Posted on Categories:信息论, 数学代写

# 数学代写|信息论代写Information Theory代考|A Baffling Experiments in Systems of Interacting Particles

avatest™

avatest信息论information theory代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。https://avatest.org/， 最高质量的信息论information theory作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于统计Statistics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此信息论information theory作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

avatest™ 为您的留学生涯保驾护航 在澳洲代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的澳洲代写服务。我们的专家在信息论information theory代写方面经验极为丰富，各种信息论information theory相关的作业也就用不着 说。

## 数学代写|信息论代写Information Theory代考|“Entropy of Mixing” of Two Different Ideal Gas

In this section we briefly mention two baffling experiment. For more details, the reader is referred to Ben-Naim [11]. In this Chapter we studied two apparently very different processes. One is an expansion of an ideal gas from $V$ to $2 V$. The second is mixing of two different gases. In both of this processes we saw that the main “driving force” is the tendency of the locational distribution function to evolve into a distribution that maximizes the SMI. In both processes, at the instant we remove the partition we have a distribution which is a step-function with a constant value in one compartment, and zero value in the second compartment. This distribution starts to evolve right after the removal of the partition until we reach a final distribution which is uniform in the entire new volume of $2 \mathrm{~V}$. This final distribution is also the equilibrium one. This is the reason why we identify the distribution which maximizes the SMI with the equilibrium distribution. This is also why we identify the tendency to evolve into the new equilibrium state as one manifestation of the Second Law of thermodynamics.

We say that these two processes are irreversible-meaning that we never observe the reversal of these processes. A gas occupying the volume $2 V$ will never be found in a smaller region, say, in one compartment of volume $V$, and a mixture of two gases will never separate into two places each containing only one component.

All that has been said above regarding ideal gases is true. It is, in general not true when there are strong intermolecular interactions. Here, we describe in a qualitative manner how a seemingly reversal of both of the processes of expansion and mixing can occur. A more quantitative discussion of such processes is available in Ben-Naim $[11]$

Supposed we start with a system of two compartments, Fig. 4.11. The left contains $N_A$ molecules of type $A$ in a solvent $\alpha$. The right contains $N_A$ molecules of type $A$ in a solvent $\beta$. The two solvents are transparent, but $\mathrm{A}$ is blue.

Initially, all the $2 N_A$ molecules are distributed in the entire system of volume $2 \mathrm{~V}$. We now remove the constraint that allows the passage of A molecules between the two compartments. If $\mathrm{A}$ interacts very strongly and attractively with the solvent $\alpha$, then we shall observe that most of the A molecules will move into the left compartment. The “driving force” for this process is the strong solvation of A in $\alpha$. Since the two solvents are transparent, an observer who is not aware of the existence of the solvents $\alpha$ and $\beta$ will “see” that the blue molecules A move from occupying the entire volume $2 V$ into the smaller region of volume $V$. This will seem to the observer as a “reversal” of the expansion process.

## 数学代写|信息论代写Information Theory代考|Communal SMI and Communal Entropy

The concept of communal entropy has featured within the lattice models of liquids and mixtures. In this section we show first that this communal SMI is due to a combination of assimilation and expansion. And second, that this communal SMI turns to entropy only when the number of particles in each cell is very large.

Figure 4.12 depicts a process which called delocalization. Initially, we have $N$ particles, each confined to a cell of size $v$. We remove all the partitions and the particles are allowed to occupy the entire volume $V$. The change in entropy in this process is:
\begin{aligned} \Delta S= & S(\text { ideal gas })-S(\text { localized }) \ = & k N \ln \left(\frac{V}{\Lambda^3}\right)-k \ln N !+\frac{3 k N}{2} \ & -k N \ln \left(v / \Lambda^3\right)-\frac{3 k N}{2} \end{aligned}
Here $v=V / N$ is the volume of each call. Application of the Stirling approximation to $\ln N$ ! yields:
$$\Delta S=k N$$
The quantity $k N$, in Eq. (4.27) is known as the communal entropy, or the “delocalization” entropy. In fact, $\Delta S$ in Eq. (4.27) is the sum of two effects; the increase of the accessible volume, for each particle from $v$ to $V$, and the assimilation of $N$ particles. These two contributions are explicitly given by:

$$\Delta S=k \ln \left(\frac{V}{v}\right)-k \ln N !$$
Note that the volume per particle is denoted by $v=V / N$. Clearly, first term on the right hand side of (4.28) is the dominating one, i.e., the contribution of the volume change accessible to each particle is larger than the assimilation contribution.
Using the Stirling approximation, we get approximately:
$$\Delta S=k N \ln N-(k N \ln N-k N)=k N$$

## 数学代写|信息论代写Information Theory代考|Communal SMI and Communal Entropy

$$\Delta S=S(\text { ideal gas })-S(\text { localized })=\quad k N \ln \left(\frac{V}{\Lambda^3}\right)-k \ln N !+\frac{3 k N}{2}-k N \ln \left(v / \Lambda^3\right)-\frac{3 k N}{2}$$

$$\Delta S=k N$$

$$\Delta S=k \ln \left(\frac{V}{v}\right)-k \ln N !$$

$$\Delta S=k N \ln N-(k N \ln N-k N)=k N$$

## 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。