## 1. Introduction

Water-soluble materials often have surface chemical groups that are dissociated in a polar solvent. Examples of such materials include not only mesoscopic particles, such as viruses, proteins, polyelectrolytes, membranes, and micelles, but also macroscopic objects like a glass plate of sample cell [

1,

2]. Both mesoscopic and macroscopic particles will be referred to here as ‘macroions’. The macroions are likely to carry a total surface charge exceeding thousands of elementary charges

e, surrounded by oppositely charged counterions that are dissociated from the macroions [

1,

2]; counterions are electrostatically bound around macroions, due to the high asymmetry between counterions and macroions in valence of charges.

Focusing on the counterions, the macroion systems can be rephrased as inhomogeneous one-component ionic fluids in the presence of external fields—the one-component fluids of counterions can be regarded as the one-component plasma (OCP) [

3,

4,

5]. Systems of charged particles immersed in a smooth neutralizing medium are commonly observed in nature, such as a suspension of dust grains in plasmas, as well as colloidal solutions, which can be modeled by the OCP in the unscreened limit of Yukawa fluids [

3,

4,

5]. The 2D OCP has been used for the description of dusty plasmas confined by external fields, where the motion of particles interacting via 3D electrostatic interaction potential is restricted to a 2D surface [

3,

4,

5]. There is, however, a crucial difference between counterion systems and the OCP, due to the localization of the electrically neutralizing background. While the whole space in the OCP is filled with a smooth background in either the 2D or 3D systems, counterions form a 3D electric double layer and are not neutralized unless they are localized on the macroion surfaces [

6,

7,

8].

This paper will address the strong coupling systems of counterions in the presence of one charged plate, focusing especially on the transverse dynamics of density fluctuations around the ground state that will be specified in

Section 2.1 [

8,

9,

10,

11]. Turning our attention to the dynamics, the strongly coupled counterion system is distinguished from the 2D OCP by one extra dimension, vertical to the macroion surface. Accordingly, density fluctuations occur not only along the 2D plane parallel to the macroion surface, but also along the one extra dimension. Furthermore, the strong coupling regime in the counterion systems may be realized at room temperature, and therefore dynamics due to thermal fluctuations need to be considered; however, there are few studies on the counterion dynamics in the ground state.

Thus, the main aim of this paper is to investigate the anisotropic fluctuation field

$n(\mathbf{r},t)$ of counterion density due to the coarse-grained dynamics of counterion density

$\rho (\mathbf{r},t)={\rho}_{\infty}(\mathbf{r})+n(\mathbf{r},t)$ around a ground state distribution

${\rho}_{\infty}(\mathbf{r})$, using the stochastic density functional equation. The stochastic density functional theory is based on the so-called Dean–Kawasaki (DK) equation that describes the evolution of the instantaneous microscopic density field of overdamped Brownian particles [

12,

13,

14,

15,

16,

17,

18,

19,

20]. The stochastic density functional theory has been used as one of the most powerful tools for describing slowly fluctuating and/or intermittent phenomena [

16,

17,

18,

19,

20], such as glassy dynamics, nucleation or pattern formation of colloidal particles, stochastic thermodynamics of colloidal suspensions, dielectric relaxation of Brownian dipoles, and even tumor growth.

The original DK equation includes nonlinear terms of dynamic origin due to the kinetic coefficient that is proportional to fluctuating field

$\rho (\mathbf{r},t)$ [

12,

13,

14,

15,

16,

17,

18,

19,

20]. While the nonlinearility of the original DK equation leads to the above successful descriptions of various phenomena [

16,

17,

18,

19,

20], a more tractable form is required. It has been recently demonstrated that the DK equation can be linearized with respect to

$n(\mathbf{r},t)$ when

$n(\mathbf{r},t)/{\rho}_{\infty}(\mathbf{r})\ll 1$, and that the linear stochastic equation of density fluctuations is of great practical use [

21,

22,

23,

24,

25,

26,

27]. The density fluctuations of fluids near equilibrium are surprisingly well described by model-B dynamics of a Gaussian field theory whose effective quadratic Hamiltonian for the density fluctuation field is constructed to yield the exact form of the static density-density correlation function [

25]. Furthermore, we have demonstrated that the DK equation can be directly linearized in the first approximation of the driving force due to the free energy functional

$F\left[\rho \right]$ of an instantaneous density distribution

$\rho $, when small density fluctuations around a metastable state are considered [

21].

