### 物理代写|统计力学代写Statistical mechanics代考|PHYS3034

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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|INTERNAL ENERGY: WORK AND HEAT

In the performance of adiabatic work $W_{\text {ad }}$-work on adiabatically isolated systems-it’s found that transitions $i \rightarrow f$ produced between reproducible equilibrium states ${ }^{8}(i, f)$ depend only on the amount of work and not on how it’s performed. Regardless of how the proportions of the types of work are varied, the same total amount of adiabatic work results in the same transition $i \rightarrow f$. This discovery is of fundamental importance. If the transition $i \rightarrow f$ produced by adiabatic work is independent of the means by which it’s brought about, it can depend only on the initial and final states $(i, f)$. That implies the existence of a physical quantity associated with equilibrium states that “couples” to adiabatic work, the internal energy $U$, such that ${ }^{9}$
$$\Delta U=U_{f}-U_{i}=W_{\text {ad } .} .$$
Internal energy is a state variable-one that depends only on the state of equilibrium of the system and not on how the system was prepared in that state. Adiabatic work done on a system increases its internal energy and is taken as a positive quantity. Adiabatic work done by a system (somewhere in the environment a weight is higher, a spring is compressed) is accompanied by a decrease in internal energy, and is taken as a negative quantity. Changes in internal energy come at the expense of adiabatic work done on or by a system. Equation (1.1) expresses conservation of energy: If we don’t let heat escape, work performed on the system is stored in its internal energy, energy that can be recovered by letting the system do adiabatic work on the environment. We’ll say that internal energy is the storehouse of adiabatic work.

Now, let work $W$ be performed under nonadiabatic conditions. It’s found, for the same transition $i \rightarrow f$ produced by $W_{\text {ad }}$, that $W \neq \Delta U$. The energy of mechanical work is not conserved in systems with diathermic boundaries. Energy conservation is one of the sacred principles of physics, and we don’t want to let go of it. The principle can be restored by recognizing different forms of energy. ${ }^{10}$ The heat transferred to or from the system, $Q$, is the difference in work
$$Q \equiv \Delta U-W$$
that effects the same change in state of systems with the two types of boundaries, $Q=W_{\text {ad }}-W$. If it takes more work $W$ to produce the same change of state as that under adiabatic conditions, $Q<0$ : heat leaves the system by flowing through the boundary; $Q>0$ corresponds to heat entering the system. ${ }^{11}$ Equation (1.2) is the first law of thermodynamics. One might think it applies to closed systems only, based on how we’ve formulated it. It applies to open systems when we introduce another kind of work-chemical work – the energy required to change the amount of matter in the system (see Section 1.6). The point here is that the nature of the boundaries allows us to classify different types of energy.

## 物理代写|统计力学代写Statistical mechanics代考| IRREVERSIBILITY, DISORGANIZATION

Entropy-discovered through an analysis of the second law of thermodynamics-is an unexpected yet significant development. ${ }^{17}$ The Clausius inequality is a consequence of the second law:[3, p33]
$$\oint \frac{\mathrm{d} Q}{T} \leq 0,$$
where the integral is over all steps of a cyclic process (one that returns a system to its initial state), $T$ is the absolute temperature 18 at which heat transfer $₫ Q$ occurs, and where equality in (1.6) holds for reversible heat transfers, ${ }^{19}$
$$\oint \frac{(\mathrm{d} Q){\mathrm{rev}}}{T}=0 .$$ Differentials of state variables are exact. ${ }^{20}$ Exact differentials $\mathrm{d} g$ have the property that $\oint{C} \mathrm{~d} g=$ 0 for any integration path ${ }^{21} C$. We infer from Eq. (1.7) the existence of a state variable, entropy, the differential of which is ${ }^{22}$
$$\mathrm{d} S \equiv(\mathbb{\pi} Q){\mathrm{rev}} / T .$$ The quantity $T^{-1}$ is the integrating factor ${ }^{23}$ for $(\mathrm{dQ}){\text {rev }}$. As a state variable, entropy is defined only in equilibrium. Changes in entropy between equilibrium states $(A, B)$ are found by integrating its differential, Eq. (1.8),
$$S(B)-S(A)=\int_{A}^{B}(\mathrm{~d} Q)_{\mathrm{rev}} / T$$ for any path connecting $A$ and $B$, such that heat transfers occur reversibly at all stages. ${ }^{24}$

## 物理代写|统计力学代写Statistical mechanics代考|INTERNAL ENERGY: WORK AND HEAT

$$\Delta U=U_{f}-U_{i}=W_{\mathrm{ad} . .} .$$

$$Q \equiv \Delta U-W$$

## 物理代写|统计力学代写Statistical mechanics代考| IRREVERSIBILITY, DISORGANIZATION

$$\oint \frac{\mathrm{d} Q}{T} \leq 0$$

$$\oint \frac{(\mathrm{d} Q) \mathrm{rev}}{T}=0 .$$

$$\mathrm{d} S \equiv(\pi Q) \mathrm{rev} / T .$$

$$S(B)-S(A)=\int_{A}^{B}(\mathrm{~d} Q)_{\mathrm{rev}} / T$$

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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。