数学代写|信息论代写information theory代考|FEO3350

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论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|The SMI of a System of Interacting Particles in Pairs Only

In this section we consider a special case of a system of interacting particles. We start with an ideal gas-i.e. system for which we can neglect all intermolecular interactions. Strictly speaking, such a system does not exist. However, if the gas is very dilute such that the average intermolecular distance is very large the system behaves as if there are no interactions among the particle.

Next, we increase the density of the particles. At first we shall find that pairinteractions affect the thermodynamics of the system. Increasing further the density, triplets, quadruplets, and so on interactions, will also affect the behavior of the system. In the following we provide a very brief description of the first order deviation from ideal gas; systems for which one must take into account pair-interactions but neglect triplet and higher order interactions. The reader who is not interested in the details of the derivation can go directly to the result in Eq. (2.51) and the following analysis of the MI.

We start with the general configurational PF of the system, Eq. (2.31) which we rewrite in the form:
$$Z_N=\int d R^N \prod_{i<j} \exp \left[-\beta U_{i j}\right]$$
where $U_{i j}$ is the pair potential between particles $i$ and $j$. It is assumed that the total potential energy is pairwise additive.
Define the so-called Mayer $f$-function, by:
$$f_{i j}=\exp \left(-\beta U_{i j}\right)-1$$
We can rewrite $Z_N$ as:
$$Z_N=\int d R^N \prod_{i<j}\left(f_{i j}+1\right)=\int d R^N\left[1+\sum_{i<j} f_{i j}+\sum f_{i j} f_{j k}+\cdots\right]$$
Neglecting all terms beyond the first sum, we obtain:
$$Z_N=V^N+\frac{N(N-1)}{2} \int f_{12} d R^N=V^N+\frac{N(N-1)}{2} V^{N-2} \int f_{12} d R_1 d R_2$$

数学代写|信息论代写information theory代考|Entropy-Change in Phase Transition

In this section, we shall discuss the entropy-changes associated with phase transitions. Here, by entropy we mean thermodynamic entropy, the units of which are cal/(deg $\mathrm{mol}$ ). However, as we have seen in Chap. 5 of Ben-Naim [1]. The entropy is up to a multiplicative constant an SMI defined on the distribution of locations and velocities (or momenta) of all particles in the system at equilibrium. To convert from entropy to SMI one has to divide the entropy by the factor $k_B \log _e 2$, where $k_B$ is the Boltzmann constant, and $\log _e 2$ is the natural $\log$ arithm of 2 , which we denote by $\ln 2$. Once we do this conversion from entropy to SMI we obtain the SMI in units of bits. In this section we shall discuss mainly the transitions between gases, liquids and solids. Figure 2.9 shows a typical phase diagram of a one-component system. For more details on phase diagrams, see Ben-Naim and Casadei [8].

It is well-known that solid has a lower entropy than liquid, and liquid has a lower entropy of a gas. These facts are usually interpreted in terms of order-disorder. This interpretation of entropy is invalid; more on this in Ben-Naim [6]. Although, it is true that a solid is viewed as more ordered than liquid, it is difficult to argue that a liquid is more ordered or less ordered than a gas.

In the following we shall interpret entropy as an SMI, and different entropies in terms of different MI due to different intermolecular interactions. We shall discuss changes of phases at constant temperature. Therefore, all changes in SMI (hence, in entropy) will be due to locational distributions; no changes in the momenta distribution.

The line SG in Fig. 2.9 is the line along in which solid and gas coexist. The slope of this curve is given by:
$$\left(\frac{d P}{d T}\right)_{e q}=\frac{\Delta S_s}{\Delta V_s}$$
In the process of sublimation ( $s$, the entropy-change and the volume change for both are always positive. We denoted by $\Delta V_s$ the change in the volume of one mole of the substance, when it is transferred from the solid to the gaseous phase. This volume change is always positive. The reason is that a mole of the substance occupies a much larger volume in the gaseous phase than in the liquid phase (at the same temperature and pressure).

