### 物理代写|统计物理代写Statistical Physics of Matter代考|KYA322

statistics-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
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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 Van der Waals Equation of State

We now make an approximation that is useful for non-dilute fluids and derive the van-der Waals equation by statistical mechanical methods. The intermolecular pair potential $\varphi(r)$ can in many cases be separated into two parts, a harsh, short-range (hard-core) repulsion for $r<\sigma$ and a smooth, relatively long-range attraction for $r>\sigma$, where $\sigma$ is the hard-core size or the diameter of molecules. A typical example is the Lennard-Jones potential (Fig. 4.3).
$$\varphi_{L J}(r)=4 \epsilon\left{\left(\frac{r}{\sigma}\right)^{-12}-\left(\frac{r}{\sigma}\right)^{-6}\right}$$
Then the second virial coefficient (4.55) is expressed as the sum of two integrals, each representing the hard-repulsion and soft-attraction part:
$$B_{2}=2 \pi\left[\int_{0}^{\sigma} d r r^{2}\left{1-e^{-\beta \varphi(r)}\right}+\int_{\sigma}^{\infty} d r r^{2}\left{1-e^{-\beta \varphi(r)}\right}\right] .$$

In the first integral, the exponent $e^{-\beta \varphi(r)}$ is negligible for $r<\sigma$ where the potential sharply rises to infinity, so that the integral is evaluated as $2 \pi \sigma^{3} / 3 \equiv b$. For $r>\sigma$, $\varphi(r)$ is a weak attraction effectively so that $e^{-\beta \varphi(r)} \approx 1-\beta \varphi(r)$, yielding the second integral as $-a /\left(k_{B} T\right)$, where
$$a=-2 \pi \int_{\sigma}^{\infty} r^{2} \varphi(r) d r$$
Then, the second virial coefficient is given as
$$B_{2}=b-\frac{a}{k_{B} T}=b\left(1-\frac{\Theta}{T}\right),$$
where the $\Theta=a /\left(k_{B} b\right)$ is the parameter called the Boyle temperature. If $T>\Theta$, then $B_{2}>0$; the repulsion dominates the attraction overall, contributing positively to the pressure and free energy. If $T=\Theta$, then $B_{2}=0$ and the gas behaves ideally. For $T<\Theta$ and $B_{2}<0$, the attraction dominates the repulsion, contributing negatively to them.

## 物理代写|统计物理代写Statistical Physics of Matter代考|Solvent-Averaged Solute Particles

We have been considering a simple fluid of one-component particles moving in a vacuum. However, in biology we consider solute particles such as ions, and macromolecules immersed in water, which itself is a complex liquid that undergoes anisotropic molecular interactions. We remind ourselves that at equilibrium the arated and become irrelevant. Yet the statistical mechanics involves complex situations in which the configurations of all particles in mixtures (i.e., solutions), solute as well as solvent, must be considered, including all interactions.

A simple approach to bypass this formidable task is to highlight the solute particles while treating the solvent as the continuous background whose degrees of freedom are averaged (Fig. 4.6). To describe this formally, we write the total interaction energy as the sum, $\Phi_{V}\left{\boldsymbol{r}{V}\right}+\Phi{U}\left{\boldsymbol{r}{U}\right}+\Phi{V U}\left{\boldsymbol{r}{V}, \boldsymbol{r}{U}\right}$. Here $\Phi_{V}, \Phi_{U}$ are the interaction energies among the solvent particles and solute particles respectively, and $\Phi_{V U}$ is the interaction energy between the solvent and solute particles with $\left{\boldsymbol{r}{V}\right},\left{\boldsymbol{r}{U}\right}$ representing the solvent and solute particle positions. The configuration partition function is given by
$$Q=\iint d\left{\boldsymbol{r}{V}\right} d\left{\boldsymbol{r}{U}\right} \exp \left(-\beta\left[\Phi_{V}\left{\boldsymbol{r}{V}\right}+\Phi{U}\left{\boldsymbol{r}{U}\right}+\Phi{V U}\left{\boldsymbol{r}{V}, \boldsymbol{r}{U}\right}\right]\right)$$
where $d\left{\boldsymbol{r}{V}\right}=d \boldsymbol{r}{v}^{1} d \boldsymbol{r}{v}^{2}, \ldots, d\left{\boldsymbol{r}{U}\right}=d \boldsymbol{r}{u}^{1} d \boldsymbol{r}{u}^{2} \ldots .$ Then we can write
\begin{aligned} Q &=\int d\left{\boldsymbol{r}{U}\right} \exp \left(-\beta\left[\Phi{U}{r}\right]\right) \int d\left{\boldsymbol{r}{V}\right} \exp \left(-\beta\left[\Phi{V}\left{\boldsymbol{r}{V}\right}+\Phi{V U}\left{\boldsymbol{r}{V}, \boldsymbol{r}{U}\right}\right]\right) \ &=\int d\left{\boldsymbol{r}{U}\right} \exp \left(-\beta\left[\Phi{e f f}\left{\boldsymbol{r}_{U}\right}\right]\right) \end{aligned}

## 物理代写|统计物理代写Statistical Physics of Matter代考|The Van der Waals Equation of State

$$a=-2 \pi \int_{\sigma}^{\infty} r^{2} \varphi(r) d r$$

$$B_{2}=b-\frac{a}{k_{B} T}=b\left(1-\frac{\Theta}{T}\right)$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|Solvent-Averaged Solute Particles

. 这里 $\Phi_{V}, \Phi_{U}$ 分别是溶剂粒子和溶质粒子之间的相互作用能，和 $\Phi_{V U}$ 是溶剂和溶质粒子之间的相互作用能

d \left } { \backslash \text { boldsymbol{r} } { V } \backslash \text { right } } = d \backslash \text { boldsymbol } { r } { V } \wedge { 1 } d \backslash \text { boldsymbol } { r } \vee V } \wedge { 2 } , \backslash \text { dots, } d \backslash \text { left } { \backslash \text { boldsymbol } { r } U } \backslash r i g h t } = d

## 有限元方法代写

tatistics-lab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。