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统计力学是一个数学框架，它将统计方法和概率理论应用于大型微观实体的集合。它不假设或假定任何自然法则，而是从这种集合体的行为来解释自然界的宏观行为。
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我们提供的统计力学Statistical mechanics及其相关学科的代写，服务范围广, 其中包括但不限于:
- Statistical Inference 统计推断
- Statistical Computing 统计计算
- Advanced Probability Theory 高等概率论
- Advanced Mathematical Statistics 高等数理统计学
- (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|Large Deviation Theory
We want to generalize the theory of the previous chapter to particles with more than two energy levels. To this end we shall first generalize Laplace’s theorem to a result more suited to the general situation. This generalization is called large-deviation theory. In chapter 21 we apply this theory in rederiving equation (19.10) for the entropy of a paramagnet. In chapter 22 we consider the more general situation. Large-deviation theory also has important applications in other areas of mathematics, for example information theory, dynamical systems, and queueing theory. See problems II-19 to II-23.
The main theorem of large-deviation theory is a generalization of theorem $19.1$, where the integral over $x$ is replaced by an integral with respect to the distribution function $F_{N}$ of a general random variable $X_{N}$. Remember that, if $X_{N}$ is a random variable then its distribution function $F_{N}$ is defined by $F_{N}(x)=\mathbb{P}\left(X_{N} \in(-\infty, x]\right)$, where $\mathbb{P}$ denotes the probability. Similarly, if $X_{N}$ is a random variable with values in $\mathbb{R}^{d}$ then
$$
F_{N}\left(x_{1}, \ldots, x_{d}\right)=\mathbb{P}\left(X_{N} \in\left(-\infty, x_{1}\right] \times \cdots \times\left(-\infty, x_{d}\right]\right) .
$$
To formulate the theorem we first need some more definitions.
Definition 20.1 A function $I: \mathbb{R}^{d} \rightarrow[0,+\infty]$ (which may also take the value $+\infty)$ is called a rate function if, for every $B \in[0,+\infty)$, the set $I^{-1}(B)=$ $\left{x \in \mathbb{R}^{d} \mid I(x) \leqslant B\right}$ is compact.
Definition 20.2 A sequence of random variables $X_{N}(N \in \mathbb{N})$ with values in $\mathbb{R}^{d}$ satisfies the large-deviation principle $(L D P)$ if there exists a rate function $I: \mathbb{R}^{d} \rightarrow[0, \infty]$ such that the following hold:
LD 1. For all closed subsets $F \subset \mathbb{R}^{d}$,
$$
\limsup {N \rightarrow \infty} \frac{1}{N} \ln \mathbb{P}\left(X{N} \in F\right) \leqslant-\inf {x \in F} I(x) $$ LD 2. For all open subsets $G \subset \mathbb{R}^{d}$, $$ \liminf {N \rightarrow \infty} \frac{1}{N} \ln \mathbb{P}\left(X_{N} \in G\right) \geqslant-\inf _{x \in G} I(x) .
$$
REMARK 20.1: The one-dimensional case.
In the one-dimensional case $(d=1)$ which we shall mostly consider, it suffices to prove LD 1 and LD 2 for closed and open intervals respectively. Moreover, if $F_{N}$ is continuous, in particular if $X_{N}$ has a density,
$$
\mathbb{P}\left(X_{N} \in(a, b)\right)=\mathbb{P}\left(X_{N} \in[a, b]\right)=F_{N}(b)-F_{N}(a)
$$
If $I(x)$ is also continuous, then the two inequalities (20.2) and (20.3) combine and become
$$
\lim {N \rightarrow \infty} \frac{1}{N} \ln \mathbb{P}\left(X{N} \in(a, b)\right)=-\inf _{x \in(a, b)} I(x)
$$
物理代写|统计力学代写Statistical mechanics代考|The Paramagnet Revisited
Let us now apply large-deviation theory to the problem of chapter 19 . We introduce the following probability distribution for the random variable $u_{N}=$ $E / N:$
$$
F_{N}(x)=\frac{\sum_{M: E_{M} / N \leqslant x} \Omega\left(E_{M}\right)}{\sum_{M=0}^{N} \Omega\left(E_{M}\right)},
$$
where $E_{M}$ is defined by equation (19.1). Note that the normalization constant
$$
\sum_{M=0}^{N} \Omega\left(E_{M}\right)=2^{N}
$$
is the total number of microstates. The probability distribution (21.1) is the uniform distribution giving equal probabilities to every possible microstate. We shall use this distribution in the large deviation principle to derive the entropy density (19.10). Before embarking on this, note that we have been slightly inaccurate in the definition of $s(u)$ in chapter 19 : the limit (19.3) can only be defined for $u=E / N$ with $E$ given by equation (19.1), i.e. if $u / \epsilon$ is a rational number! Clearly this is an undesirable artefact of the definition (19.3) so we replace it with the more accurate
