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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}

物理代写|统计力学代写Statistical mechanics代考|Large Deviation Theory

Fñ(X1,…,Xd)=磷(Xñ∈(−∞,X1]×⋯×(−∞,Xd]).

LD 1. 对于所有封闭子集F⊂Rd,

物理代写|统计力学代写Statistical mechanics代考|The Paramagnet Revisited

Fñ(X)=∑米:和米/ñ⩽XΩ(和米)∑米=0ñΩ(和米),

∑米=0ñΩ(和米)=2ñ

s(在)=林Δ→0林ñ→∞ķ乙ñln⁡ΩΔ(ñ在)

ΩΔ(和)=∑−ñΔ<d和<ñΔΩ(和+d和)

ΩΔ(和)=2ñ磷(在ñ∈(和ñ−Δ,和ñ+Δ))

C(吨)=ln⁡∫−∞∞和吨εX dF(X)

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：

物理代写|统计力学代写Statistical mechanics代考|Identical Localized Free Particles

s(在)=林Δ→0林ñ→∞ķ乙ñln⁡ΩΔ(ñ,在)

ΩΔ(ñ,在)=∑−ñΔ<d和<ñΔΩ(ñ,ñ在+d和)

∑一世=0∞n一世=ñ 和 ∑一世=0∞n一世ε一世=和

\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。）

\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

\left 的分隔符缺失或无法识别