### 英国补考|统计力学代写Statistical mechanics代考|PHYS 3006

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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

## 英国补考|统计力学代写Statistical mechanics代考|THERMODYNAMIC POTENTIALS

Equilibrium is specified by the values of state variables. Whether measured or postulated by the laws of thermodynamics, state variables are not independent of each other (peek ahead to Eq. (1.53)). Depending on the system, there may be variables that are not readily subject to experimental control and yet others that are. Legendre transformations (defined in Appendix C) provide a way of obtaining equivalent descriptions of the energy of a system known as thermodynamic potentials, which involve variables that may be easier to control.

Three Legendre transformations of $U(S, V)$ can be formed from the products of variables with the dimension of energy (when the number of particles is fixed): $T S$ and $P V$ (our old friends heat and work). ${ }^{40}$ They are:
$$\begin{array}{ll} F \equiv U-T S & \text { (Helmholtz free energy) } \ H \equiv U+P V & \text { (enthalpy) } \ G \equiv U-T S+P V=F+P V=H-T S . & \text { (Gibbs free energy) } \end{array}$$
The relationships amongst these functions are shown in Fig. 1.5. Their physical interpretation is
Figure 1.5: The four ways to say energy: Legendre transformations of the internal energy function (for fixed particles numbers).

## 英国补考|统计力学代写Statistical mechanics代考| FREE ENERGY AND DISSIPATED ENERGY

By rewriting (1.15), we have the inequality $-\mathrm{d} W \leq T \mathrm{~d} S-\mathrm{d} U$. For there to be work done by the system (counted as a negative quantity), we must have $\mathrm{d} U<T \mathrm{~d} S$, a generalization of $\Delta U=W_{\text {ad }}$ : $\Delta U<0$ if $W_{\text {ad }}<0$. The maximum value of $\mathrm{d} W$ (counted as a negative quantity) is therefore $|\mathrm{d} W|{\max }=T \mathrm{~d} S-\mathrm{d} U$. Undoing the minus sign, $\mathrm{d} W{\max }=\mathrm{d} U-T \mathrm{~d} S=[\mathrm{d} F]{T}$. The Helmholtz energy is the maximum obtainable work at constant $T: W{\max }-[\Delta F]_{T}$.

Thus, not all of the energy change $\Delta U$ is available for work if $\Delta S>0$, which is the origin of the term free energy: the amount of energy available for work. For this reason $F$ is called the work function. Enthalpy is the heat function, $[\Delta H]{P}=Q$, and the Helmholtz energy is the work function, $[\Delta F]{T}=W$. It’s straightforward to show that $[\Delta H]{P}+[\Delta F]{T}=\Delta U$.

The Gibbs energy also specifies a free energy. With $\mathrm{d} W=-P \mathrm{~d} V+\mathrm{d} W^{\prime}$, (1.15) implies $-\mathrm{d} W^{\prime} \leq T \mathrm{~d} S-(\mathrm{d} U+P \mathrm{~d} V)$. To obtain other work from the system, we must have that $T \mathrm{~d} S>$ $\mathrm{d} U+P \mathrm{~d} V$. The maximum value of $\mathrm{d} W^{\prime}$ (counted as a negative quantity) is therefore $\mathrm{d} W_{\max }^{\prime}=$ $\mathrm{d} U+P \mathrm{~d} V-T \mathrm{~d} S=[\mathrm{d} G]{T, P}$. The Gibbs energy is the maximum work obtainable from the system in a form other than $P \mathrm{~d} V$ work: $W{\max }^{\prime}=[\Delta G]_{T, P}$.

## 英国补考|统计力学代写Statistical mechanics代考|THERMODYNAMIC POTENTIALS

$F \equiv U-T S \quad$ (Helmholtz free energy) $H \equiv U+P V \quad$ (enthalpy) $G \equiv U-T S+P V=F+$

## 英国补考|统计力学代写Statistical mechanics代考| FREE ENERGY AND DISSIPATED ENERGY

$W \max ^{\prime}=[\Delta G]_{T, P}$

## 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。