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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

## 物理代写|统计力学代写Statistical mechanics代考|Phase-space probability density, ensemble average

Members of ensembles are prepared in the same way subject to macroscopic constraints. Microscopically, however, systems (elements of an ensemble), even though prepared as identically as possible, have system points in $\Gamma$-space that are not identical. While the same macroscopic constraints are met hy all memhers of an ensemble, they differ in their phase points. ${ }^{39}$ Finsemhles are represented as collections of system points in $\Gamma$-space (see Fig. 2.4), where, we emphasize, each point is the phase point of an $N$-particle system. The ensemble is thus a swarm of points in $\Gamma$-space. As time progresses, the swarm moves under the natural motion of phase space (indicated in Fig. 2.4). The flow of the ensemble turns out to be that of an incompressible fluid (Exercise 2.12). We can now pose the analogous question we asked about fluctuations: What is the probability that the phase point of a randomly selected member of an ensemble lies in the range $(p, q)$ to $(p+\mathrm{d} p, q+\mathrm{d} q)$ (where $p$ and $q$ denote $\left(p_{1}, \cdots, p_{3 N}\right)$ and $\left(q_{1}, \cdots, q_{3 N}\right)$, with $\mathrm{d} p \equiv \mathrm{d} p_{1} \cdots \mathrm{d} p_{3 N}$ and $\left.\mathrm{d} q \equiv \mathrm{d} q_{1} \cdots \mathrm{d} q_{3 N}\right)$ ? Such a probability will be proportional to $\mathrm{d} p$ and $\mathrm{d} q$, which we can write as
$$\rho(p, q) \frac{\mathrm{d} p \mathrm{~d} q}{h^{3 N}} \equiv \rho(p, q) \mathrm{d} \Gamma$$
where the factor of $h^{3 N}$ ensures that the phase-space probability density, $\rho(p, q)$, is dimensionless. Whatever the form of $\rho(p, q)$, it must be normalized,
$$\int_{\Gamma} \rho(p, q) \mathrm{d} \Gamma=1,$$
which simply indicates that any element of the ensemble has some phase point. If $A(p, q)$ is a generic phase-space function, its ensemble average is, by definition (see Chapter 3 ),
$$\bar{A} \equiv \int_{\Gamma} A(p, q) \rho(p, q) \mathrm{d} \Gamma .$$

## 物理代写|统计力学代写Statistical mechanics代考|Liouville’s theorem and constants of the motion

Our goal is to find the phase-space probability density $\rho(p, q)$, a task greatly assisted by Liouville’s theorem. If at time $t$ we denote a given volume in $\Gamma$-space as $\Omega_{t}$, such as the ellipse indicated in Fig. 2.4, and $\mathcal{N}{t}$ as the number of ensemble points within $\Omega{t}$, then the density of ensemble points in $\Omega_{t}$ is $\mathcal{N}{t} / \Omega{t}$. An important property of ensembles is that the phase trajectories of system points never cross (Section 2.1). Assume that the $\mathcal{N}{t}$ ensemble points are within the interior of $\Omega{t}$, i.e., not on the boundary on $\Omega_{t}$. As time progresses, the number of ensemble points interior to $\Omega_{t+\mathrm{d} t}$ is the same as in $\Omega_{t}$ (see Fig. 2.4), $\mathcal{N}{t+\mathrm{d} t}=\mathcal{N}{t}$. Points interior to $\Omega_{t}$ (not necessarily ensemble system points) evolve to points interior to $\Omega_{t+\mathrm{d} t}$, because points on the boundary of $\Omega_{t}$ evolve to points on the boundary of $\Omega_{t+\mathrm{d} t}$ and trajectories never cross. The volumes are also equal, $\Omega_{t+\mathrm{d} t}=\Omega_{t}$; as shown in Appendix $C$, the volume element $\mathrm{d} \Gamma$ is invariant under the time evolution imposed by Hamilton’s equations, ${ }^{40} \mathrm{~d} \Gamma_{t+\mathrm{d} t}=\mathrm{d} \Gamma_{t}$. The density $\mathcal{N}{t} / \Omega{t}$ is thus a dynamical invariant.

The density of ensemble points, however, is none other than the phase-space probability density for ensemble points, $\rho(p, q)$. We therefore arrive at Liouville’s theorem: $\rho(p, q)$ is constant along phase trajectories,
$$\frac{\mathrm{d}}{\mathrm{d} t} \rho(p, q)=0 .$$
The function $\rho(p, q)$ is certainly a phase-space function, and thus, combining Eqs. (2.2) and (2.54),
$$\frac{\partial \rho}{\partial t}+[\rho, H]=0$$
where we allow for a possible explicit time dependence to $\rho(p, q, t)$. Equation (2.55), Liouville’s equation, is the fundamental equation of classical statistical mechanics, on par with the status of the Schrödinger equation in quantum mechanics.

# 统计力学代考

## 物理代写|统计力学代写Statistical mechanics代考|Phase-space probability density, ensemble average

$$\rho(p, q) \frac{\mathrm{d} p \mathrm{~d} q}{h^{3 N}} \equiv \rho(p, q) \mathrm{d} \Gamma$$

$$\int_{\Gamma} \rho(p, q) \mathrm{d} \Gamma=1,$$

$$\bar{A} \equiv \int_{\Gamma} A(p, q) \rho(p, q) \mathrm{d} \Gamma .$$

## 物理代写|统计力学代写Statistical mechanics代考|Liouville’s theorem and constants of the motion

$$\frac{\mathrm{d}}{\mathrm{d} t} \rho(p, q)=0 .$$

$$\frac{\partial \rho}{\partial t}+[\rho, H]=0$$

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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。

assignmentutor™您的专属作业导师
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