物理代写|统计力学代写Statistical mechanics代考|PHYS3934

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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|统计力学代写Statistical mechanics代考|The Second Law

The second law of thermodynamics concerns the nonexistence of a perpetuum mobile. This has long been a contentious issue but it is based on one of the most obvious facts of everyday experience, namely that certain processes are irreversible. This means simply that they cannot be reversed in time. Thus, e.g. the breaking of a glass on the floor is an irreversible process because the reverse process: the spontaneous creation of a glass from pieces on the floor simply does not happen. Characteristic of a process like that is that an ordered structure is destroyed which cannot be recreated without “doing something” to the system. It makes sense, therefore, to postulate the existence of a measure for the order in the system. Traditionally, one has chosen to introduce a measure of disorder instead, which is called the entropy. The second law is then a formulation of the creation of entropy in processes where the system is undisturbed. The first precise formulation of the second law was given by Lord Kelvin and Clausius. Kelvin’s formulation reads as follows.
There exists no thermodynamic transformation of which the only result is to convert heat from a heat reservoir to work.
Clausius’ formulation is slightly different.
There exists no thermodynamic transformation of which the only result is the transfer of heat from a colder heat reservoir to a hotter heat reservoir.
Clausius deduced from these basic statements the existence of a function of state which can only increase if the system is thermally isolated. This is the entropy which we simply postulate to exist. We then show in chapter 5 that our formulation implies their original statements.
To formulate the law in its entirety we need the following observation:
All thermodynamic parameters can be divided into two categories, namely intensive and extensive parameters. Intensive parameters are thermodynamic parameters that are independent of the size of the system; extensive parameters are thermodynamic parameters that are directly proportional to the size of the system.

物理代写|统计力学代写Statistical mechanics代考|Thermal Engines and Refrigerators

Thermodynamics was developed in a study of the efficient operation of thermal engines, the steam engine in particular. It is still important for these applications, e.g. the car engine, refrigerators, and steam turbines in electricity plants. The principle of operation of a heat engine is pictured schematically in figure 5.1.

Heat is absorbed from a heat reservoir at temperature $T_{A}$. The engine $E$ converts part of this heat into useful work $W$ and rejects the surplus heat $Q_{B}$ to a second heat reservoir at temperature $T_{B}$. Clearly,
$$W=Q_{A}-Q_{B}$$

The efficiency of the engine is defined as
$$\eta=\frac{W}{Q_{A}} .$$
Obviously, $\eta \leqslant 1$, but the second law gives a more stringent bound on the efficiency. Indeed, since
$$\Delta S=-\frac{Q_{A}}{T_{A}}+\frac{Q_{B}}{T_{B}} \geqslant 0$$
we find
$$\eta \leqslant 1-\frac{T_{B}}{T_{A}} \equiv \eta_{C}$$
The maximum attainable efficiency $\eta_{C}$ is called the Carnot efficiency. It also follows that the most efficient engines are reversible, $\Delta S=0$. An example of a process in which the Carnot efficiency is attained is the Carnot process. For an ideal gas this process is represented by the cycle abcd in the following $p-V$ diagram of figure $5.2$.

$\mathrm{ab}$ and cd are isotherms, bc and da are adiabatics. In practice this process is very difficult to carry out. Most practical electricity plants use steam as working fluid. The process can then take place inside the coexistence region of steam and liquid water. This is called wet steam. It is then much easier to keep the temperature constant during the transfer of heat as the evaporation occurs at constant temperature. The process is most conveniently represented in a $T-s$ diagram (figure $5.3$ ).

In fact, the actual process used in practical power plants is considerably more complicated. Let us consider two of the most important modifications.

