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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|Quantum Gases
In chapter 22 we considered a system of identical, localized particles. These particles can therefore be distinguished according to their position in space. When identical particles are not localized they can no longer be distinguished. This situation occurs in the case of a gas since the particles of a gas are unconstrained in their motion except by the walls of the container. It also occurs when the particles are so close together that their wave functions overlap, that is when they are free to exchange positions. The indistinguishability of particles has an important consequence for the counting of microstates:
The microstates of indistinguishable particles are completely determined by the number of particles in each individual particle eigenstate; one says that the statistics of indistinguishable particles is different from that of localized particles.
It should be stressed that this is a typical quantum-mechanical phenomenon, formulated as QM 3 in chapter 18. Classical particles are always distinguishable because one can in principle follow each particle along its trajectory. According to quantum mechanies this is impossible even in principle. According to principle QM 3, there are in fact two different kinds of statistics depending on the type of particle: Bose-Einstein statistics for bosons and Fermi-Dirac statistics for fermions. This is related to the fact that fermions satisfy the Pauli exclusion principle; there can be no more than one particle in each eigenstate. (As explained in chapter 18, the degeneracy of each energy level contains a factor of $(2 s+1)$ due to the spin degrees of freedom; each energy level can then contain as many particles as its degeneracy.)
For bosons then, the number of particles $n_{i, \alpha}$ in each energy state is in principle unbounded except that at fixed total energy $E$ and total particle number $N$ the restrictions (22.3) have to be satisfied:
$$
\sum_{i=0}^{\infty} \sum_{\alpha=1}^{g_{i}} n_{i, \alpha}=N \text { and } \sum_{i=0}^{\infty} \sum_{\alpha=1}^{g_{i}} n_{i, \alpha} \epsilon_{i}=E .
$$
A microstate is completely determined by the set of numbers $\left{n_{i, \alpha}\right}$ since individual particles cannot be distinguished. In defining the entropy density we must now take the thermodynamic limit in the proper sense: $V \rightarrow \infty$ but $N / V \rightarrow \rho$, where the particle number density $\rho$ is given. Indeed, while for localized particles the limits $N \rightarrow \infty$ and $V \rightarrow \infty$ are the same thing, for gaseous systems the number of particles, and hence the density, is a variable independent of $V$. We therefore define the entropy density (now per unit volume; see the definition (8.17) of the grand potential) $\tilde{s}(\tilde{u}, \rho)$ as follows:
$$
\tilde{s}(\tilde{u}, \rho)=\lim {\Delta, \delta \rightarrow 0} \lim {V \rightarrow \infty} \frac{k_{\mathrm{B}}}{V} \ln \Omega_{\Delta, \delta}(\tilde{u}, \rho)
$$
where $\Omega_{\Delta, \delta}(\tilde{u}, \rho)$ is the number of microstates with density $N / V \in(\rho-\delta, \rho+\delta)$ and energy density $E / V \in(\tilde{u}-\Delta, \tilde{u}+\Delta)$.
To evaluate the limit (23.2), we need to use the large-deviation theory of chapter 20 for random variables with values in $\mathbb{R}^{2}$. We define two-dimensional random variables
$$
X_{V}^{\beta, \lambda}=\left(\frac{1}{V} \sum_{i, \alpha} n_{i, \alpha}, \frac{1}{V} \sum_{i, \alpha} n_{i, \alpha} \epsilon_{i}\right)
$$
with distribution function
$$
F_{V}^{\beta, \lambda}(x, y)=\frac{1}{\mathcal{Z}{V}(\beta, \lambda)} \sum{\left{n_{i, \alpha}\right}: N / V \leqslant x, E / V \leqslant y} e^{-\lambda N-\beta E}
$$
物理代写|统计力学代写Statistical mechanics代考|Ensembles