$$
s(u)=\lim {\Delta \rightarrow 0} \lim {N \rightarrow \infty} \frac{k_{\mathrm{B}}}{N} \ln \Omega_{\Delta}(N u)
$$
where
$$
\Omega_{\Delta}(E)=\sum_{-N \Delta<\delta E<N \Delta} \Omega(E+\delta E)
$$
This is well defined for all values of $u$ and we now prove that it is the same as equation (19.10). For this we use the fact that
$$
\Omega_{\Delta}(E)=2^{N} \mathbb{P}\left(u_{N} \in\left(\frac{E}{N}-\Delta, \frac{E}{N}+\Delta\right)\right)
$$
Since the spins are all independent we can write
$$
u_{N}=\frac{1}{N} E \sum_{k=1}^{N} s^{(k)}
$$
where the $s^{(k)}$ are independent random variables, equal to $\pm 1$ with probability $\frac{1}{2}$. By theorem $20.2, u_{N}$ satisfies the LDP with rate function given by (20.12) and
$$
C(t)=\ln \int_{-\infty}^{\infty} e^{t \epsilon x} \mathrm{~d} F(x)
$$
where $F(x)$ is the distribution function for $s^{(k)}$, that is
$$
F(x)= \begin{cases}0 & \text { if } x<-1 \ \frac{1}{2} & \text { if }-1 \leqslant x<1 \ 1 & \text { if } x \geqslant 1\end{cases}
$$
$C(t)$ can be computed as follows:
$$
\begin{aligned}
e^{C(t)} &=\lim {L \rightarrow \infty} \int{-L}^{L} e^{t \epsilon x} \mathrm{~d} F(x) \
&=\lim {L \rightarrow \infty}\left(\left.e^{t \epsilon x} F(x)\right|{-L} ^{L}-\epsilon t \int_{-L}^{L} e^{\epsilon t x} F(x) \mathrm{d} x\right) \
&=\lim {L}\left(e^{\epsilon t L}-\frac{1}{2} \epsilon t \int{-1}^{1} e^{t \epsilon x} \mathrm{~d} x-\epsilon t \int_{1}^{L} e^{\epsilon t x} \mathrm{~d} x\right) \
&=\lim _{L \rightarrow \infty}\left(e^{\epsilon t L}-\frac{1}{2}\left(e^{\epsilon t}-e^{-\epsilon t}\right)-\left(e^{\epsilon t L}-e^{\epsilon t}\right)\right) \
&=\frac{1}{2}\left(e^{\epsilon t}+e^{-\epsilon t}\right)=\cosh (\epsilon t)
\end{aligned}
$$
物理代写|统计力学代写Statistical mechanics代考|Identical Localized Free Particles
In general, a particle in an energy well will have infinitely many energy levels (See for instance example 18.1). In this chapter, we extend the analysis of the previous chapter to this case. We shall again assume that the particles are localized so that the volume does not play a role in the analysis. We also assume again that there is no interaction between the particles. In chapter 24 we shall see that this analysis forms the basis of the analysis of general statistical mechanical systems, even when there is interaction! In chapter 23 we consider the case of non-localized particles without interaction, i.e. gases.
Let us denote the energy levels of a single particle by $\epsilon_{0}<\epsilon_{1}<\epsilon_{2}<\ldots$, each with a certain degeneracy $g_{0}, g_{1}, g_{2}, \ldots$. Again we define the entropy density $s(u)$ by
$$
s(u)=\lim {\Delta \rightarrow 0} \lim {N \rightarrow \infty} \frac{k_{\mathrm{B}}}{N} \ln \Omega_{\Delta}(N, u)
$$
where $\Omega_{\Delta}(N, u)$ is defined as in equation (21.4):
$$
\Omega_{\Delta}(N, u)=\sum_{-N \Delta<\delta E<N \Delta} \Omega(N, N u+\delta E)
$$
and $\Omega(N, E)$ is the total number of microstates with total energy $E$. The total number of microstates can be evaluated as follows. Let $n_{i}$ denote the number of particles in a state with energy $\epsilon_{i}$. Since the total number of particles is $N$ and we have fixed the total energy $E$, the numbers $n_{i}$ must satisfy the conditions
$$
\sum_{i=0}^{\infty} n_{i}=N \text { and } \sum_{i=0}^{\infty} n_{i} \epsilon_{i}=E
$$
if a microstate with the given values of $n_{i}$ is to be allowed. The number of microstates with given $n_{0}, n_{1}, \ldots$ is:
$$
\Omega\left(N,\left{n_{i}\right}_{i=0}^{\infty}\right)=\frac{N !}{\prod_{i=0}^{\infty} n_{i} !} \prod_{i=0}^{\infty} g_{i}^{n_{i}} .
$$
$\left(N ! /\left(\prod n_{i} !\right)\right.$ is the number of ways to choose which particles go into which energy level $f_{i}$ and $g_{i}^{n_{i}}$ is the mumher of whys the $n_{i}$ proticles in level $i$ can he subdivided over the $g_{i}$ degenerate states.)
