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# 数学代写|概率论代考Probability Theory代写|Bose-Einstein and Fermi-Dirac statistics

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## 数学代写|概率论代考Probability Theory代写|Bose-Einstein and Fermi-Dirac statistics

Consider a mechanical system of $r$ indistinguishable particles. In statistical mechanics it is usual to subdivide the phase space into a large number, $n$, of small regions or cells so that each particle is assigned to one cell. In this way the state of the entire system is described in terms of a random distribution of the $r$ particles in $n$ cells. Offhand it would seem that (at least with an appropriate definition of the $n$ cells) all $n^r$ arrangements should have equal probabilities. If this is true, the physicist speaks of Maxwell-Boltzmann statistics (the term “statistics” is here used in a sense peculiar to physics). Numerous attempts have been made to prove that physical particles behave to accordance with Maxwell-Boltzmann statistics, but modern theory has shown beyond doubt that this statistics does not apply to any known particles; in no case are all $n^r$ arrangements approximately equally probable. Two different probability models have been introduced, and each describes satisfactorily the behavior of one type of particle. The justification of either model depends on its success. Neither claims universality, and it is possible that some day a third model may bc introduced for certain kind of particles.

Remember that we are here concerned only with indistinguishable particles. We have $r$ particles and $n$ cells. By Bose-Einstein statistics we mean that only distinguishable arrangements are considered and that each is assigned probability $1 / A_{r, n}$ with $A_{r, n}$ defined in (5.2). It is shown in statistical mechanics that this assumption holds true for photons, nuclei, and atoms containing an even number of elementary particles. ${ }^8$ To describe other particles a third possible assignment of probabilities must be introduced. Fermi-Dirac statistics is based on these hypotheses: (1) it is impossible for two or more particles to be in the same cell, and (2) all distinguishable arrangements satisfying the first condition have equal probabilities. The first hypothesis requires that $r \leq n$. An arrangement is then completely described by stating which of the $n$ cells contain a particle; and since there are $r$ particles, the corresponding cells can be chosen in $\left(\begin{array}{l}n \ r\end{array}\right)$ ways. Hence, with Fermi-Dirac statistics there are in all $\left(\begin{array}{l}n \ r\end{array}\right)$ possible arrangements, each having probability $\left(\begin{array}{l}n \ r\end{array}\right)^{-1}$. This model applies to electrons, neutrons, and protons. We have here an instructive example of the impossibility of selecting or justifying probability models by a priori arguments. In fact, no pure reasoning could tell that photons and protons would not obey the same probability laws. (Essential differences between Maxwell-Boltzmann and Bose-Einstein statistics are discussed in section 11 , problems $14-19$. )

## 数学代写|概率论代考Probability Theory代写|Application to Runs

In any ordered sequence of elements of two kinds, each maximal subsequence of elements of like kind is called a run. For example, the sequence $\alpha \alpha \alpha \beta \alpha \alpha \beta \beta \beta \alpha$ opens with an alpha run of length 3 ; it is followed by runs of length $1,2,3,1$, respectively. The alpha and beta runs alternate so that the total number of runs is always one plus the number of conjunctions of unlike neighbors in the given sequence.

Examples of applications. The theory of runs is applied in statistics in many ways, but its principal uses are connected with tests of randomness or tests of homogeneity.
(a) In testing randomness, the problem is to decide whether a given observation is attributable to chance or whether a search for assignable causes is indicated. As a simple example suppose that an observation ${ }^9$ yielded the following arrangement of empty and occupied seats along a lunch counter: EOEEOEEEOEEEOEOE. Note that no two occupied seats are adjacent. Can this be due to chance? With five occupied and eleven empty seats it is impossible to get more than eleven runs, and this number was actually observed. It will be shown later that if all arrangements were equally probable the probability of eleven runs would be $0.0578 \ldots$. This small probability to some extent confirms the hunch that the separations observed were intentional. This suspicion cannot be proved by statistical methods, but further evidence could be collected from continued observation. If the lunch counter were frequented by families, there would be a tendency for occupants to cluster together. and this would lead to relatively small numbers of runs. Similarly counting runs of boys and girls in a classroom might disclose the mixing to be better or worse than random. Improbable arrangements give clues to assignable causes; an excess of runs points to intentional mixing, a paucity of runs to intentional clustering. It is true that these conclusions are never foolproof, but efficient statistical techniques have been developed which in actual practice minimize the risk of incorrect conclusions.

The theory of runs is also useful in industrial quality control as introduced by Shewhart. As washers are produced, they will vary in thickness. Long runs of thick washers may suggest imperfections in the production process and lead to the removal of the causes; thus oncoming trouble may be forestalled and greater homogeneity of product achieved.

# 概率论代写

## 数学代写|概率论代考Probability Theory代写|Application to Runs

(a)在检验随机性时，问题是决定一个给定的观察结果是归因于偶然还是表明要寻找可分配的原因。举个简单的例子，假设通过观察${ }^9$得到了一个午餐柜台上的空位和有人的座位排列:eoeeoeeeeeeeoeoe。请注意，没有两个被占用的座位是相邻的。这是偶然吗?5个座位坐满，11个座位空着，不可能超过11个座位，而这个数字实际上是观察到的。稍后将说明，如果所有排列都是等概率的，则11次运行的概率为$0.0578 \ldots$。这个小概率在某种程度上证实了观察到的分离是故意的预感。这种怀疑不能用统计方法证明，但可以从继续观察中收集进一步的证据。如果午餐柜台经常有家庭光顾，那么就会有住户聚集在一起的趋势。这将导致相对较少的运行次数。同样，计算一个教室里男孩和女孩的人数可能会揭示混合比随机更好或更差。不可能的安排为可分配的原因提供了线索;过多的运行表明有意混合，运行不足表明有意集群。的确，这些结论从来都不是万无一失的，但是有效的统计技术已经被开发出来，在实际操作中可以最大限度地减少得出错误结论的风险。

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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。