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• 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|Interfaces and Interfacial Surface Tensions

In the case where the repulsive energy between two different species predominates over the entropy of mixing, two phases well separate forming domain boundaries or interfaces as shown in Fig. 8.5b. If there are $n$ molecules of species $A$ and $B$ each at the two dimensional interface, $N_A-n$ and $N_B-n$ molecules are within the three dimensional bulk phases of $A$ and $B$. The interfacial surface tension, which is the free energy derivative with respect to the change of the surface area, is given as follows. Since each phase are ordered, their entropies are zero, while the internal energy is

$$E=-\frac{q}{2}\left{\left(N_A-n\right) b_{A A}+\left(N_B-n\right) b_{B B}\right}-n\left{\frac{q-1}{2}\left(b_{A A}+b_{B B}\right)+b_{A B}\right}$$
where the first term is from the bulk and the second from the interface.
The surface tension is given by
\begin{aligned} \gamma &=\left(\frac{\partial F}{\partial A}\right)=\frac{\partial E}{a \partial n} \ &=\frac{1}{a}\left{\left(b_{A A}+b_{B B}\right) / 2-b_{A B}\right}=\frac{b}{2 a} \end{aligned}
where $A=n a$ is the total area of the interface and $a$ is the area per site. The surface tension, which is positive for this case, describes the energy of transferring a molecule from the two bulk media into the interface. For the surface of a pure media composed of $A$ molecules in contact with the vacuum or a gas, one may apply (8.33) to find the surface tension; with $b_{B B}=0=b_{A B}$, it yields
$$\gamma=b_{A A} /(2 a)$$
These results can be adapted to line tension of a domain in two dimensions, with $a$ interpreted as the size of a molecule.

## 物理代写|统计物理代写Statistical Physics of Matter代考|Exact Solution of 1-D Ising Model

The traced $(T r)$ quantities are invariant under a transformation of the basis; in the basis $\boldsymbol{P}$ is diagonal (8.40) can be expressed in terms of the two eigenvalues $\lambda_{\pm}$ of $\boldsymbol{P}$ with $\lambda_{+}>\lambda_{-}$:
\begin{aligned} Z &=\lambda_{+}^N+\lambda_{-}^N \ &=\lambda_{+}^N\left(1+\left(\frac{\lambda_{-}}{\lambda_{+}}\right)^N\right) \approx \lambda_{+}^N, \end{aligned}
where the last can be an excellent approximation provided that $N$ is very large. The eigenvalues are obtained by the secular determinant $|\boldsymbol{P}-\lambda \boldsymbol{I}|=0$ :
$$\lambda_{\pm}=e^{\beta J}\left{\cosh \beta h \pm\left(\sinh ^2 \beta h+e^{-4 \beta J}\right)^{1 / 2}\right} .$$
The free energy then is
\begin{aligned} F &=-k_B T \ln Z \ &=-N k_B T \ln \left[e^{\beta J}\left{\cosh \beta h+\left(\sinh ^2 \beta h+e^{-4 \beta J}\right)^{1 / 2}\right}\right] . \end{aligned}
The average magnetization per site is proportional to
\begin{aligned} m &=\sigma=-\frac{\partial F}{N \partial h} \ &=\frac{\sinh \beta h}{\left(\sinh ^2 \beta h+e^{-4 \beta J}\right)^{1 / 2}}, \end{aligned}
In the absence of an external field $(h=0), m=0$, i.e., spontaneous magnetization does not occur at any finite temperature, i.e., no ferromagnetic phase transition occurs in one dimensional spin systems. The reason is that the entropy associated with randomizing the spins dominates over the internal energy associated with aligning the spins at any temperature. This domination occurs because the number of nearest neighbors is too small to enable formation of a sufficient number of attractive pairs in one dimension. However, in higher dimensions, the number of nearest neighbor attractions is large enough to induce ferromagnetic transition. As temperature approaches zero, $\sinh \beta h \gg e^{-2 \beta J}, F=-N J$, and $m=\pm 1$; this result suggests that ferromagnetic transition to perfectly aligned spins occurs only at $T=0$. At a finite temperature, this perfect alignment occurs only when $h$ is very high.

# 统计物理代考

## 物理代写|统计物理代写Statistical Physics of Matter代考|Interfaces and Interfacial Surface Tensions

$$\gamma=b_{A A} /(2 a)$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|Exact Solution of 1-D Ising Model

$$Z=\lambda_{+}^N+\lambda_{-}^N \quad=\lambda_{+}^N\left(1+\left(\frac{\lambda_{-}}{\lambda_{+}}\right)^N\right) \approx \lambda_{+}^N,$$

\left 的分隔符缺失或无法识别

\left 的分隔符缺失或无法识别

$$m=\sigma=-\frac{\partial F}{N \partial h} \quad=\frac{\sinh \beta h}{\left(\sinh ^2 \beta h+e^{-4 \beta J}\right)^{1 / 2}},$$

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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。

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