### 物理代写|统计力学代写Statistical mechanics代考|PHYC30017

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• 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|统计力学代写Statistical mechanics代考|CHEMICAL PQTENTIAL AND OPEN SYSTEMS

Work is the energy to change the macroscopically observable, extensive properties of systems. What work is required to change the number of particles, $N$ ? Open systems allow the exchange of matter as well as energy with the environment. The first law for open systems is 48
$$\mathrm{d} U=T \mathrm{~d} S-P \mathrm{~d} V+\mu \mathrm{d} N,$$
where $\mu$ is the chemical potential -roughly the energy to add another particle of given chemical species to the system. ${ }^{49,50} \mathrm{We}$ can now answer the question posed at the end of Section $1.2$. We see from Eq. (1.21) that $U$ is a function of $S, V, N: U=U(S, V, N)$, and more generally $U=$ $U\left(S, X_{*}, N\right)$. Wc can thercforc identify the chemical potential as the dcrivative
$$\mu \equiv\left(\frac{\partial U}{\partial N}\right)_{S, V},$$ the change in internal energy upon adding a particle, holding $S, V$ fixed. ${ }^{51}$ The chemical potential is often. but not always. a negative quantity [3. p39]. We require $\mu \leq 0$ for bosons. whereas there is no restriction on the sign of $\mu$ for fermions (Section 5.5). Only if inter-particle interactions are sufficiently repulsive does $\mu$ become positive. The Fermi energy is an example; the repulsive interaction in that case is the requirement of the Pauli exclusion principle. We’ll examine the effects on $\mu$ of a hard-core, short-range repulsive inter-particle potential in Sections $6.4$ and 7.3.
It’s often better to write the first law in terms of entropy, ${ }^{52}$
$$\mathrm{d} S=\frac{1}{T} \mathrm{~d} U-\frac{1}{T} \sum_{j} Y_{j} \mathrm{~d} X_{j}-\frac{\mu}{T} \mathrm{~d} N .$$

## 物理代写|统计力学代写Statistical mechanics代考| Mathematical interlude

Consider three variables connected through a functional relation, $f(x, y, z)=0$. Any two can be taken as independent, and each can be considered a function of the other two: $x=x(y, z)$, $z=z(x, y)$, or $y=y(x, z)$. Form the differential of $x$ in terms of the differentials of $y$ and $z$ :
$$\mathrm{d} x=\left(\frac{\partial x}{\partial y}\right){z} \mathrm{~d} y+\left(\frac{\partial x}{\partial z}\right){y} \mathrm{~d} z .$$
Now form the differential of $z$ in terms of $x$ and $y$,
$$\mathrm{d} z=\left(\frac{\partial z}{\partial x}\right){y} \mathrm{~d} x+\left(\frac{\partial z}{\partial y}\right){x} \mathrm{~d} y .$$
Suhstitute d $z$ in Fq . (1.28) for that in Fr. (1.27). We find:
$$0=\left[\left(\frac{\partial x}{\partial z}\right){y}\left(\frac{\partial z}{\partial x}\right){y}-1\right] \mathrm{d} x+\left[\left(\frac{\partial x}{\partial y}\right){z}+\left(\frac{\partial x}{\partial z}\right){y}\left(\frac{\partial z}{\partial y}\right){x}\right] \mathrm{d} y .$$ For Eq. (1.29) to hold for arbitrary $\mathrm{d} x, \mathrm{~d} y$, we have $$1=\left(\frac{\partial x}{\partial z}\right){y}\left(\frac{\partial z}{\partial x}\right)_{y}$$

$$-1=\left(\frac{\partial x}{\partial y}\right){z}\left(\frac{\partial y}{\partial z}\right){x}\left(\frac{\partial z}{\partial x}\right)_{y}$$
Equation (1.30) is the reciprocity relation and Eq. (1.31) the cyclic relation. ${ }^{55}$

## 物理代写|统计力学代写Statistical mechanics代考|CHEMICAL PQTENTIAL AND OPEN SYSTEMS

$$\mathrm{d} U=T \mathrm{~d} S-P \mathrm{~d} V+\mu \mathrm{d} N,$$

$$\mu \equiv\left(\frac{\partial U}{\partial N}\right){S, V},$$ 添加粒子时内能的变化，保持 $S, V$ 固定的。 ${ }^{51}$ 化学势通常是。但不总是。负数 [3. 第 39 页]。我们需要 $\mu \leq 0$ 对 于玻色子。而对的符号没有限制 $\mu$ 对于费米子 (第 $5.5$ 节) 。只有当粒子间的相互作用足够排后时 $\mu$ 变得积极。费 米能量就是一个例子；这种情况下的排斥相互作用是泡利不相容原理的要求。我们将检查对 $\mu$ 截面中的硬核、短 程排斥粒子间势 $6.4$ 和 $7.3$ 。 用樀的形式写出第一定律通常会更好， ${ }^{52}$ $$\mathrm{d} S=\frac{1}{T} \mathrm{~d} U-\frac{1}{T} \sum{j} Y_{j} \mathrm{~d} X_{j}-\frac{\mu}{T} \mathrm{~d} N .$$

## 物理代写|统计力学代写Statistical mechanics代考| Mathematical interlude

$$\mathrm{d} x=\left(\frac{\partial x}{\partial y}\right) z \mathrm{~d} y+\left(\frac{\partial x}{\partial z}\right) y \mathrm{~d} z .$$

$$\mathrm{d} z=\left(\frac{\partial z}{\partial x}\right) y \mathrm{~d} x+\left(\frac{\partial z}{\partial y}\right) x \mathrm{~d} y$$

$$0=\left[\left(\frac{\partial x}{\partial z}\right) y\left(\frac{\partial z}{\partial x}\right) y-1\right] \mathrm{d} x+\left[\left(\frac{\partial x}{\partial y}\right) z+\left(\frac{\partial x}{\partial z}\right) y\left(\frac{\partial z}{\partial y}\right) x\right] \mathrm{d} y .$$

$$\begin{gathered} 1=\left(\frac{\partial x}{\partial z}\right) y\left(\frac{\partial z}{\partial x}\right){y} \ -1=\left(\frac{\partial x}{\partial y}\right) z\left(\frac{\partial y}{\partial z}\right) x\left(\frac{\partial z}{\partial x}\right){y} \end{gathered}$$

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## 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。