### 物理代写|统计力学代写Statistical mechanics代考|PHYSICS2520

statistics-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|A Simple Example

A way to establish a connection between the Bayesian and the frequentist views on probability relies on the law of large numbers: the calculus of probabilities-viewed now as part of deductive reasoning-leads one to ascribe subjective probabilities close to one for certain events that are precisely those that the objective approach deals with, namely the frequencies of some events, when the ‘same’ experiment is repeated many times. So, rather than opposing the two views, one should carefully distinguish between them, but regard the objective one as, in a sense, derived from the subjective one (i.e. when the law of large numbers leads to subjective probabilities sufficiently close to one). Let us state the law of large numbers, using a terminology that will be useful when we turn to statistical mechanics later.

Consider first the simple example of coin flipping. Let 0 denote ‘head’ and 1 , ‘tail’. The ‘space’ of results of any single flip, ${0,1}$, will be called the ‘individual phase space’ while the space of all possible results of $N$ flips, ${0,1}^{N}$, will be called the ‘total phase space’.

The variables $N_{0}, N_{1}$ that count the number of heads $(0)$ or tails (1) will be called macroscopic, in anticipation for their later use in statistical mechanics.

Here we introduce a distinction which will be essential throughout this book between the macroscopic variables, or the macrostate, and the microstate. The microstate, for $N$ flips, is the sequence of results for all the flips, while the macrostate simply specifies the values of $N_{0}$ and $N_{1}$.

Now, define a sequence of sets of microstates $\mathcal{T}{N} \subset{0,1}^{N}$ to be typical relative to a given sequence of probability measures $P{N}$ on ${0,1}^{N}$, if
$$P_{N}\left(\mathcal{T}{N}\right) \rightarrow 1$$ as $N \rightarrow \infty$. If the typical sets $\mathcal{T}{N}$ are defined by a property, we will also call that property typical. ${ }^{13}$
Let $G_{N}(\epsilon)$ be the set of microstates such that

$$\left|\frac{N_{0}}{N}-\frac{1}{2}\right| \leq \epsilon$$
Here the letter $G$ stand for “good”, because we will use the same expression later in the context of statistical mechanics.
Then, (a weak form of) the law of large numbers states that $\forall \epsilon>0$
$$P_{N}\left(G_{N}(\epsilon)\right) \rightarrow 1$$
as $N \rightarrow \infty$, where $P_{N}$ the product measure on ${0,1}^{N}$ that assigns independent probabilities $\frac{1}{2}$ to each outcome of each flip. This is the measure that one would assign on the basis of the indifference principle: give an equal probability to all possible sequences of results. In words, (2.3.3) says that the set of sequences $G_{N}(\epsilon)$ is typical in the sense of definition (2.3.1), $\forall \epsilon>0$.

## 物理代写|统计力学代写Statistical mechanics代考|A More General Result

This subsection is somewhat more mathematical than the rest of this chapter and we refer to the Appendix 2.A for the notations, definitions and results mentioned here.

Let $\boldsymbol{\Omega}, \boldsymbol{\Sigma}$ be a product space $x_{i=1}^{\infty} \Omega_{i}$ and $\boldsymbol{\mu}$ be a product measure $x_{i=1}^{\infty} \mu_{i}$ on $\boldsymbol{\Omega}, \boldsymbol{\Sigma}$ (where all $\Omega_{i}$ ‘s are copies of a given $\Omega$ and all $\mu_{i}$ ‘s are copies of a given measure $\mu$ on $\Omega$ ). Let $f_{i}: \Omega_{i} \rightarrow \mathbb{R}$ be a sequence of identical random variables (all $f_{i}$ ‘s are copies of a given $f: \Omega \rightarrow \mathbb{R})$ and form the sum $S_{N}(\mathbf{x}): \Omega \rightarrow \mathbb{R}$ :
$$S_{N}(\mathbf{x})=\frac{1}{N} \sum_{i=1}^{N} f_{i}\left(x_{i}\right)$$
Assume for simplicity that the function $f$ is bounded. Then, if $G_{N}(\epsilon)$ denotes the set of microstates $\mathbf{x}$ such that
$$\left|S_{N}(\mathbf{x})-\mathbb{E}(f)\right| \leq \epsilon,$$
with $\mathbb{E}(f)=\int_{\Omega} f(x) d \mu(x)$ the expectation value of $f$, we have $\forall \epsilon>0$
$$\boldsymbol{\mu}\left(G_{N}(\epsilon)\right) \rightarrow 1$$
as $N \rightarrow \infty$. This is proven in Appendix $2 . B$, with explicit bounds on $\boldsymbol{\mu}\left(G_{N}(\epsilon)\right)$.
Formula (2.3.8) is called the weak law of large numbers because there is also a “strong” formulation of the law of large numbers:
$$\boldsymbol{\mu}\left(\left{\mathbf{x}\left|\lim {N \rightarrow \infty}\right| S{N}(\mathbf{x})-\mathbb{E}(f) \mid=0\right}\right)=1,$$
or, in words, the convergence of $\lim {N \rightarrow \infty}\left|S{N}(\mathbf{x})-\mathbb{E}(f)\right|$ to 0 holds $\boldsymbol{\mu}$ almost everywhere.