The entropy-change $\Delta S_s$ is also positive. This entropy-change is traditionally interpreted in terms of transition from an ordered phase (solid) to a disordered (gaseous) phase. However, the more correct interpretation is that the entropy-change is due to two factors; the huge increase in the accessible volume available to each particle and the decrease in the extent of the intermolecular interaction. Note that the slope of the SG curve is quite small (but positive) due to the large $\Delta V_s$.

信息论代写

数学代写|信息论代写information theory代考|The SMI of a System of Interacting Particles in Pairs Only

$$Z_N=\int d R^N \prod_{i<j} \exp \left[-\beta U_{i j}\right]$$

$$f_{i j}=\exp \left(-\beta U_{i j}\right)-1$$

$$Z_N=\int d R^N \prod_{i<j}\left(f_{i j}+1\right)=\int d R^N\left[1+\sum_{i<j} f_{i j}+\sum f_{i j} f_{j k}+\cdots\right]$$

$$Z_N=V^N+\frac{N(N-1)}{2} \int f_{12} d R^N=V^N+\frac{N(N-1)}{2} V^{N-2} \int f_{12} d R_1 d R_2$$

数学代写|信息论代写information theory代考|Entropy-Change in Phase Transition

$$\left(\frac{d P}{d T}\right)_{e q}=\frac{\Delta S_s}{\Delta V_s}$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

数学代写|信息论代写information theory代考|EE430

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论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|The Forth Step: The SMI of Locations and Momentaof N Independent Particles in a Box of Volume V.Adding a Correction Due to Indistinguishabilityof the Particles

The final step is to proceed from a single particle in a box, to $N$ independent particles in a box of volume $V$, Fig. 2.4.

We say that we know the microstate of the particle, when we know the location $(x, y, z)$, and the momentum $\left(p_x, p_y, p_z\right)$ of one particle within the box. For a system of $N$ independent particles in a box, we can write the SMI of the system as $N$ times the SMI of one particle, i.e., we write:
$$\mathrm{SMI}(N \text { independent particles })=N \times \mathrm{SMI} \text { (one particle) }$$
This is the SMI for $N$ independent particles. In reality, there could be correlation among the microstates of all the particles. We shall mention here correlations due to the indistinguishability of the particles, and correlations is due to intermolecular interactions among all the particles. We shall discuss these two sources of correlation separately. Recall that the microstate of a single particle includes the location and the momentum of that particle. Let us focus on the location of one particle in a box of volume $V$. We write the locational SMI as:
$$H_{\max }(\text { location })=\log V$$
For $N$ independent particles, we write the locational SMI as:
$$H_{\max } \text { (locations of N particles) }=\sum_{i=1}^N H_{\max }(\text { one particle })$$
Since in reality, the particles are indistinguishable, we must correct Eq. (2.22). We define the mutual information corresponding to the correlation between the particles as:

$$I(1 ; 2 ; \ldots ; N)=\ln N !$$
Hence, instead of (2.22), for the SMI of $N$ indistinguishable particles, will write:
$$H(\text { Nparticles })=\sum_{i=1}^N H(\text { oneparticle })-\ln N !$$
A detailed justification for introducing $\ln N$ ! as a correction due to indistinguishability of the particle is discussed in Sect. 5.2 of Ben-Naim [1]. Here we write the final result for the SMI of $N$ indistinguishable (but non-interacting) particles as:
$$H(N \text { indistinguishable particles })=N \log V\left(\frac{2 \pi m e k_B T}{h^2}\right)^{3 / 2}-\log N !$$

数学代写|信息论代写information theory代考|The Entropy of a System of Interacting Particles. Correlations Due to Intermolecular Interactions