The stochastic thermodynamics around a metastable state has been investigated using the stochastic density functional equation (the DK equation), showing that the heat dissipated into the reservoir is generally negligible [

21]. The linear stochastic density functional theory has also been found relevant to investigate out-of-equilibrium phenomena, including the formulations of the full Onsager theory of electrolyte conductivity [

22].

The remainder of this paper is organized as follows:

Section 2 provides formal background in the case of a single charged plate system. We give the linear DK equation as a stochastic density functional equation, after specifying a general form of the free energy functional

$F\left[\rho \right]$ of a given density. In

Section 3, the linear DK equation is applied to the strongly coupled counterion system by considering density fluctuations

$n=\rho -{\rho}_{\infty}$ around

${\rho}_{\infty}$. We can verify that the first derivative of

$F\left[\rho \right]$ in the ground state (i.e.,

${\delta F\left[\rho \right]/\delta \rho |}_{\rho ={\rho}_{\infty}}$) produces a constant, similar to the above metastable state. Accordingly, the DK equation of the counterion system can be linearized around the ground state. We will also see the underlying physics of anisotropic fluctuations (vertical to the plate) in terms of the general form of the linear DK equation. In

Section 4, we focus on the transverse dynamics along the plate surface, assuming the absence of the gradient of the fluctuating density field vertical to the plate. First, we derive the frozen dynamics over a long-range scale beyond the Wigner–Seitz cell, reflecting the formation of the Wigner crystal on the charged plate. Furthermore, the linear DK equation determines a crossover length

${l}_{c}$, below which we can observe diffusive behaviors of counterions condensed on the plate. Our main result in this study is the quantitative evaluation of

${l}_{c}$, yielding

${l}_{c}\sim a$ for

$\Xi \sim {10}^{3}$.

Section 5 contains a summary and conclusions.

## 4. Density-Density Correlations Due to Transverse Dynamics Along the Plate Surface

Supposing that

${\partial}_{{z}_{0}}n({\mathbf{r}}_{0},t)=0$, we can focus on the transverse dynamics parallel to the charged plate at

${z}_{0}=0$, which we will investigate quantitatively. With the use of the Fourier transform

$n(k,t)$ of

$n({\mathbf{r}}_{0},t)$, Equation (

37) becomes

given that

${\partial}_{{z}_{0}}n({\mathbf{r}}_{0},t)=0$. In Equation (

39), the conventional coupling constant

$\Xi ={q}^{2}{l}_{B}/\lambda $ given by Equation (

2) appears using

${\rho}_{\infty}(0)=\sigma /\lambda $. In both of the strong coupling regime (

$\Xi \gg 1$) and the low wavenumber region (

$ka\ll 1$), the propagator

$\mathcal{G}(k)$ is approximated by

using the relation

$\sigma =q/(\pi {a}^{2})$.

Now we have the analytical solution to Equation (

38) in the following form [

25]:

Considering the real space representation that

${D}_{0}\Xi \mathcal{G}({\mathbf{r}}_{0})\approx 4\pi {D}_{0}\Xi \sigma $ in the above approximation of Equation (

40), the above exponential factors,

${e}^{-t{D}_{0}\Xi \phantom{\rule{0.166667em}{0ex}}\mathcal{G}}$ and

${e}^{-(t-s){D}_{0}\Xi \phantom{\rule{0.166667em}{0ex}}\mathcal{G}}\phantom{\rule{0.222222em}{0ex}}(s<t)$, are negligible due to

$\Xi \gg 1$. Hence, Equation (

41) is reduced to

$n({\mathbf{r}}_{0},t)=\zeta [{\rho}_{\infty}({z}_{0}),\eta ({\mathbf{r}}_{0},s)]$, thereby providing

It is found from Equation (

42) that there is no correlation of transverse density fluctuations, which represents a coarse-grained frozen dynamics in the strong coupling regime of coarse-grained 2D OCP.