Here is another version of the law of large numbers: let $\left(A_{1}, A_{2}, \ldots, A_{k}\right)$ be a partition of $\mathbb{R}$. Given a sequence $\left(x_{1}, \ldots, x_{N}\right) \in \Omega^{N}$ define the histogram $\left(n_{\alpha}\right){\alpha=1}^{k}$ of the random variables $\left(f{1}, \ldots, f_{N}\right)$ by:
$$n_{\alpha}(\mathbf{x})=\frac{\left|\left{i \in{1, \ldots, N} \mid f_{i}\left(x_{i}\right) \in A_{\alpha}\right}\right|}{N}$$
where $|E|$ is the cardinality of the set $E$. The numbers $n_{\alpha}(\mathbf{x})$ give the fractions of $x_{i}$ ‘s, $i=1, \ldots, N$, for which the random variables $f_{i}\left(x_{i}\right) \in A_{\alpha}$.

Let $P_{\alpha}=\mu\left(\left{x \in \Omega \mid f(x) \in A_{\alpha}\right}\right)$ be the probability that the random variable $f$ takes values in $A_{\alpha}$.

Let $G_{N}^{\prime}(\epsilon)$ denote the set of microstates $\mathbf{x}$ such that the fractions in the histogram are close to the corresponding probabilities:
\begin{aligned} &\left|n_{\alpha}(\mathbf{x})-P_{\alpha}\right| \leq \epsilon \ &\forall \alpha=1, \ldots, k \end{aligned}

## 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 $X_{N}=\frac{\sum_{i=1}^{N} f_{i}\left(x_{1}\right)-N E(f)}{\sqrt{N}}$, with $f_{i}$ as in $(2.3 .6)$.
Then, $\forall a, b \in \mathbb{R}, a<b$,
$$\lim {N \rightarrow \infty} \boldsymbol{\mu}\left(a \leq X{N} \leq b\right)=\frac{1}{\sqrt{2 \pi} \sigma} \int_{a}^{b} \exp \left(-\frac{x^{2}}{2 \sigma^{2}}\right)$$
where $\sigma^{2}=\mathbb{E}\left(f^{2}\right)-\mathbb{E}(f)^{2}$.

|ñ0ñ−12|≤ε

## 物理代写|统计力学代写Statistical mechanics代考|A More General Result

|小号ñ(X)−和(F)|≤ε,

μ(Gñ(ε))→1

\boldsymbol{\mu}\left(\left{\mathbf{x}\left|\lim {N \rightarrow \infty}\right| S{N}(\mathbf{x})-\mathbb{E}( f) \mid=0\right}\right)=1,\boldsymbol{\mu}\left(\left{\mathbf{x}\left|\lim {N \rightarrow \infty}\right| S{N}(\mathbf{x})-\mathbb{E}( f) \mid=0\right}\right)=1,

n_{\alpha}(\mathbf{x})=\frac{\left|\left{i \in{1, \ldots, N} \mid f_{i}\left(x_{i}\right) \在 A_{\alpha}\right}\right|}{N}n_{\alpha}(\mathbf{x})=\frac{\left|\left{i \in{1, \ldots, N} \mid f_{i}\left(x_{i}\right) \在 A_{\alpha}\right}\right|}{N}

|n一个(X)−磷一个|≤ε ∀一个=1,…,ķ

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

∑一世=1ñF一世(X一世)≈ñ和(F)

∑一世=1ñF一世(X一世)≈ñ和(F)+ñX

## 有限元方法代写

tatistics-lab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。