In this section we derive the most general relationship between the SMI (or the entropy) of a system of interacting particles, and the corresponding mutual information (MI). Later on in this chapter we shall apply this general result to some specific cases. The implication of this result is very important in interpreting the concept of entropy in terms of SMI. In other words, the “informational interpretation” of entropy is effectively extended for all systems of interacting particles at equilibrium.
We start with some basic concepts from classical statistical mechanics [7]. The classical canonical partition function (PF) of a system characterized by the variable $T, V, N$, is:
$$Q(T, V, N)=\frac{Z_N}{N ! \Lambda^{3 N}}$$
where $\Lambda^3$ is called the momentum partition function (or the de Broglie wavelength), and $Z_N$ is the configurational PF of the system”
$$Z_N=\int \cdots \int d R^N \exp \left[-\beta U_N\left(R^N\right)\right]$$
Here, $U_N\left(R^N\right)$ is the total interaction energy among the $N$ particles at a configuration $R^N=R_1, \cdots, R_N$. Statistical thermodynamics provides the probability density for finding the particles at a specific configuration $R^N=R_1, \cdots, R_N$, which is:
$$P\left(R^N\right)=\frac{\exp \left[-\beta U_N\left(R^N\right)\right]}{Z_N}$$
where $\beta=\left(k_B T\right)^{-1}$ and $T$ the absolute temperature. In the following we chose $k_B=1$. This will facilitate the connection between the entropy-change and the change in the SMI. When there are no intermolecular interactions (ideal gas), the configurational $\mathrm{PF}$ is $Z_N=V^N$, and the corresponding partition function is reduced to:
$$Q^{i g}(T, V, N)=\frac{V^N}{N ! \Lambda^{3 N}}$$
Next we define the change in the Helmholtz energy $(A)$ due to the interactions as:
$$\Delta A=A-A^{i g}=-T \ln \frac{Q(T, V, N)}{Q^{i g}(T, V, N)}=-T \ln \frac{Z_N}{V^N}$$
This change in Helmholtz energy corresponds to the process of “turning-on” the interaction among all the particles at constant $(T, V, N)$, Fig. 2.5.
The corresponding change in the entropy is:
\begin{aligned} \Delta S & =-\frac{\partial \Delta A}{\partial T}=\ln \frac{Z_N}{V^N}+T \frac{1}{Z_N} \frac{\partial Z_N}{\partial T} \ & =\ln Z_N-N \ln \mathrm{V}+\frac{1}{T} \int d R^N P\left(R^N\right) U_N\left(R^N\right) \end{aligned}
We now substitute $U_N\left(R^N\right)$ from (2.36) into (2.35) to obtain the expression for the change in entropy corresponding to “turning on” the interactions:
$$\Delta S=-N \ln V-\int P\left(R^N\right) \ln P\left(R^N\right) d R^N$$

信息论代写

数学代写|信息论代写information theory代考|The Forth Step: The SMI of Locations and Momentaof N Independent Particles in a Box of Volume V.Adding a Correction Due to Indistinguishabilityof the Particles

$$\mathrm{SMI}(N \text { independent particles })=N \times \mathrm{SMI} \text { (one particle) }$$

$$H_{\max }(\text { location })=\log V$$

$$H_{\max } \text { (locations of N particles) }=\sum_{i=1}^N H_{\max }(\text { one particle })$$

$$I(1 ; 2 ; \ldots ; N)=\ln N !$$

$$H(\text { Nparticles })=\sum_{i=1}^N H(\text { oneparticle })-\ln N !$$

$$H(N \text { indistinguishable particles })=N \log V\left(\frac{2 \pi m e k_B T}{h^2}\right)^{3 / 2}-\log N !$$

数学代写|信息论代写information theory代考|The Entropy of a System of Interacting Particles. Correlations Due to Intermolecular Interactions

$$Q(T, V, N)=\frac{Z_N}{N ! \Lambda^{3 N}}$$

$$Z_N=\int \cdots \int d R^N \exp \left[-\beta U_N\left(R^N\right)\right]$$

$$P\left(R^N\right)=\frac{\exp \left[-\beta U_N\left(R^N\right)\right]}{Z_N}$$

$$Q^{i g}(T, V, N)=\frac{V^N}{N ! \Lambda^{3 N}}$$

$$\Delta A=A-A^{i g}=-T \ln \frac{Q(T, V, N)}{Q^{i g}(T, V, N)}=-T \ln \frac{Z_N}{V^N}$$

\begin{aligned} \Delta S & =-\frac{\partial \Delta A}{\partial T}=\ln \frac{Z_N}{V^N}+T \frac{1}{Z_N} \frac{\partial Z_N}{\partial T} \ & =\ln Z_N-N \ln \mathrm{V}+\frac{1}{T} \int d R^N P\left(R^N\right) U_N\left(R^N\right) \end{aligned}

$$\Delta S=-N \ln V-\int P\left(R^N\right) \ln P\left(R^N\right) d R^N$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