We can determine a crossover scale

${l}_{c}$. or associated crossover wavenumber

${k}_{c}=2\pi /{l}_{c}$ by comparing two terms on the rhs of Equation (

39). In the above approximation, the first term on the rhs of Equation (

39) has been neglected based on the condition that

$\Xi \gg 1$ and

$ka\ll 1$. While increasing the wavenumber and maintaining the strong coupling of

$\Xi \gg 1$, we arrive at the crossover wavenumber

${k}_{c}$ that is defined by the following relation:

stating that the two terms on the rhs of Equation (

39) are comparable to each other. We now introduce the main branch

${W}_{0}(x)$ of the Lambert

W-function [

34], so that Equation (

43) is converted to

based on another expression of Equation (

43) as follows:

The approximate form of

${W}_{0}(x)\approx lnx$ for

$x\gg 1$ [

34] applies to Equation (

44) because of

$qm\Xi \gg 1$. It follows that Equation (

44) reads

which is our main result in this study. Below this scale specified by the crossover length

${l}_{c}$, we can observe the diffusive behavior of counterions, instead of frozen correlations represented by Equation (

42). Taking

$qm\Xi ={10}^{4}$ (or

$\Xi \sim {10}^{3}$ for

$q\sim 10$) as an example of the strong coupling regime, Equation (

46) provides

The latter relation implies that the transverse dynamics of strongly-coupled counterions still retain diffusive behaviors within each Wigner–Seitz cell (i.e.,

${l}_{c}\sim a$), which is physically plausible (see also

Figure 3).

## 5. Summary and Conclusions

We have investigated stochastic density fluctuations

$n=\rho -{\rho}_{\infty}$ around the ground state distribution (

${\rho}_{\infty}\propto {e}^{-{z}_{0}/\lambda}$) of strongly coupled counterions near a single charged plate, focusing especially on the transverse dynamics parallel to the charged plate at

${z}_{0}=0$. The key to treating the stochastic dynamics is to use the DK equation of overdamped Brownian particles [

13,

14,

15,

16,

17,

18,

19,

20,

21,

22,

23,

24,

25,

26,

27] that is linearized by expanding the first derivative of a free energy functional of given density, (

$\delta F\left[\rho \right]/\delta \rho $) around the ground state density

${\rho}_{\infty}$. As a result, we have obtained the linear DK Equation (

37), which is applicable to the longitudinal and transverse dynamics of counterions in the strong coupling regime where the stationary density distribution has been investigated using Monte Carlo simulations [

8,

9,

10,

11].

The linear DK equation allows us to quantitatively investigate the dynamical crossover of transverse density fluctuations along the plate surface. Accordingly, we have found a crossover scale, given by Equations (

46) and (

47), above which the transverse density dynamics appear frozen, generating white noise that is uncorrelated with respect to time and space. Below the crossover scale, on the other hand, diffusive behavior of counterions can be observed along the plate surface, as illustrated in

Figure 3. For instance, the crossover length

${l}_{c}$ is of the order of the Wigner–Seitz radius

a when

$\Xi \sim {10}^{3}$. Furthermore, the longitudinal dynamics vertical to the plate arises from the gradient of a fluctuating density field along the

z-axis, producing additional contributions to the transverse dynamics, such as electrical reverse-flow as well as advective flow (see

Figure 2). The electrical reverse-flow would be crucial in experimental situations where mobile ions, including not only counterions but also added salt, are affected considerably by the longitudinal dynamics. This remains to be addressed in a quantitative manner, by extending the present formulation to multi-component systems.