assignmentutor™您的专属作业导师

assignmentutor-lab™ 为您的留学生涯保驾护航 在代写统计力学Statistical mechanics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。我们的专家在代写统计力学Statistical mechanics代写方面经验极为丰富，各种代写统计力学Statistical mechanics相关的作业也就用不着说。

• 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|Single-particle density of states function

Counting is a big part of statistical physics-the number of ways that various physical quantities can be arranged or processes occur. A fundamental quantity in statistical mechanics (see Section 2.6) is the density of energy levels, the number of energy levels per unit energy range. ${ }^{12}$ We derive in Chapter 4 , the general properties of density-of-state functions without regard to their specific form. We reserve $\Omega(E)$ to denote the density of states function for interacting particles. In many cases, however, it turns out that the single-particle density of states function is all we need to know; we’ll use $g(E)$ to denote the density of states for noninteracting, i.e., free particles.

A free particle of mass $m$ is described by wavefunctions satisfying the time-independent Schrödinger equation $-\left(\hbar^{2} /(2 m)\right) \nabla^{2} \psi(\boldsymbol{r})=E \psi(\boldsymbol{r})$ in the form of plane waves, $\psi(\boldsymbol{r})=$ $\exp (\mathrm{i} k \cdot r)$, where $k \equiv \sqrt{2 m E / \hbar^{2}}$, with the direction of the wavevector $k$ the direction of propagation. How many single-particle energy levels are available between $E$ and $E+\mathrm{d} E$ ?

To answer that question, boundary conditions must be specified on $\psi(\boldsymbol{r})$. For electrons in solids (an important case), there are confining potentials at the edges of the system keeping them within the system. Confining potentials present a complication to analyzing the density of states. Unless we’re specifically interested in the physics of surface phenomena, we’d rather not have to take into account the modification of allowed energies engendered by confining potentials. If $L$ is a characteristic length of a macroscopic sample, and if there are $N$ particles in volume $\approx L^{3}$, and if the ratio of surface area to volume scales as $L^{-1}$, then the ratio of the number of atoms near a surface to the number in bulk scales as $N^{-1 / 3}\left(10^{-8}\right.$ for Avogadro’s number). Is there a way to ignore surface effects while retaining the finite volume $V$ of a sample?

Periodic boundary conditions are a way to do that which is easy to implement and give accurate results. Assume a particle is confined to a cube of volume $V$ of side $L=V^{1 / 3}$. Imagine that each face of the cube “wraps around” to adjoin the face opposite to it, so that a particle approaching one of the faces appears at the corresponding point on the opposite face. In one dimension, the point at $x=L$ coincides with the point at $x=0$; the system is a circle of circumference $L$. A square of side $L$ embodying these boundary conditions is a torus (show this). A three-dimensional system satisfying periodic boundary conditions can’t be visualized by denizens of three dimensions such as us, but we can easily write down the mathematical requirements:
$$\left.\begin{array}{l} \psi(x, y, z+L) \ \psi(x, y+L, z) \ \psi(x+L, y, z) \end{array}\right}=\psi(x, y, z) .$$

## 物理代写|统计力学代写Statistical mechanics代考|STATE VARIABLES-THE FEW

Whereas microstates incessantly evolve in time, ${ }^{18}$ equilibrium is described by the time independent state variables of thermodynamics. ${ }^{19}$ The first law of thermodynamics, Eq. (1.2), $\Delta U=Q+W$, indicates the two ways by which internal energy is changed: heat transfers and/or the performance of work. In differential form, Eq. (1.3), $\mathrm{d} U=\mathrm{d} Q+\mathrm{d} W$, where đ signifies an inexact differential. The notation indicates we’re not implying small changes in previously existing quantities $Q$ and $W$ (which would be written $\mathrm{d} Q$ and $\mathrm{d} W$ ), but rather that $\mathrm{\AA} Q$ and $\mathrm{d} W$ denote process-dependent modes of energy transfer. ${ }^{20}$

Work involves observable changes in the extensive properties of a system, ${ }^{21} X_{i}$. For small amounts of work, $\mathrm{d} W=\sum_{i} Y_{i} \mathrm{~d} X_{i}$, the sum of products of generalized displacements $\mathrm{d} X_{i}$ with their conjugate generalized forces $Y_{i}$ (see Section 1.2). The energy to change the number of particles by $\mathrm{d} N$ is expressed in terms of the chemical potential $\mu, \mathrm{d} W=\mu \mathrm{d} N$ (see Section 1.6). Thus,
$$\mathrm{d} U=\mathrm{d} Q+\sum_{i} Y_{i} \mathrm{~d} X_{i}+\mu \mathrm{d} N .$$
Relatively, few variables appear in thermodynamic descriptions, yet there are myriad microscopic degrees of freedom that contribute to the energy of a system (its Hamiltonian). Where are microscopic degrees of freedom represented in thermodynamics? The division of internal energy into work and heat lines up with the distinction between macroscopic and non-macroscopic, i.e., microscopic. What is not work is heat, energy transfers to microscopic degrees of freedom. Heat is extensive (see Section 1.2), yet it’s not a state variable (see Section 1.3). Entropy, an extensive state variable related to heat transfers, is defined through a differential relation $\mathrm{d} S \equiv(\mathrm{d} Q)_{\text {rev }} / T$, Eq. (1.8). Equation (2.21) can thercforc be written (including only $P \mathrm{~d} V$ work), $\mathrm{d} U=T \mathrm{~d} S-P \mathrm{~d} V+\mu \mathrm{d} N$. To answer our question, microscopic degrees of freedom are represented as entropy. Entropy is a proxy for the remaining degrees of freedom after the observable state variables are accounted for.

# 统计力学代考

## 物理代写|统计力学代写Statistical mechanics代考|STATE VARIABLES-THE FEW

$$\mathrm{d} U=\mathrm{d} Q+\sum_{i} Y_{i} \mathrm{~d} X_{i}+\mu \mathrm{d} N$$

$\mathrm{d} U=T \mathrm{~d} S-P \mathrm{~d} V+\mu \mathrm{d} N$. 为了回答我们的问题，微观自由度表示为樀。樀是考虑可观察状态变量后剩余自由度的代表。

## 有限元方法代写

assignmentutor™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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师