数学代写|信息论代写information theory代考|COMP2610

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论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|Third Step: Combining the SMI for the Location and Momentum of a Particle in a $1 D$ System. Addition of Correction Due to Uncertainty

If the location and the momentum (or velocity) of the particles were independent events, then the joint SMI of location and momentum would be the sum of the two SMIs in Eqs. (2.4) and (2.12). Therefore, for this case we write:
\begin{aligned} H_{\max }(\text { location and momentum }) & =H_{\max }(\text { location })+H_{\max }(\text { momentum }) \ & =\log \left[\frac{L \sqrt{2 \pi e m k_B T}}{h_x h_p}\right] \end{aligned}
It should be noted that in the very writing of Eq. (2.14), the assumption is made that the location and the momentum of the particle are independent. However, quantum mechanics imposes restriction on the accuracy in determining both the location $x$ and the corresponding momentum $p_x$. Originally, the two quantities $h_x$ and $h_p$ that we defined above, were introduced because we did not care to determine the location and the momentum with an accuracy better than $h_x$ and $h_p$, respectively. Now, we must acknowledge that quantum mechanics imposes upon us the uncertainty condition, about the accuracy with which we can determine simultaneously both the location and the corresponding momentum of a particle. This means that in Eq. (2.14), $h_x$ and $h_p$ cannot both be arbitrarily small; their product must be of the order of Planck constant $h=6.626 \times 10^{-34} \mathrm{Js}$. Therefore, we introduce a new parameter $h$, which replaces the product:
$$h_x h_p \approx h$$
Accordingly, we modify Eq. (2.14) to:
$$H_{\max }(\text { location and momentum })=\log \left[\frac{L \sqrt{2 \pi e m k_B T}}{h}\right]$$

数学代写|信息论代写information theory代考|The SMI of One Particle in a Box of Volume $\mathrm{V}$

Figure 2.3 shows one simple particle in a cubic box of volume $V$.
To proceed from the 1D to the 3D system, we assume that the locations of the particle along the three axes $x, y$ and $z$ are independent. With this assumption, we can write the SMI of the location of the particle in a cube of edges $L$, as a sum of the SMI along $x, y$, and $z$, i.e.
$$H(\text { location in } 3 \mathrm{D})=3 H_{\max } \text { (location in 1D) }$$
We can do the same for the momentum of the particle if we assume that the momentum (or the velocity) along the three axes $x, y$ and $z$ are independent. Hence, we can write the SMI of the momentum as:
$$H_{\max }(\text { momentum in } 3 \mathrm{D})=3 H_{\max }(\text { momentum in 1D) }$$
We can now combine the SMI of the locations and momenta of one particle in a box of volume $V$, taking into account the uncertainty principle, to obtain the result:
$$H_{\max }(\text { location and momentum in } 3 \mathrm{D})=3 \log \left[\frac{L \sqrt{2 \pi e m k_B T}}{h}\right]$$

信息论代写

数学代写|信息论代写information theory代考|Third Step: Combining the SMI for the Location and Momentum of a Particle in a $1 D$ System. Addition of Correction Due to Uncertainty

$$H[f(x)]=-\int f(x) \log f(x) d x$$

$$f_{e q}(x)=\frac{1}{L}$$

$$H(\text { locations in } 1 D)=\log L$$

$$H\left(\text { locations in 1D) }=\log L-\log h_x\right.$$

数学代写|信息论代写information theory代考|The SMI of One Particle in a Box of Volume $\mathrm{V}$

$$H(\text { location in } 3 \mathrm{D})=3 H_{\max } \text { (location in 1D) }$$

$$H_{\max }(\text { momentum in } 3 \mathrm{D})=3 H_{\max }(\text { momentum in 1D) }$$

$$H_{\max }(\text { location and momentum in } 3 \mathrm{D})=3 \log \left[\frac{L \sqrt{2 \pi e m k_B T}}{h}\right]$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。