## 3 Fluctuation Correlations and Thermodynamics of Binary Symmetric Electrolyte Solutions
### Balance Equations  
We start with balancing the mass of dissolved electrolytes as well as the potential in the solution. The mass balance Nernst-Planck equations read  
\[\begin{split}\frac{\partial c_{+}}{\partial t}&=- \
abla\cdot\mathbf{j}_{+}+ze\beta_{+}\
abla\cdot(c_{+}\
abla\psi)\\ \frac{\partial c_{-}}{\partial t}&=-\
abla\cdot \mathbf{j}_{-}-ze\beta_{-}\
abla\cdot(c_{-}\
abla\psi).\end{split} \tag{24}\]  
The coefficients \(\beta_{+}\) and \(\beta_{-}\) are the hydrodynamic mobilities of the positive and negative ions, while \(e\) is the elementary charge. The electrostatic potential \(\psi\) is related to the charge density \(\rho_{e}=ze(c_{+}-c_{-})\) by means of the Poisson equation [10]  
\[\
abla^{2}\psi=-\frac{\rho_{e}}{\varepsilon\varepsilon_{0}}, \tag{25}\]  
where \(\varepsilon_{0}\) is the dielectric constant in vacuo, and \(\varepsilon\) is the relative dielectric permittivity.  
In order to account for the fluctuations in the solution, we define the fluxes, the concentrations, and the local potential as  
\[\mathbf{j}_{+} =-D_{+}\
abla c_{+}+\delta\mathbf{j}_{+} \tag{26}\] \[\mathbf{j}_{-} =-D_{-}\
abla c_{-}+\delta\mathbf{j}_{-},\] (27) \[c_{+} =c+\delta c_{+}\] \[c_{-} =c+\delta c_{-},\]  
and  
\[\psi=0+\delta\psi, \tag{28}\]  
where \(D_{+}=k_{B}T/\beta_{+}\) and \(D_{-}=k_{B}T/\beta_{-}\) are the diffusion coefficients for the positive and negative ions, \(\delta c_{+}\) and \(\delta c_{-}\) are the concentration fluctuations for the positive and negative ions, and \(c\) is the uniform average concentration in units of number per volume. The resultant local electrostatic potential fluctuation is \(\delta\psi\).  
The balance equations for all fluctuating quantities become  
\[\begin{split}\frac{\partial\delta c_{+}}{\partial t}& =\
abla^{2}\delta c_{+}-\
abla\delta\cdot\mathbf{j}_{+}+ze\beta_{ +}c\
abla^{2}\delta\psi\\ \frac{\partial\delta c_{-}}{\partial t}&=\
abla^{2} \delta c_{-}-\
abla\delta\cdot\mathbf{j}_{-}-ze\beta_{-}c\
abla^{2}\delta \psi.\end{split} \tag{29}\]  
\[\
abla^{2}\delta\psi=-\frac{\delta\rho_{e}}{\varepsilon\varepsilon_{0}}, \tag{30}\]  
The charge fluctuation is  
\[\delta\rho_{e}=ze(\delta c_{+}-\delta c_{-})=\delta\rho_{+}+\delta\rho_{-}, \tag{31}\]  
where  
\[\delta\rho_{+}=ze\delta c_{+},\text{ and }\delta\rho_{-}=-ze\delta c_{-}. \tag{32}\]  
Rearranging the above equations yields  
\[\begin{split}\frac{\partial\delta\rho_{+}}{\partial t}& =D_{+}\left(\
abla^{2}\delta\rho_{+}-\frac{\kappa^{2}}{2}\delta \rho_{e}\right)-ze\
abla\cdot\delta\mathbf{j}_{+}\\ \frac{\partial\delta\rho_{-}}{\partial t}&=D_{-} \left(\
abla^{2}\delta\rho_{-}-\frac{\kappa^{2}}{2}\delta\rho_{e}\right)+ze \
abla\cdot\delta\mathbf{j}_{-},\\ \kappa^{2}&=\frac{2z^{2}e^{2}c}{\varepsilon\varepsilon _{0}k_{B}T}=\frac{2z^{2}e^{2}c\delta_{+}}{\varepsilon\varepsilon_{0}D_{+}}= \frac{2z^{2}e^{2}c\delta_{-}}{\varepsilon\varepsilon_{0}D_{-}}.\end{split} \tag{33}\]The parameter \(\kappa\) is the [PERSON] screening parameter (inverse length) [11], which is a measure of potential magnitude reduction with distance around an ion in the solution. A significant simplification of Equation (33) can be achieved by setting \(D_{+}=D_{-}=D\), and noting that for symmetric electrolytes, the overall concentrations are \(c_{+}=c_{-}=c\). This seems like a very strong limitation, but it does not affect the majority of the final results. Combining the first two equations in (33) leads to an equation for the charge density fluctuation  
\[\frac{\partial\delta\rho_{e}}{\partial t}=D\Big{(}\
abla^{2}\delta\rho_{e}- \kappa^{2}\delta\rho_{e}\Big{)}-ze\
abla\cdot(\delta\mathbf{j}_{+}-\delta \mathbf{j}_{-}). \tag{34}\]