## 3 Model Definition
\begin{array}{rl}\frac{\partial^{2}\delta\hat{\rho}({\bf k},t)}{\partial t^{2 }}=-<\Psi_{ion}|\sum\limits_{j}e^{-i{\bf k}\cdot{\bf r}_{j}}\left\{-\frac{ \hbar^{2}}{m^{2}}\left({\bf k}\cdot\
abla_{j}\right)^{2}+\frac{\hbar^{2}k^{2}}{ m^{2}}i{\bf k}\cdot\
abla_{j}+\frac{\hbar^{2}k^{4}}{4m^{2}}\right\}|\Psi_{ion}>\\ -\frac{4\pi q^{{}^{\prime}2}}{m(2\pi)^{3}}\int d^{3}p\bar{n}({\bf k}-{\bf p}) \frac{{\bf k}\cdot{\bf p}}{p^{2}}\delta\hat{\rho}({\bf p},t)-\frac{4\pi q^{{}^{ \prime}2}}{m(2\pi)^{3}}\int d^{3}p\frac{{\bf k}\cdot{\bf p}}{p^{2}}<\Psi_{ ion}|\delta\hat{\rho}({\bf k}-{\bf p})\delta\hat{\rho}({\bf p})|\Psi_{ion}>.\end{array}  \tag{11}\]For small \(k\), in analogy to the semiclassical approximation for electrons [1; 2], the contributions of the second and third components to the first term on the right-hand-side of Equation (11) can be neglected as small in comparison to the first component. Small and thus negligible is also the third term in the right-hand-side of Equation (11), as involving a product of two \(\delta\hat{\rho}\) (which we assumed small, \(\delta\hat{\rho}/n<<1\)). This approach corresponds to the RPA formulated for bulk metal [2] (note that \(\delta\hat{\rho}(0)=0\) and the coherent RPA contribution of interaction is comprised by the last but one term in Equation (11)). Within the RPA, Equation (11) thus attains the following shape,  
\[ \begin{array}{ll}\frac{\partial^{2}\delta\hat{\rho}({\bf k},t)}{\partial t^{ 2}}=&\frac{2k^{2}}{3m}<\Psi_{ion}|\sum\limits_{j}e^{-i{\bf k}\cdot{\bf r}_{j}} \frac{\hbar^{2}\
abla_{j}^{2}}{2m}|\Psi_{ion}>\\ &-\frac{4\pi{q^{2}}}{m(2\pi)^{3}}\int d^{3}p\hat{n}({\bf k}-{\bf p})\frac{{\bf k }\cdot{\bf p}}{p^{2}}\delta\hat{\rho}({\bf p},t),\end{array}  \tag{12}\]  
and due to the spherical symmetry,  
\[<\Psi_{ion}|\sum\limits_{j}e^{-i{\bf k}\cdot{\bf r}_{j}}\frac{\hbar^{2}}{m^{2 }}\left({\bf k}\cdot\
abla_{j}\right)^{2}|\Psi_{ion}>\simeq\frac{2k^{2}}{3m}< \Psi_{ion}|\sum\limits_{j}e^{-i{\bf k}\cdot{\bf r}_{j}}\frac{\hbar^{2}\
abla_{ j}^{2}}{2m}|\Psi_{ion}>.\]  
One can rewrite Equation (12) in the position representation,  
\[ \begin{array}{ll}\frac{\partial^{2}\delta\hat{\rho}({\bf r},t)}{\partial t^{ 2}}=-\frac{2}{3m}\
abla^{2}<\Psi_{ion}|\sum\limits_{j}\delta({\bf r}-{\bf r}_{ j})\frac{\hbar^{2}\
abla_{j}^{2}}{2m}|\Psi_{ion}>\\ +\frac{\omega_{2}^{2}}{4\pi}\
abla\left\{\Theta(a-r)\
abla\int d^{3}r_{1}\frac {1}{|{\bf r}-{\bf r}_{1}|}\delta\hat{\rho}({\bf r}_{1},t)\right\}.\end{array}  \tag{13}\]  
In the case of metals it was next used the Thomas-Fermi formula to assess the averaged kinetic energy [2]:  
\[ \begin{array}{ll}<\Psi_{ion}|-\sum\limits_{j}\delta({\bf r}-{\bf r}_{j}) \frac{\hbar^{2}\
abla_{j}^{2}}{2m}|\Psi_{ion}>\simeq\frac{3}{5}(3\pi^{2})^{2/3 }\frac{\hbar^{2}}{2m}(\rho({\bf r},t))^{5/3}\\ =\frac{3}{5}(3\pi^{2})^{2/3}\frac{\hbar^{2}}{2m}n^{5/3}\Theta(a-r)\left[1+ \frac{5}{3}\frac{\delta\hat{\rho}({\bf r},t)}{n}+...\right].\end{array}  \tag{14}\]  
The Thomas-Fermi formula is addressed, however, to fermionic and degenerate quantum systems, as electrons in metals. For ionic systems such estimation of kinetic energy is inappropriate, because the ion concentration is usually much lower than that of electrons in metals and the system is not degenerated even if ions are fermions. The Maxwell-Boltzmann distribution should be applied instead of the Fermi-Dirac or Bose-Einstein ones. Independently of fermionic of bosonic statistics of ions, the Maxwell-Boltzmann distribution allows for the estimation of the averaged kinetic energy of ions located inside the spherical electrolyte with the radius \(a\), in the following form,  
\[<\Psi_{ion}|-\sum\limits_{j}\delta({\bf r}-{\bf r}_{j})\frac{\hbar^{2}\
abla _{j}^{2}}{2m}|\Psi_{ion}>\simeq(n+\delta\rho({\bf r},t))\Theta(a-r)\frac{3k_{ B}T}{2}, \tag{15}\]  
where \(k_{B}\) is the Boltzmann constant and \(T\) the temperature.