## 3 Model Definition  
The Hamiltonian for the binary ion system has the form,  
\[ \begin{array}{ll}\hat{H}_{ion}=&-\sum\limits_{i=1}^{N}\frac{\hbar^{2}\
abla_ {i}^{2}}{2m}-\sum\limits_{j=1}^{N}\frac{\hbar^{2}\
abla_{j}^{2}}{2m^{-}}-\sum \limits_{i,j}^{N-N}\frac{\hbar^{-}q^{+}}{\varepsilon|\mathbf{r}_{i}-\mathbf{r} _{j}|}\\ &+\frac{1}{2}\sum\limits_{i,j^{\prime}\
eq i\
eq j}^{N}\frac{(q^{-})^{2}}{ \varepsilon|\mathbf{r}_{i}-\mathbf{r}_{j^{\prime}}|}+\frac{1}{2}\sum\limits_{ j,j^{\prime}\
eq j^{\prime}}^{N}\frac{(q^{+})^{2}}{\varepsilon|\mathbf{r}_{j}- \mathbf{r}_{j^{\prime}}|},\end{array}  \tag{1}\]  
where \(q^{-(+)}\), \(M^{-(+)}\), \(N^{-(+)}\) are the charge, the mass and the number of the \(-(+)\) ions, respectively. To analyze this complicated system we propose the following approximation, assuming, for simplicity, \(q^{-}=-q^{+}=q\), \(N^{-}=N^{+}=N\), \(m^{-}=m^{+}=m\). We next add and subtract the same terms as written below,  
\[ \begin{array}{ll}\hat{H}_{ion}=-\sum\limits_{i=1}^{N}\frac{\hbar^{2}\
abla_ {i}^{2}}{2m}-\sum\limits_{j=1}^{N}\frac{\hbar^{2}\
abla_{j}^{2}}{2m}-\sum \limits_{i,j}\frac{q^{2}}{\varepsilon|\mathbf{r}_{i}-\mathbf{r}_{j}|}+\frac{ 1}{2}\sum\limits_{i,j^{\prime}\
eq i^{\prime}}^{N}\frac{q^{2}}{\varepsilon| \mathbf{r}_{i}-\mathbf{r}_{j}|}+\frac{1}{2}\sum\limits_{j,j^{\prime}\
eq j^{ \prime}}^{N}\frac{q^{2}}{\varepsilon|\mathbf{r}_{j}-\mathbf{r}_{j^{\prime}}|}\\ -q^{2}\sum\limits_{j}\int\frac{n(\mathbf{r})d^{3}\mathbf{r}}{\varepsilon| \mathbf{r}_{j}-\mathbf{r}_{j}|}-q^{2}\sum\limits_{i}\int\frac{n(\mathbf{r})d^ {3}\mathbf{r}}{\varepsilon|\mathbf{r}_{i}-\mathbf{r}|}+q^{2}\sum\limits_{j} \int\frac{n(\mathbf{r})d^{3}\mathbf{r}}{\varepsilon|\mathbf{r}_{j}-\mathbf{r} _{j}|}+q^{2}\sum\limits_{i}\int\frac{n(\mathbf{r})d^{3}\mathbf{r}}{\varepsilon| \mathbf{r}_{j}-\mathbf{r}_{j}|},\end{array}  \tag{2}\]  
where we have introduced formally the fictitious jellium of the spherical shape for both types of ions, with the density \(n\) ideally compensating opposite charges of uniformly distributed ions, \(n(\mathbf{r})=n\Theta(a-r)\), \(a\) is the sphere radius (the positive with the negative jellium mutually cancel themselves). Assuming that,  
\[q^{2}\sum\limits_{j}\int\frac{n(\mathbf{r})d^{3}\mathbf{r}}{\varepsilon| \mathbf{r}_{j}-\mathbf{r}|}+q^{2}\sum\limits_{i}\int\frac{n(\mathbf{r})d^{3} \mathbf{r}}{\varepsilon|\mathbf{r}_{i}-\mathbf{r}|}-\sum\limits_{i,j}\frac{q^{ 2}}{\varepsilon|\mathbf{r}_{i}-\mathbf{r}_{j}|}\simeq 0, \tag{3}\]  
which is fulfilled for not too strong ion fluctuations beyond the uniform distribution (in dilute ion systems), we can separate the Hamiltonian into the sum, \(\hat{H}_{ions}=\hat{H}^{-}+\hat{H}^{+}\), where,  
\[\hat{H}^{-(+)}=\sum\limits_{j}\left[-\frac{\hbar^{2}\
abla_{j}^{2}}{2m}-q^{2} \int\frac{n(\mathbf{r})d^{3}\mathbf{r}}{\varepsilon|\mathbf{r}_{j}-\mathbf{r} |}\right]+\frac{1}{2}\sum\limits_{j\
eq j^{\prime}}\frac{q^{2}}{\varepsilon| \mathbf{r}_{j}-\mathbf{r}_{j^{\prime}}|}. \tag{4}\]  
The latter term in r.h.s. of Equation (4) corresponds to the interaction between ions of the same sign, whereas the second term in the first sum describes the interaction of these ions with the effective jellium (of opposite sign), \(\varepsilon\) is the dielectric permittivity of the electrolyte medium. The confined electrolyte system can be shaped by appropriately formed membrane, as frequently happens in biological systems. Because of separation of the Hamiltonian (1) one can consider the single Hamiltonian (4). The second one is analogous. The ion wave function corresponding to Hamiltonian (4) we will denote by \(\Psi_{ion}(t)\). The form of Hamiltonian (4) allows for the repetition of its further discussion along the scheme applied to electrons in metals [1], which we will recall below, for the sake of completeness.