The total number of allowed states with total energy $E$ is therefore
$$
\Omega(E, N)=\sum_{\left{n_{i}\right}}^{} \Omega\left(N,\left{n_{i}\right}\right) $$ where the ${ }^{}$ indicates that the sum is to be restricted to those sequences $\left{n_{i}\right}$ satisfying equation (22.3). It is now not so easy to compute the limit (22.1) directly using Stirling’s formula. Instead, we use the methods of largedeviation theory. This also leads to a difficulty, however. If we assign equal probabilities to all microstates as we did in chapter 21 (see equation (21.1)) then the probability of each individual microstate will be zero because there are infinitely many microstates. To avoid this problem we employ a trick: we give decreasing weights $e^{-\beta E}$ to higher energy states and compensate for this factor in the expression for $\Omega_{\Delta}$. We choose therefore an arbitrary constant $\beta>0$ and define random variables $X_{N}^{\beta}$ with distribution functions
$$
F_{N}(x)=\frac{1}{Z_{N}(\beta)} \sum_{\left{n_{i}\right}: \sum n_{i}=N, E / N \leqslant x} \Omega\left(N,\left{n_{i}\right}\right) e^{-\beta E}
$$
where $E=\sum_{i=0}^{\infty} n_{i} \epsilon_{i}$ and
$$
\begin{aligned}
Z_{N}(\beta) &=\sum_{\left{n_{i}\right}: \sum_{n_{i}=N}} \Omega\left(N,\left{n_{i}\right}\right) e^{-\beta E} \
&=\left(\sum_{i=0}^{\infty} g_{i} e^{-\beta \epsilon_{i}}\right)^{N}=\left(Z_{\beta}\right)^{N}
\end{aligned}
$$
统计力学代考
物理代写|统计力学代写Statistical mechanics代考|Large Deviation Theory
我们想将上一章的理论推广到具有两个以上能级的粒子。为此，我们首先将拉普拉斯定理推广到更适合一般情况的结果。这种概括称为大偏差理论。在第 21 章中，我们应用这个理论来重新推导顺磁体熵的方程（19.10）。在第 22 章中，我们考虑更一般的情况。大偏差理论在其他数学领域也有重要应用，例如信息论、动力系统和排队论。见习题 II-19 至 II-23。
大偏差理论的主要定理是定理的推广19.1, 其中积分超过X被关于分布函数的积分代替Fñ一般随机变量的Xñ. 请记住，如果Xñ是一个随机变量，那么它的分布函数Fñ定义为Fñ(X)=磷(Xñ∈(−∞,X])， 在哪里磷表示概率。同样，如果Xñ是一个随机变量，其值为Rd然后
Fñ(X1,…,Xd)=磷(Xñ∈(−∞,X1]×⋯×(−∞,Xd]).
为了制定这个定理，我们首先需要更多的定义。
定义 20.1 一个函数我:Rd→[0,+∞]（也可能取值+∞)称为速率函数，如果，对于每个乙∈[0,+∞), 集合我−1(乙)= \left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别紧凑。
定义 20.2 随机变量序列Xñ(ñ∈ñ)与值Rd满足大偏差原则(大号D磷)如果存在比率函数我:Rd→[0,∞]使得以下成立：
LD 1. 对于所有封闭子集F⊂Rd,
林汤ñ→∞1ñln磷(Xñ∈F)⩽−信息X∈F我(X)LD 2. 对于所有开放子集G⊂Rd,林信息ñ→∞1ñln磷(Xñ∈G)⩾−信息X∈G我(X).