物理代写|统计力学代写Statistical mechanics代考|The Fundamental Equation

In this chapter, we show that the thermodynamics of a (simple) system is completely determined by a single function: the entropy density as a function of the internal energy density and the specific volume. This function has an important property, namely concavity, which enables us to define a number of other thermodynamic functions in chapter 8. Let us recall the definition of convex and concave functions: $A$ region $D$ in $\mathbb{R}^{k}$ is called convex if for every
two points $\vec{x}, \vec{y} \in D$ and every $t \in[0,1]$, also $t \vec{x}+(1-t) \vec{y} \in D$. A function $g: D \rightarrow \mathbb{R} \cup{+\infty}$ from a convex region $D \subset \mathbb{R}^{k}$ to the real line united with $+\infty$ is called convex if, for every $t \in[0,1]$ and all $\vec{x}, \vec{y} \in D$,
$$g(t \vec{x}+(1-t) \vec{y}) \leqslant \operatorname{tg}(\vec{x})+(1-t) g(\vec{y})$$
$g$ is called concave if $-g$ is convex. A region $D$ is convex when it does not have dents. Figures $6.1$ (a) and $6.1$ (b) show a convex region and a non-convex region respectively.

Similarly, a function is convex if its graph is everywhere bending upwards. In chapter 7 , we state and prove various useful facts about convex functions. Here, we first outline the significance for thermodynamic functions.

In the introduction and also in chapter 4 we remarked that, in order to describe a macroscopic system mathematically we have to take the thermodynamic limit, i.e. the limit of an infinitely large system with a finite particle number density: $N \rightarrow \infty, V \rightarrow \infty ; \quad \rho=N / V$ fixed. We also write $v=1 / \rho$ for the specific volume. In this limit all surface effects disappear and we are left with the pure bulk properties of the system. For intensive parameters $X(N, V, T)$ we expect the limit
$$\lim {N, V \rightarrow \infty ; V / N=v} X(N, V, T)=x(v, T)$$ to exist, while for extensive variables $Y(N, V, T)$ we expect the limit $$\lim {N, V \rightarrow \infty ; V / N=v} \frac{Y(N, V, T)}{N}=y(v, T)$$
to exist. In other words, $X(N, V, T)=x(v, T)+o(1)$ and $Y(N, V, T)=$ $N y(v, T)+o(N)$ as $N \rightarrow \infty$. In fact, in real systems the values of $X$ and $Y / N$ fluctuate around their mean values $x(v, T)$ and $y(v, T)$.

Let us now concentrate in particular on the entropy density for a simple system defined by
$$s(u, v)=\lim {N, V \rightarrow \infty ; V / N=v} \frac{1}{N} S(N u, V, N)$$ We shall argue that this function is concave in both its arguments. To show this we consider two systems $\Sigma{1}$ with parameters $\left(U_{1}, V_{1}, N_{1}\right)$ and $\Sigma_{2}$ with parameters $\left(U_{2}, V_{2}, N_{2}\right)$, which are brought into thermal and mechanical contact so that they can exchange energy and their relative volumes ean adjust. Figure $6.2$ gives an impression of this situation.

物理代写|统计力学代写Statistical mechanics代考|Thermal Engines and Refrigerators

Δ小号=−问一个吨一个+问乙吨乙⩾0

物理代写|统计力学代写Statistical mechanics代考|The Fundamental Equation

G(吨X→+(1−吨)是→)⩽tg⁡(X→)+(1−吨)G(是→)
G称为凹如果−G是凸的。一个地区D没有凹痕时是凸的。数字6.1(a) 和6.1(b) 分别表示凸区域和非凸区域。

s(在,在)=林ñ,在→∞;在/ñ=在1ñ小号(ñ在,在,ñ)我们将论证这个函数在它的两个参数中都是凹的。为了证明这一点，我们考虑两个系统Σ1带参数(在1,在1,ñ1)和Σ2带参数(在2,在2,ñ2)，它们被带入热和机械接触，以便它们可以交换能量并且它们的相对体积可以调整。数字6.2给人这种情况的印象。

有限元方法代写

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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。