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# 物理代写|热力学代写Thermodynamics代考|CHE713 Amount of Information

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## 物理代写|热力学代写Thermodynamics代考|Amount of Information

So far, our information definition of entropy is only qualitative-and therefore, according to Lord Kelvin, unscientific (see quote on p. 75) because we have not yet specified exactly what is meant by “amount of information.” Fortunately, the field of information theory provides us with exactly the quantitative formula that we need.
$\triangleright \triangleright \triangleright$ To Ponder… In fact, it is the same basic formula that is used in computer science-to determine, e.g., how much information can be stored on your computer, or how many bytes are needed to store the MP3 file for your favorite song.

The framework is that of a hypothetical observer, who gains information by performing a measurement that yields one outcome from a number of possibilities.

Definition $10.2$ (quantitative) The amount of information gained from any measurement is the logarithm of $\bar{\Omega}$, where $\Omega$ is the number of possible outcomes.

Definition $10.2$ is technically only valid if the possible outcomes are all equally likely; if not, we need a means of estimating the “effective” number of possible or available outcomes – e.g., by ignoring those that are extremely unlikely.

It makes intuitive sense that the amount of information should increase with $\Omega$. Why is it that $\log (\Omega)$ is the amount of information, though, and not simply $\Omega$ itself? This is so that the amount of information gained from separate, independent measurements is additive. By “additive,” we mean that the total amount of information gained from independent measurements is just the sum of the amounts of information gained from each measurement separately.

## 物理代写|热力学代写Thermodynamics代考|Application to Thermodynamics

Combining the qualitative Definition $10.1$ (p. 79) with the information theory results from Section 10.3, we are now in a position to provide a quantitative information definition of the entropy for a thermodynamic system, as follows:
Definition $10.3$ (quantitative) The entropy, denoted ‘ $S$ ‘, is defined to be
$$S=k \ln (\Omega),$$
where ‘ln’ denotes the natural $\log , \Omega$ is the number of possible or available molecular states (of the whole system) associated with the thermodynamic state, and $k$ is the Boltzmann constant.

Equation $10.2$ is the famous Boltzmann entropy formula-i.e., the statistical definition developed by Boltzmann and by Gibbs.

In principle, computing the entropy for a given thermodynamic state reduces to simply counting the number of corresponding molecular states, $\Omega$. However, this is not at all easy to do in practice. For one thing, $\Omega$ is absolutely enormous – much, much larger than Avogadro’s number, $N_{\mathrm{A}}$, and in fact much closer to $\exp \left(N_{\mathrm{A}}\right)$. The logarithm in Equation (10.2) helps a lot, by reducing the mind-bogglingly huge $\Omega$ down to something that is “merely” on the order of $N_{\mathrm{A}}$. The factor of $k$ then converts this molecularscale number to macroscopic units, leaving us in the end with typically quite reasonable SI values for $S$.

Nevertheless, there is still the problem of actually counting $\Omega$. This is essentially impossible in the non-ideal case when $N$ is on the order of $N_{\mathrm{A}}$; we must therefore make do with the largest $N$ values that can be achieved using the latest computers, and hope that these are “large enough” to obtain reasonable results.

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## 物理代写|热力学代写THERMODYNAMICS代考|POROUS MEDIA

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$$\mathbf{q} w=-K_w \nabla H_w \quad ; \quad \mathbf{q}w \equiv \mathbf{v} \varphi_p \quad ; \quad H_w \equiv z+\frac{p}{\rho g}$$ 在饱和的各向同性多孔介质中。这里 $\mathbf{v}, g, p, z, \rho, \varphi_p \equiv \frac{V_p}{V \text { tot }}, V_p$ 和 $V{\text {tot }}$ 是水流速度 $\mathbf{v}$ ，重力加速度的绝对值，压力，垂直坐标，水的质量密度，介质的孔隙率 whichissupposeduniformbelow，分别为空隙体积和总体积； $\mathbf{q}_w, H_w, K_w>0$ 和 $\frac{1}{K_w}$ 分别被称为“体积水通量”、水头”、水力传导率”和“水力电阻率”。
Ifthemediumisanisotropic\$K\$isreplacedbyatensor; nothingessentialchangesinthe following. 这里的“饱和”是指所有孔隙都被填充和传导，因此水力 传导率最大。有两点值得注意：首先， $\rho g H_w$ 是对焓的贡南 $h$ Sect能量平衡中的每单位质量。4.2.7在一个均匀、恒定、垂直的引力场中一-一个明显的例子就是地球 的场。其次，达西定律在形式上类似于静止的刚性导体中的欧姆定律 Sect. 5.3.1: $\mathbf{q}_w, \frac{1}{K_w}$ 和 $H_w$ 分别起电流密度、电阻率和静电势的作用。这个类比表明潜伏着一 些变分原理。 太低，则达西定律不再是线性的，并且 $K_w$ 只有当 $\left|\mathbf{q}_w\right|$ 超过一个阈值。另一个非线性的例子发生在大范围内 $\left|\mathbf{q}_w\right|:$ 过多的水倒入系统中，确实会破坏固体的多孔结 构并形成微裂缝；后者可能合并’ fingering 进入较大的裂缝，这些裂缝作为水流穿过系统的优先通道。在这种情况下， $\frac{d K_a}{d\left|\mathbf{q}_w\right|}>0$.

## 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。