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assignmentutor-lab™ 为您的留学生涯保驾护航 在代写宇宙学cosmology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。我们的专家在代写宇宙学cosmology代写方面经验极为丰富，各种代写宇宙学cosmology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|宇宙学代写cosmology代考|Distribution functions

A key concept in statistical mechanics is the probability that a particle chosen at random has a momentum p. In general, this (probability) distribution function, $f(\mathbf{p}, t)$, can be very complicated. ${ }^{4}$ However, if we wait long enough (relative to the typical interaction timescale), then the system will reach equilibrium and is characterized by a time-independent distribution function. At this point, the gas has reached a state of maximum entropy in which the distribution function is given by either the Fermi-Dirac distribution (for fermions) or the Bose-Einstein distribution (for bosons)
$$f(p, T)=\frac{1}{e^{(E(p)-\mu) / T} \pm 1}$$
where the $+\operatorname{sign}$ is for fermions and the $-\operatorname{sign}$ for bosons. A derivation of these equilibrium distribution functions can be found in any textbook on statistical mechanics. Thé function in (3.4) has twô parameters: thé témperaturee, $T$, and thé chemical potential, $\mu$. The latter describes the response of a system to a change in particle numbers (see Section 3.1.5). The chemical potential may be temperature dependent, and since the temperature changes in an expanding universe, even the equilibrium distribution functions depend implicitly on time.

## 物理代写|宇宙学代写cosmology代考|Density of states

To relate this microscopic description of the gas to its macroscopic properties, we must sum over all possible momentum states of the particles weighted by their probabilities. For example, the number density of particles in the gas is
$$n=\sum_{\mathrm{p}} f(p, T) .$$
To define this sum over states as an integral over the continuous variable p, requires the density of states. It is easiest to derive this density of states by considering the gas as a quantum system. Recall that, in quantum mechanics, the momentum eigenstates of a particle in a box of side length $L$ have a discrete spectrum. Solving the Schrödinger equation with periodic boundary conditions gives
$$\mathbf{p}=\frac{h}{L}\left(r_{1} \hat{\mathbf{x}}+r_{2} \hat{\mathbf{y}}+r_{3} \hat{\mathbf{z}}\right)$$
where $r_{i}=0, \pm 1, \pm 2, \cdots$ and $h=4.14 \times 10^{-15} \mathrm{eV} \mathrm{s}$ is Planck’s constant. In momentum space, the states of the particle are therefore represented by a discrete set of points. The density of states in momentum space ${\mathbf{p}}$ then is $L^{3} / h^{3}=V / h^{3}$, and the state density in phase space ${\mathbf{x}, \mathbf{p}}$ is $1 / h^{3}$. If the particle has $g$ internal degrees of freedom (for example, due to the intrinsic spin of elementary particles), then the density of states becomes
$$\frac{g}{h^{3}}=\frac{g}{(2 \pi)^{3}},$$
where in the second equality we have used natural units with $\hbar=h /(2 \pi) \equiv 1$.

# 宇宙学代考

## 物理代写|宇宙学代写cosmology代考|Distribution functions

$$f(p, T)=\frac{1}{e^{(E(p)-\mu) / T} \pm 1}$$

## 物理代写|宇宙学代写cosmology代考|Density of states

$$n=\sum_{\mathrm{p}} f(p, T) .$$

$$\mathbf{p}=\frac{h}{L}\left(r_{1} \hat{\mathbf{x}}+r_{2} \hat{\mathbf{y}}+r_{3} \hat{\mathbf{z}}\right)$$

$$\frac{g}{h^{3}}=\frac{g}{(2 \pi)^{3}},$$

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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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