19th Ave New York, NY 95822, USA

# 物理代写|统计力学代写Statistical Mechanics代考|PHYS521 Corrections to the Law of Large Numbers

my-assignmentexpert™提供最专业的一站式服务：Essay代写，Dissertation代写，Assignment代写，Paper代写，Proposal代写，Proposal代写，Literature Review代写，Online Course，Exam代考等等。my-assignmentexpert™专注为留学生提供Essay代写服务，拥有各个专业的博硕教师团队帮您代写，免费修改及辅导，保证成果完成的效率和质量。同时有多家检测平台帐号，包括Turnitin高级账户，检测论文不会留痕，写好后检测修改，放心可靠，经得起任何考验！

## 物理代写|统计力学代写Statistical Mechanics代考|Corrections to the Law of Large Numbers

A informal way to state (2.3.9) is
$$\sum_{i=1}^N f_i\left(x_i\right) \approx N \mathbb{E}(f)$$
which holds for typical configurations when $N \rightarrow \infty$.
One may ask: what is the correction to that approximation? It turns out that this correction is of order $\sqrt{N}$ :
$$\sum_{i=1}^N f_i\left(x_i\right) \approx N \mathbb{E}(f)+\sqrt{N} X$$
where $X$ is a Gaussian random variable. The precise formulation of (2.3.16) is:
Theorem 2.1 The central limit theorem.
Let $X_N=\frac{\sum_{i=1}^N f_i\left(x_i\right)-N \mathbb{E}(f)}{\sqrt{N}}$, with $f_i$ as in (2.3.6).
Then, $\forall a, b \in \mathbb{R}, a<b$,

## 物理代写|统计力学代写Statistical Mechanics代考|The Law of Large Numbers and the Frequentist Interpretation

A frequentist might want to use the law of large numbers, specially in its strong form, (2.3.9) or (2.3.13), in order to define the concept of probability. That might answer the objection that, if one defines that concept as a frequency of results in the repetition of a large number of the “same” experiment, the notion of “large” is imprecise. But, if we take the limit $N \rightarrow \infty$, then “large” becomes precise.

There are two obvious objections to that answer: first of all, it is true that many idealizations are made in physics by taking appropriate limits, of infinite time intervals or infinite spatial extension, ${ }^{14}$ but it is difficult to see how a concept can be defined only through such a limit. Since the limit is never reached in nature and if the concept of probability makes no sense for finite sequences of experiments, then it cannot be used in the natural sciences.

The next objection is that statements like (2.3.9) or (2.3.13) are probabilistic statements even if they refer to events having probability one. But it is circular to define a concept by using a formula that involves that very concept.

The mathematician and defender of the frequentist interpretation, Richard von Mises, avoids referring to the law of large numbers and defines probabilities as limits of frequencies of particular attributes (like falling heads for a coin or landing on a 5 for a die) within what he calls a “collective.” [324]
A collective is defined as an unlimited sequence of observations so that:

1. The limits of frequencies of particular attributes within the collective exist.
2. These limits are not affected by place selection, which means that the same limit would be obtained if we choose a subsequence of the original sequence of the collective according to some rule, for example the subsequence of events indexed by even numbers or by prime numbers or by numbers that are squares of integers. Of course, the rule must be specified independently of the results of the sequence of observations.
This is a way of guaranteeing that the collective is “random”. Consider the sequence $0,1,0,1,0,1,0,1, \ldots$. The limits of frequencies of 0 ‘s and 1 ‘s is obviously $\frac{1}{2}$, but it would not be so if we chose the subsequence of events indexed by even numbers or by odd numbers. And that is a way of characterizing the sequence as nonrandom.

## 物理代写|统计力学代写STATISTICAL MECHANICS 代考|THE INDIFFERENCE PRINCIPLE

$$P(i)=\frac{1}{N}, \forall i=1, \ldots, N \text {. }$$

## 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。