t_{2}\)) and positions (\(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\)) variables, or [16]  
\[ \begin{split}&\frac{1}{(2\pi)^{4}}\int_{-\infty}^{\infty}d\mathbf{r}_{1} \int_{-\infty}^{\infty}dt_{1}\int_{-\infty}^{\infty}d\mathbf{r}_{2}\int_{- \infty}^{\infty}dt_{2}[2c\mathbf{l}\delta(t_{1}-t_{2})\delta(\mathbf{r}_{1}- \mathbf{r}_{2})]\times\\ &\quad\quad e^{i(\omega_{1}t_{1}-\mathbf{k}_{1}\cdot\mathbf{r}_{1 })}e^{i(\omega_{2}t_{2}-\mathbf{k}_{2}\cdot\mathbf{r}_{2})}=2c\mathbf{l}D \delta(\omega_{1}+\omega_{2})\delta(\mathbf{k}_{1}+\mathbf{k}_{2}).\end{split}  \tag{12}\]  
The mass fluxes fluctuation correlation result (12) is used to find the charge-potential fluctuation correlations and ionic concentration-fluctuation correlations as shown below. Appendix B Electrostatic Charge-Potential Correlations and the Electrolyte Contribution to the Osmotic Pressure  
Below, we provide details on the Fourier inversion of the potential-charge correlation expression (47) with respect to \(\omega_{1}\) and \(\mathbf{k}_{1}\). Starting with the frequency, we write  
\[ \begin{split}&\frac{1}{(2\pi)^{1/2}}\int_{-\infty}^{\infty}d \omega_{1}\langle\delta\hat{\mathbf{j}}(\mathbf{k}_{1},\omega_{1})\delta\hat{ \rho}_{\epsilon}(\mathbf{k}_{2},\omega_{2})\rangle e^{-it(\omega_{1}+\omega_{ 2})}=\\ &\quad\quad\quad-\int_{-\infty}^{\infty}d\omega_{1}\frac{4(ze)^{2 }cD\mathbf{k}_{1}\cdot\mathbf{k}_{2}\delta(\omega_{1}+\omega_{2})\delta( \mathbf{k}_{1}+\mathbf{k}_{2})e^{-it(\omega_{1}+\omega_{2})}}{(2\pi)^{1/2} \varepsilon\varepsilon_{0}k_{1}^{2}[i\omega_{1}+D(k_{1}^{2}+\kappa^{2})][i \omega_{2}+D(k_{2}^{2}+\kappa^{2})]}.\end{split}  \tag{13}\]  
The right-hand side can be further rearranged to read  
\[ \begin{split}-\frac{4(ze)^{2}cD\mathbf{k}_{1}\cdot\mathbf{k}_{2} }{(2\pi)^{1/2}\varepsilon\varepsilon_{0}k_{1}^{2}}\int_{-\infty}^{\infty}d \omega_{1}\frac{\delta(\omega_{1}+\omega_{2})\delta(\mathbf{k}_{1}+\mathbf{k} _{2})e^{-it(\omega_{1}+\omega_{2})}}{[i\omega_{1}+D(k_{1}^{2}+\kappa^{2})][i \omega_{2}+D(k_{2}^{2}+\kappa^{2})]}=\\ \frac{4(ze)^{2}cD\mathbf{k}_{1}\cdot\mathbf{k}_{2}}{(2\pi)^{1/2} \varepsilon\varepsilon_{0}k_{1}^{2}}\frac{\delta(\mathbf{k}_{1}+\mathbf{k}_{2} )}{[i\omega_{2}-D(k_{1}^{2}+\kappa^{2})][i\omega_{2}+D(k_{2}^{2}+\kappa^{2}) ]}.\end{split}  \tag{14}\]Next, we perform the inverse Fourier transform over the wave vectors starting with \(\mathbf{k}_{1}\)  
\[ \begin{split}\frac{4(ze)^{2}cD}{(2\pi)^{2}\varepsilon\epsilon_{0}} \int_{-\infty}^{\infty}d\mathbf{k}_{1}\frac{\mathbf{k}_{1}\cdot\mathbf{k}_{2} \delta(\mathbf{k}_{1}+\mathbf{k}_{2})e^{-i(\mathbf{k}_{1}\cdot\mathbf{r}_{1}- \mathbf{k}_{2}\cdot\mathbf{r}_{2})}}{k_{1}^{2}[i\omega_{2}-D(k_{1}^{2}+ \kappa^{2})][i\omega_{2}+D(k_{2}^{2}+\kappa^{2})]}=\\ \frac{4(ze)^{2}cD}{(2\pi)^{2}\varepsilon\epsilon_{0}}\frac{e^{i \mathbf{k}_{2}\cdot\mathbf{A}\mathbf{r}}}{[\omega_{2}^{2}+D^{2}(k_{2}^{2}+ \kappa^{2})^{2}]}.\end{split}  \tag{10}\]  
where \(\mathbf{\Delta r}=\mathbf{r}_{2}-\mathbf{r}_{1}\).