## 3 Fluctuation Correlations and Thermodynamics of Binary Symmetric Electrolyte Solutions
### Charge Fluctuation Correlations and Correlation Energy  
We will use the Fourier transform technique [16] to express the charge and flux fluctuations as  
\[\delta\hat{\rho}_{e}(\mathbf{k},\omega)=\frac{1}{(2\pi)^{2}}\int_{-\infty}^{ \infty}d\mathbf{r}\int_{-\infty}^{\infty}dt\rho_{e}(\mathbf{r},t)e^{i(\omega t -\mathbf{k}\cdot\mathbf{r})} \tag{35}\]  
and  
\[\delta\hat{\mathbf{j}}(\mathbf{k},\omega)=\frac{1}{(2\pi)^{2}}\int_{-\infty}^ {\infty}d\mathbf{r}\int_{-\infty}^{\infty}dt\delta\hat{\mathbf{j}}(\mathbf{r },t)e^{i(\omega t-\mathbf{k}\cdot\mathbf{r})}. \tag{36}\]  
The inverse transforms are  
\[\delta\rho_{e}(\mathbf{r},t)=\frac{1}{(2\pi)^{2}}\int_{-\infty}^{\infty}d \mathbf{k}\int_{-\infty}^{\infty}d\omega\hat{\rho}_{e}(\mathbf{k},\omega)e^{- i(\omega t-\mathbf{k}\cdot\mathbf{r})} \tag{37}\]  
and  
\[\delta\hat{\mathbf{j}}(\mathbf{r},t)=\frac{1}{(2\pi)^{2}}\int_{-\infty}^{ \infty}d\mathbf{k}\int_{-\infty}^{\infty}d\omega\delta\hat{\mathbf{j}}( \mathbf{k},\omega)e^{-i(\omega t-\mathbf{k}\cdot\mathbf{r})} \tag{38}\]  
where \(\mathbf{k}\) is the wavevector and \(\omega\) is the frequency. The ionic fluxes and their correlations are of special interest. For low ionic concentrations, the diffusion flux of each component does not depend on any variable pertinent to the other solutes (see Equations (20)). Hence, the correlations between the fluxes for different species is zero, or  
\[ \begin{split}\langle\delta\hat{\mathbf{j}}_{+}(\mathbf{r}_{1},t_{ 1})\delta\hat{\mathbf{j}}_{+}(\mathbf{r}_{2},t_{2})\rangle&=2 cD \mathbf{I}\delta(t_{1}-t_{2})\delta(\mathbf{r}_{1}-\mathbf{r}_{2}),\\ \langle\delta\hat{\mathbf{j}}_{-}(\mathbf{r}_{1},t_{1})\delta \hat{\mathbf{j}}_{-}(\mathbf{r}_{2},t_{2})\rangle&=2 cD \mathbf{I}\delta(t_{1}-t_{2})\delta(\mathbf{r}_{1}-\mathbf{r}_{2}),\\ \langle\delta\hat{\mathbf{j}}_{+}(\mathbf{r}_{1},t_{1})\delta \hat{\mathbf{j}}_{-}(\mathbf{r}_{2},t_{2})\rangle&=\langle \delta\hat{\mathbf{j}}_{-}(\mathbf{r}_{1},t_{1})\delta\hat{\mathbf{j}}_{+}( \mathbf{r}_{2},t_{2})\rangle=0.\end{split}  \tag{39}\]  
Equations (38) and (39) allow for the derivation of the important relationship (see Appendix A)  
\[\langle\delta\hat{\mathbf{j}}(\mathbf{k}_{1},\omega_{1})\delta\hat{\mathbf{j }}(\mathbf{k}_{2},\omega_{2})\rangle=2c\mathbf{I}D\delta(\omega_{1}+\omega_{2 })\delta(\mathbf{k_{1}}+\mathbf{k_{2}}). \tag{40}\]  
Note that the right-hand sides for the first two equations in (40) are the same irrespective of whether the positive or negative ionic fluxes are correlated. Equations (35)-(38) allow Fourier transforming the whole Equation (34)  
\[i\omega\delta\hat{\rho}_{e}(\mathbf{k},\omega)=-D(k^{2}+\kappa^{2})\delta \hat{\rho}_{e}(\mathbf{k},\omega)+ize\mathbf{k}\cdot[\delta\hat{\mathbf{j}}_{+ }(\mathbf{k},\omega)-\delta\hat{\mathbf{j}}_{-}(\mathbf{k},\omega)]. \tag{41}\]  
Solving for the charge density \(\delta\hat{\rho}_{e}(\mathbf{k},\omega)\) yields  
\[\delta\hat{\rho}_{e}(\mathbf{k},\omega)=\frac{i\mathbf{k}ze[\delta\hat{\mathbf{ j}}_{+}(\mathbf{k},\omega)-\delta\hat{\mathbf{j}}_{-}(\mathbf{k},\omega)]}{i \omega+D(k^{2}+\kappa^{2})}. \tag{42}\]The charge correlation fluctuation correlation is  
\[ \begin{split}\langle\delta\hat{\rho}_{e}(\mathbf{k}_{1},\omega_{1}) \delta\hat{\rho}_{e}(\mathbf{k}_{2},\omega_{2})\rangle=-\frac{4(ze)^{2}cD \mathbf{k}_{1}\cdot\mathbf{k}_{2}\delta(\omega_{1}+\omega_{2})\delta(\mathbf{ k}_{1}+\mathbf{k}_{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}