# Transverse Density Fluctuations around the Ground State Distribution of Counterions near One Charged Plate: Stochastic Density Functional View

## Abstract

**:**

## 1. Introduction

## 2. Formal Background

#### 2.1. Ground State of Counterion System in the Strong Coupling Limit

#### 2.2. Imposing a Given Density Distribution $\rho $ on the Grand Potential $\Omega $

#### 2.3. Stochastic Density Dynamics Obeying the Dean–Kawasaki Equation

## 3. Stochastic Density Functional Equation for Fluctuations round the Ground State Distribution ${\rho}_{\infty}$

#### 3.1. Linearizing the Stochastic Dean–Kawasaki Equation (14)

#### 3.2. Implications of Longitudinal Contributions Given by Equation (33)

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

## 5. Summary and Conclusions

## Funding

## Conflicts of Interest

## Appendix A. Electrostatic Interaction Energies: General Forms When Rescaled by the Wigner–Seitz Radius a

## Appendix B. The Grand Potential Ω[J] for a One-Plate System

## Appendix C. A Remark on Equation (11)

## Appendix D. Details of Equation (19)

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**Figure 1.**Two schematics illustrating the side view of one charged plate system that consists of a positively charged plate carrying a surface charge density $+\sigma e$ and negatively charged counterions with valence q. Our scaling ($z={z}_{0}/a$) of an actual system depicted on the left side implies a coarse-grained system on the right hand side, where a denotes a mean separation between counterions, provided that all of the counterions are condensed on the oppositely charged plate uniformly: $\pi {a}^{2}\sigma =q$. The Gouy–Chapman length $\lambda \equiv 1/(2\pi q{l}_{B}\sigma )$ is also indicated. This paper adopts the coupling constant $\Gamma $, defined by $\Gamma ={q}^{2}{l}_{B}/a$ that applies to the 2D one-component plasma (OCP), instead of the conventional one, $\Xi ={q}^{2}{l}_{B}/\lambda $, used for the counterion system.

**Figure 2.**A schematic of anisotropic fluctuations due to longitudinal terms underlined in Equation (37). Here we consider the case that the fluctuating density $n({\mathbf{r}}_{0},t)$ decreases with ${z}_{0}$, and $n(0.t)$ and $n(\lambda ,t)$ are abbreviations of $n({\mathbf{r}}_{0},t){|}_{{z}_{0}=0}$ and $n({\mathbf{r}}_{0},t){|}_{{z}_{0}=\lambda}$, respectively. The advective flow, or migration of density fluctuations as a whole, is always in the negative direction along the ${z}_{0}$-axis. Meanwhile, the electrical reverse-flow is also in the negative direction, irrespective of the sign of ${\partial}_{{z}_{0}}n$, or ${\mathcal{E}}_{z}$. Accordingly, the latter flow acts as a positive feedback of density fluctuations for ${\partial}_{{z}_{0}}n<0$.

**Figure 3.**A schematic of dynamical crossover, showing that diffusive dynamics of strongly coupled counterions along the plate surface can be observed within the scale of Wigner–Seitz cell. The determining equation for the crossover scale ${k}_{c}$ is also given, based on the expression of propagator $\mathcal{G}(k)$ given by Equation (39).

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**MDPI and ACS Style**

Frusawa, H. Transverse Density Fluctuations around the Ground State Distribution of Counterions near One Charged Plate: Stochastic Density Functional View. *Entropy* **2020**, *22*, 34.
https://doi.org/10.3390/e22010034

**AMA Style**

Frusawa H. Transverse Density Fluctuations around the Ground State Distribution of Counterions near One Charged Plate: Stochastic Density Functional View. *Entropy*. 2020; 22(1):34.
https://doi.org/10.3390/e22010034

**Chicago/Turabian Style**

Frusawa, Hiroshi. 2020. "Transverse Density Fluctuations around the Ground State Distribution of Counterions near One Charged Plate: Stochastic Density Functional View" *Entropy* 22, no. 1: 34.
https://doi.org/10.3390/e22010034