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## 物理代写|统计物理代写Statistical Physics of Matter代考|Liouville Theorem

The dynamics of the considered $N$ particles is described by their Hamiltonian $\mathcal{H}(p, q)$. The equations of motion are
$$\dot{p}i=-\frac{\partial \mathcal{H}}{\partial q_i} \quad \text { and } \quad \dot{q}_i=\frac{\partial \mathcal{H}}{\partial p_i} \quad(i=1, \ldots, 3 N) .$$ Moreover, the temporal evolution of a phase space element of volume $V$ and boundary $\partial V$ is given by $$\frac{\partial}{\partial t} \int_V \rho \mathrm{d} V+\int{\partial V} \rho v \mathrm{~d} A=0,$$
where $v=\left(\dot{p}1, \ldots, \dot{p}{3 N}, \dot{q}1, \ldots, \dot{q}{3 N}\right)$ is a generalized velocity vector. By applying the divergence theorem to eq. (3.4), we find that $\rho$ satisfies the continuity equation
$$\frac{\partial \rho}{\partial t}+\nabla \cdot(\rho v)=0,$$
where $\nabla=\left(\partial / \partial p_1, \ldots, \partial / \partial p_{3 N}, \partial / \partial q_1, \ldots, \partial q_{3 N}\right)$. We can further simplify eq. (3.5) with the help of the identity
$$\nabla \cdot v=\sum_{i=1}^{3 N}\left(\frac{\partial \dot{q}i}{\partial q_i}+\frac{\partial \dot{p}_i}{\partial p_i}\right)=\sum{i=1}^{3 N} \underbrace{\left(\frac{\partial}{\partial q_i} \frac{\partial \mathcal{H}}{\partial p_i}-\frac{\partial}{\partial p_i} \frac{\partial \mathcal{H}}{\partial q_i}\right)}_{=0}=0 .$$
We use Poisson brackets ${ }^1$ to rewrite eq. (3.5) and obtain Liouville’s Theorem,
$$\frac{\partial \rho}{\partial t}={\mathcal{H}, \rho} \text {, }$$
which describes the time evolution of the phase space density $\rho$.

## 物理代写|统计物理代写Statistical Physics of Matter代考|Thermal Equilibrium

In thermal equilibrium, the system reaches a steady state in which the distribution of configurations is time independent. That is, the phase space density satisfies $\partial \rho / \partial t=0$. Liouville’s theorem (eq. (3.8)) then leads to
$$v \cdot \nabla \rho={\mathcal{H}, \rho}=0 .$$
One possibility to satisfy this equation is to take a phase space density $\rho$ that depends on quantities such as energy or the number of particles, which are conserved during the time evolution of the system. We may then use such a phase space density to replace the time average (eq. (3.1)) by a corresponding ensemble average (eq. (3.2)).

In the subsequent sections, we also consider discrete configurations $X$ for which we define the ensemble average as
$$\langle Q\rangle=\frac{1}{\Omega} \sum_X Q(X) \rho(X),$$
where $\Omega$ is the normalizing volume such that $\Omega^{-1} \Sigma_X \rho(X)=1$. With the help of ensemble averages, systems can be described by macroscopic quantities such as temperature, energy, and pressure.

## 物理代写|统计物理代写物质统计物理学代考|刘维尔定理

$$\dot{p}i=-\frac{\partial \mathcal{H}}{\partial q_i} \quad \text { and } \quad \dot{q}_i=\frac{\partial \mathcal{H}}{\partial p_i} \quad(i=1, \ldots, 3 N) .$$此外，体积$V$和边界$\partial V$的相空间元的时间演化由$$\frac{\partial}{\partial t} \int_V \rho \mathrm{d} V+\int{\partial V} \rho v \mathrm{~d} A=0,$$

$$\frac{\partial \rho}{\partial t}+\nabla \cdot(\rho v)=0,$$
，其中$\nabla=\left(\partial / \partial p_1, \ldots, \partial / \partial p_{3 N}, \partial / \partial q_1, \ldots, \partial q_{3 N}\right)$。我们可以借助等式
$$\nabla \cdot v=\sum_{i=1}^{3 N}\left(\frac{\partial \dot{q}i}{\partial q_i}+\frac{\partial \dot{p}_i}{\partial p_i}\right)=\sum{i=1}^{3 N} \underbrace{\left(\frac{\partial}{\partial q_i} \frac{\partial \mathcal{H}}{\partial p_i}-\frac{\partial}{\partial p_i} \frac{\partial \mathcal{H}}{\partial q_i}\right)}_{=0}=0 .$$

$$\frac{\partial \rho}{\partial t}={\mathcal{H}, \rho} \text {, }$$

## 物理代写|统计物理代写物质统计物理学代考|热平衡

$$v \cdot \nabla \rho={\mathcal{H}, \rho}=0 .$$

$$\langle Q\rangle=\frac{1}{\Omega} \sum_X Q(X) \rho(X),$$
，其中$\Omega$是归一化体积，使$\Omega^{-1} \Sigma_X \rho(X)=1$。在系综平均值的帮助下，系统可以用诸如温度、能量和压力等宏观量来描述

## 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。