备注 20.1：一维案例。
在一维情况下(d=1)我们将主要考虑，分别证明闭区间和开区间的 LD 1 和 LD 2 就足够了。此外，如果Fñ是连续的，特别是如果Xñ有密度，
磷(Xñ∈(一个,b))=磷(Xñ∈[一个,b])=Fñ(b)−Fñ(一个)
如果我(X)也是连续的，则两个不等式 (20.2) 和 (20.3) 结合成为
林ñ→∞1ñln磷(Xñ∈(一个,b))=−信息X∈(一个,b)我(X)
物理代写|统计力学代写Statistical mechanics代考|The Paramagnet Revisited
现在让我们将大偏差理论应用于第 19 章的问题。我们为随机变量引入以下概率分布在ñ= 和/ñ:
Fñ(X)=∑米:和米/ñ⩽XΩ(和米)∑米=0ñΩ(和米),
在哪里和米由等式（19.1）定义。注意归一化常数
∑米=0ñΩ(和米)=2ñ
是微观状态的总数。概率分布 (21.1) 是对每个可能的微观状态赋予相等概率的均匀分布。我们将在大偏差原理中使用这种分布来推导熵密度（19.10）。在开始之前，请注意我们对s(在)在第 19 章中：限制（19.3）只能定义为在=和/ñ和和由等式 (19.1) 给出，即如果在/ε是有理数！显然，这是定义（19.3）的不受欢迎的伪影，因此我们将其替换为更准确的
s(在)=林Δ→0林ñ→∞ķ乙ñlnΩΔ(ñ在)
在哪里
ΩΔ(和)=∑−ñΔ<d和<ñΔΩ(和+d和)
这对于所有的值都有很好的定义在我们现在证明它与方程（19.10）相同。为此，我们使用以下事实
ΩΔ(和)=2ñ磷(在ñ∈(和ñ−Δ,和ñ+Δ))
由于自旋都是独立的，我们可以写
在ñ=1ñ和∑ķ=1ñs(ķ)
在哪里s(ķ)是独立的随机变量，等于±1有概率12. 由定理20.2,在ñ用 (20.12) 给出的速率函数满足 LDP 和
C(吨)=ln∫−∞∞和吨εX dF(X)
在哪里F(X)是分布函数s(ķ)， 那是
F(X)={0 如果 X<−1 12 如果 −1⩽X<1 1 如果 X⩾1
C(吨)可以计算如下：
和C(吨)=林大号→∞∫−大号大号和吨εX dF(X) =林大号→∞(和吨εXF(X)|−大号大号−ε吨∫−大号大号和ε吨XF(X)dX) =林大号(和ε吨大号−12ε吨∫−11和吨εX dX−ε吨∫1大号和ε吨X dX) =林大号→∞(和ε吨大号−12(和ε吨−和−ε吨)−(和ε吨大号−和ε吨)) =12(和ε吨+和−ε吨)=科什(ε吨)
物理代写|统计力学代写Statistical mechanics代考|Identical Localized Free Particles
通常，能量阱中的粒子将具有无限多个能级（参见示例 18.1）。在本章中，我们将上一章的分析扩展到这个案例。我们将再次假设粒子是局部的，因此体积在分析中不起作用。我们还再次假设粒子之间没有相互作用。在第 24 章中，我们将看到这种分析构成了一般统计力学系统分析的基础，即使存在相互作用！在第 23 章中，我们考虑了没有相互作用的非定域粒子的情况，即气体。
让我们将单个粒子的能级表示为ε0<ε1<ε2<…, 每个都有一定的退化G0,G1,G2,…. 我们再次定义熵密度s(在)经过
s(在)=林Δ→0林ñ→∞ķ乙ñlnΩΔ(ñ,在)
在哪里ΩΔ(ñ,在)定义为等式（21.4）：
ΩΔ(ñ,在)=∑−ñΔ<d和<ñΔΩ(ñ,ñ在+d和)
和Ω(ñ,和)是具有总能量的微状态总数和. 微观状态的总数可以如下评估。让n一世表示处于有能量状态的粒子数ε一世. 由于粒子总数为ñ我们已经确定了总能量和， 号码n一世必须满足条件
∑一世=0∞n一世=ñ 和 ∑一世=0∞n一世ε一世=和
如果具有给定值的微观状态n一世是被允许的。给定的微状态数n0,n1,…是：
\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别
(ñ!/(∏n一世!)是选择哪些粒子进入哪个能级的方法的数量F一世和G一世n一世是为什么？n一世水平的 proticles一世他可以细分G一世简并状态。）
具有总能量的允许状态总数和因此
\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别在哪里表示总和仅限于那些序列\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别满足方程（22.3）。现在直接使用斯特林公式计算极限（22.1）并不容易。相反，我们使用大偏差理论的方法。然而，这也导致了困难。如果我们像第 21 章中所做的那样为所有微观状态分配相等的概率（参见方程（21.1）），那么每个单独的微观状态的概率将为零，因为存在无限多个微观状态。为了避免这个问题，我们采用了一个技巧：我们给出递减的权重和−b和到更高的能量状态并补偿表达式中的这个因素ΩΔ. 因此我们选择一个任意常数b>0并定义随机变量Xñb具有分配功能
\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别
在哪里和=∑一世=0∞n一世ε一世和
\left 的分隔符缺失或无法识别