In chapter 22 we considered a system of non-interacting localized particles with general energy levels $\epsilon_{0}<\epsilon_{1}<\epsilon_{2}<\ldots$ As first argued by Gibbs, this analysis can in fact be applied to general macroscopic systems if each particle is replaced by an entire macroscopic system! Thus one obtains a large collection of identical macroscopic systems, which is called an ensemble. The energy levels $\epsilon_{i}$ must be replaced by the energy levels of an individual macroscopic system, which we denote by $E_{i}$. One can argue that the behaviour of a given macroscopic system, averaged over a long time span compared with the motion of the individual particles, is given by the average over an ensemble of identical copies of this system. This idea of replacing the average over time with an instantaneous average over copies of the same system is called ergodicity. The theory of ergodic systems has developed into a whole new branch of mathematics which we shall not discuss further here. However, to make the concept clear, let us be a bit more specific about the definition of ergodicity. Let $X^{(k)}$ be an observable of system number $k(k=1,2, \ldots, M)$. For instance, $X^{(k)}$ could be the pressure (force per unit area) exerted on the walls. Over a short time span this observable will depend on time: $X^{(k)}(t)$, but if we measure it over a sufficiently long time, the average will be constant provided the system is in equilibrium. The system is said to be ergodic if this average equals an average over an ensemble, more precisely,
$$
\lim {T \rightarrow \infty} \frac{1}{T} \int{0}^{T} X^{(k)}(t) \mathrm{d} t=\lim {M \rightarrow \infty} \mathbb{E}\left(X{M}\right),
$$
where $X_{M}=(1 / M) \sum_{k=1}^{M} X^{(k)}(t)$ for an arbitrary point $t$ in time ! The $\mathrm{~ i d e ̄ a ́ ~ o f ~ e ̀ r g o o d i c i t y ~ h a ́ s ~ c o m e ́ n e ~ u n d e ̂ r ~ s e v e r r e ̉ ~ c r i t i c i s m ~ a n d ~ i t ~ i s ~ n o ̂}$ ‘most’mechanical (i.e. Hamiltonian) systems are not ergodic.
One can look at the problem from a different viewpoint, however. Let us consider a single very large system and subdivide it into a large number $M$ of cells which, by themselves, can still be considered macroscopic. Let us first assume that the particles of the system are localized, so that they do not wander from cell to cell. Because the cells are still very large, the energy of interaction between particles of one cell and those of all other cells will be negligibly small compared with the energy of the particles within a cell, that is,
$$
E_{k} \gg \sum_{i \neq k} E_{i k},
$$
where $E_{k}$ is the total energy of the particles in cell $k$ and $E_{i k}$ is the total energy of interaction between particles of cell $i$ and particles of cell $k$. We can therefore neglect the energy of interaction between cells and assume that the cells are independent and localized, provided they are so large that equation (24.2) holds. This means that we are almost in the situation of independent identical macroscopic systems as imagined by Gibbs. The corresponding ensemble is called the canonical ensemble. If $E_{i}(i=0,1,2, \ldots)$ are the energy levels of a single cell then, in the limit $M \rightarrow \infty$, at a given temperature $T$ the distribution of the cells over the energy levels will be given by
$$
\mathbb{P}{\beta}\left(E=E{i}\right)=\frac{1}{Z(\beta)} g_{i} e^{-\beta E_{i}}
$$
where
$$
Z(\beta)=\sum_{i=0}^{\infty} g_{i} e^{-\beta E_{i}}
$$
is called the canonical partition function. This follows immediately from the discussion of systems of localized free particles in chapter 22. As remarked below equation (22.18), the distribution is just (22.10) and the internal energy per cell is given by
$$
U(\beta)=\frac{1}{Z(\beta)} \sum_{i=0}^{\infty} E_{i} g_{i} e^{-\beta E_{i}}=-\frac{\mathrm{d}}{\mathrm{d} \beta} \ln Z(\beta)
$$
物理代写|统计力学代写Statistical mechanics代考|Existence of the Thermodynamic Limit
