assignmentutor-lab™ 为您的留学生涯保驾护航 在代写统计物理Statistical Physics of Matter方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写统计物理Statistical Physics of Matter代写方面经验极为丰富，各种代写统计物理Statistical Physics of Matter相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|统计物理代写Statistical Physics of Matter代考|The Coulomb Interaction

The water medium affects fundamentally the interaction between two ions. Phenomenologically the interaction between two ions of charges $q_{1}$ and $q_{2}$ separated by a distance $r_{12}$ is just the Coulomb interaction,
$$\varphi_{12}=\frac{q_{1} q_{2}}{4 \pi \varepsilon_{w} r_{12}}$$
where $\varepsilon_{w}$ is the electric permeability of water. As mentioned in Sect. 4.4, this effective interaction is formally obtained by integrating (averaging) over all the degrees of freedom of the water molecules surrounding the two ions, with the distance between the two ions $r_{12}$ given as fixed. The effect of the water medium, treated as a continuum, is incorporated by $\varepsilon_{w}$, which depends on $T$.

Equation (6.4) is based on a coarse continuum picture. While it neglects the microscopic details involving the water molecules at short distance, it often gives a reasonable understanding when the distance $r$ longer than the small molecule scale. At such separation the electrostatic interaction energy can be smaller than the thermal energy. If two elementary charges $e$ are separated by $\sim 1 \mathrm{~nm}$ in a vacmum, they would have the Coulomb energy in the order of $1 \mathrm{eV}$, but in water its high dielectric constant $\varepsilon_{w} / \varepsilon_{0}=78.5$ can reduce the Coulomb energy below the thermal energy which is $k_{B} T \approx 1 / 40 \mathrm{eV}$ at room temperature. A convenient scale to assess dominance of the thermal energy is
$$l_{B}=\frac{e^{2}}{4 \pi \varepsilon k_{B} T}$$
called the Bjerrum length, above which the elementary Coulomb interaction energy is less than $k_{B} T$; in water at $25^{\circ} \mathrm{C}$ it is $7.1 \AA$.

Consider a single ion of charge $q$ in water. Assuming that the ion is a sphere of radius $a$, the work necessary to charge the sphere from zero to $q$ in water continuum is
$$\varphi_{b}=\int_{0}^{q} \frac{q^{\prime} d q^{\prime}}{4 \pi \varepsilon_{w} a}=\frac{q^{2}}{8 \pi \varepsilon_{w} a}$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|Ion-Dipole Interaction

Consider an ion of charge $q$ and a molecule of permanent dipole moment $\boldsymbol{p}$ where the charges $q_{d}$ and $-q_{d}$ are separated by distance $l$ (Fig. 6.5). The energy of electrostatic interaction between the ion and the dipole at a distance $r$ is
\begin{aligned} \varphi_{i d}(r, \Omega) &=\frac{q_{d} q}{4 \pi \varepsilon_{w}|\boldsymbol{r}+\boldsymbol{l} / 2|}-\frac{q_{d} q}{4 \pi \varepsilon_{w}|\boldsymbol{r}-\boldsymbol{l} / 2|} \approx q_{d} \boldsymbol{l} \cdot \boldsymbol{\nabla} \frac{q}{4 \pi \varepsilon_{w} r} \ &=-\boldsymbol{p} \cdot \boldsymbol{E}=-p E \cos \theta \end{aligned}
where $E=q /\left(4 \pi \varepsilon_{\mathrm{w}} r^{2}\right)$ is the electric field from the ion, $\theta$ is the polar angle that the dipole vector $\boldsymbol{p}=q_{d} l$ makes with the field direction (Fig. 6.5).

The dipole is undergoing thermal fluctuations (rotation). The induced polarization along the field observed at equilibrium is the thermal average over the orientation,
\begin{aligned} P_{d}=p\langle\cos \theta\rangle &=p \frac{\int d \Omega \cos \theta e^{-\beta \varphi_{i d}(r, \Omega)}}{\int d \Omega e^{-\beta \varphi_{i d}(r, \Omega)}}=p \frac{\int d \Omega \cos \theta e^{\beta p E \cos \theta}}{\int d \Omega e^{\beta p E \cos \theta}} \ &=p \frac{\partial}{\partial(\beta p E)} \ln Z_{i d}=p\left[\operatorname{coth}(\beta p E)-\frac{1}{\beta p E}\right] \equiv p \mathcal{L}(\beta p E) \end{aligned}
where $d \Omega=d(\cos \theta) d \phi$ is the solid angle element, $Z_{i d}=\int d \Omega e^{\beta p E \cos \theta}=$ $(4 \pi \sinh \beta p E) / \beta p E$, and $\mathcal{L}(x)$ is the Langevin’s function earlier seen in polymer extension problem (3.57).

Now we note that $\beta p E=p q /\left(4 \pi \varepsilon_{w} r^{2} k_{B} T\right) \sim l l_{B} / r^{2}$ where the $l_{B}$ is about $7 \AA$ at room temperature. If, for example, the dipole is due to a small polar molecule like a water molecule, the charge separation length $l$ is also a length in the order of $1 \AA$, so, at a distance $r$ over the nanoscale, $\beta p E(r) \ll 1$. We will consider this case throughout.

# 统计物理代考

## 物理代写|统计物理代写Statistical Physics of Matter代考|The Coulomb Interaction

$$\varphi_{12}=\frac{q_{1} q_{2}}{4 \pi \varepsilon_{w} r_{12}}$$

$$l_{B}=\frac{e^{2}}{4 \pi \varepsilon k_{B} T}$$

$$\varphi_{b}=\int_{0}^{q} \frac{q^{\prime} d q^{\prime}}{4 \pi \varepsilon_{w} a}=\frac{q^{2}}{8 \pi \varepsilon_{w} a}$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|Ion-Dipole Interaction

$$\varphi_{i d}(r, \Omega)=\frac{q_{d} q}{4 \pi \varepsilon_{w}|\boldsymbol{r}+\boldsymbol{l} / 2|}-\frac{q_{d} q}{4 \pi \varepsilon_{w}|\boldsymbol{r}-\boldsymbol{l} / 2|} \approx q_{d} \boldsymbol{l} \cdot \boldsymbol{\nabla} \frac{q}{4 \pi \varepsilon_{w} r} \quad=-\boldsymbol{p} \cdot \boldsymbol{E}=-p E \cos \theta$$

$$P_{d}=p\langle\cos \theta\rangle=p \frac{\int d \Omega \cos \theta e^{-\beta \varphi \text { id }(r, \Omega)}}{\int d \Omega e^{-\beta \varphi \operatorname{id}(r, \Omega)}}=p \frac{\int d \Omega \cos \theta e^{\beta p E \cos \theta}}{\int d \Omega e^{\beta p E \cos \theta}} \quad=p \frac{\partial}{\partial(\beta p E)} \ln Z_{i d}=p\left[\operatorname{coth}(\beta p E)-\frac{1}{\beta p E}\right] \equiv p \mathcal{L}(\beta p E)$$

## 有限元方法代写

assignmentutor™作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师