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# 物理代写|统计物理代写Statistical Physics of Matter代考|PHYC40650 Classical Statistical Mechanics

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## 物理代写|统计物理代写Statistical Physics of Matter代考|Phase Space

In the first part of this chapter, our goal is to provide an overview of methods to computationally study systems that can be described within the framework of equilibrium statistical mechanics. We therefore first introduce the most important concepts from classical statistical mechanics that enable us to mathematically capture the macroscopic properties of interacting microscopic units. Historically, important contributions to the microscopic formulation of thermodynamics were made by Boltzmann and Gibbs $[105,106]$. In particular, Gibbs’s notion of a statistical ensemble enables us to interpret macroscopic physical quantities as averages over a large number of different configurations of a system’s interacting units.

## 物理代写|统计物理代写Statistical Physics of Matter代考|Dijkstra’s Algorithm

Let us consider a three-dimensional classical physical system that consists of $N$ particles. We need a three-component vector to describe the position of a certain particle and another three-component vector to describe its momentum. Therefore, we denote the canonical coordinates and corresponding conjugate momenta of all $N$ particles by $q_{1}, \ldots, q_{3 N}$ and $p_{1}, \ldots, p_{3 N}$, respectively. The $6 N$-dimensional space $\Gamma$ that results from the set of canonical coordinates defines the phase space. This concept has been introduced by Ludwig Boltzmann (see Figure 3.1). The considered $N$ particles could simply be uncoupled harmonic oscillators. In this case, the phase space of each single particle would look like the one we show in Figure 3.2. By keeping certain external parameters such as temperature and pressure constant, we can measure a macroscopic physical quantity by computing the time average
$$\langle Q\rangle_{T}=\lim {T \rightarrow \infty} \frac{1}{T} \int{0}^{T} Q(p(t), q(t)) \mathrm{d} t$$
over different realizations of the underlying microscopic states. However, computing the time average of a certain macroscopic quantity makes it necessary to determine the time evolution of all microscopic states. If the number of involved particles is large, this approach would be computationally infeasible. Instead of computing time averages, another possibility is to consider an average over an ensemble of systems in different microstates (i. e., specific microscopic configurations of a certain system) under the same macroscopic conditions.

## 物理代写|统计物理代写Statistical Physics of Matter代考|Dijkstra’s Algorithm

$$\langle Q\rangle_{T}=\lim T \rightarrow \infty \frac{1}{T} \int 0^{T} Q(p(t), q(t)) \mathrm{d} t$$

## 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。