In this chapter, following the exposition by Hugenholtz (1982), we shall prove that the thermodynamic limit of the free energy density, that is the limit (24.8) exists in the case of a general class of models: classical spin systems with finite-range pair interaction. These are models of systems of localized particles as in chapter 21 , where each particle can assume only a finite number of different states which we label by $s_{x} \in{1,2, \ldots, q}$, where $x \in \mathbb{Z}^{\nu}$ is the position of the particle (spin). Here the particles are not assumed to be independent, however, but are allowed to have some mutual interaction which for simplicity we take to be given by a pair potential with finite range. This means that the energy levels of the system are given by
$$
E_{\Lambda}\left(\left{s_{x}\right}_{x \in \Lambda}\right)=\sum_{x \in \Lambda}\left(\Psi\left(s_{x}\right)+\sum_{y:|x-y| \leqslant R} \Phi_{x-y}\left(s_{x}, s_{y}\right)\right)
$$
Here $\Lambda$ is some finite region of $\nu$-dimensional space and the sum is over all lattice points inside this region (the particles or spins are assumed to be localized on a lattice, $\mathbb{Z}^{\nu}$ for definiteness). $R$ is the range of the interaction and $\Phi$ and $\Psi$ are functions determining the interaction. The model is called classical because quantum spins are operators and if one diagonalizes an operator of the form (25.1), the energy eigenvalues are in general not of the form (25.1). One can nevertheless prove the existence of the thermodynamic limit (see for example the book by Ruelle (1969)) but we shall not go into that here. (The proof is very similar!) Specific models of the type (25.1) will be discussed in Part III. Note that we assume that the interaction potential $\Phi_{x-y}$ depends only on the difference vector $x-y$. This means that the interaction is translation invariant. We now want to prove the existence of the limit (24.8), where $N=|\Lambda|$ is the number of lattice sites in $\Lambda$. We therefore take a sequence $\Lambda_{l}(l=1,2, \ldots)$ of regions with volume tending to infinity and define the finite-volume free energy
$$
F_{\Lambda_{l}}(\beta)=-\frac{1}{\beta} \ln Z_{\Lambda_{l}}(\beta),
$$
where $Z_{\Lambda_{l}}(\beta)$ is the partition function
$$
Z_{\Lambda_{l}}(\beta)=\sum_{\left{s_{\perp}\right.} \sum_{\left.\mid x \in \Lambda_{l}\right}} \exp \left[-\beta E_{\Lambda_{l}}\left(\left{s_{x}\right}\right)\right]
$$
We want to show that the limit $\lim {l \rightarrow \infty} F{\Lambda_{l}}(\beta) /\left|\Lambda_{l}\right|$ exists and is independent of the sequence $\Lambda_{l}$. However, this can obviously not be true without restrictions on this sequence. We could for example take a sequence of volumes that grows only in one direction. In fact, we may expect that the regions must cover the whole $\nu$-dimensional space in the limit and moreover, that they grow in each direction at a comparable rate. A sufficient condition was formulated by Van Hove.
统计力学代考
物理代写|统计力学代写Statistical mechanics代考|Quantum Gases
在第 22 章中，我们考虑了一个由相同的局部粒子组成的系统。因此可以根据它们在空间中的位置来区分这些粒子。当相同的粒子没有被定位时，它们就无法再被区分。这种情况发生在气体的情况下，因为气体颗粒的运动不受容器壁的限制。当粒子如此靠近以至于它们的波函数重叠时也会发生这种情况，即当它们可以自由交换位置时。粒子的不可区分性对微观状态的计数具有重要影响：
不可区分粒子的微观状态完全由每个单独粒子本征态中的粒子数决定；有人说，不可区分粒子的统计数据与局域粒子的统计数据不同。
应该强调的是，这是一种典型的量子力学现象，在第 18 章中被表述为 QM 3。经典粒子总是可以区分的，因为原则上可以沿着每个粒子的轨迹跟随每个粒子。根据量子力学，这即使在原则上也是不可能的。根据 QM 3 原理，实际上有两种不同的统计量，具体取决于粒子的类型：玻色子的 Bose-Einstein 统计量和费米子的 Fermi-Dirac 统计量。这与费米子满足泡利不相容原理有关；每个本征态中不能有超过一个粒子。（如第 18 章中所解释的，每个能级的简并性包含一个因子(2s+1)由于自旋自由度；然后每个能级可以包含与其简并度一样多的粒子。）
对于玻色子，粒子的数量n一世,一个在每个能量状态中，原则上是无界的，除了在固定的总能量下和和总粒子数ñ必须满足限制（22.3）：
∑一世=0∞∑一个=1G一世n一世,一个=ñ 和 ∑一世=0∞∑一个=1G一世n一世,一个ε一世=和.
微观状态完全由一组数字决定\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别因为无法区分单个粒子。在定义熵密度时，我们现在必须采用正确意义上的热力学极限：在→∞但ñ/在→ρ, 其中粒子数密度ρ给出。事实上，虽然对于局部粒子，限制ñ→∞和在→∞是相同的，对于气态系统，粒子的数量，因此密度，是一个独立的变量在. 因此，我们定义了熵密度（现在是每单位体积；参见大势的定义（8.17））s~(在~,ρ)如下：
s~(在~,ρ)=林Δ,d→0林在→∞ķ乙在lnΩΔ,d(在~,ρ)
在哪里ΩΔ,d(在~,ρ)是具有密度的微状态数ñ/在∈(ρ−d,ρ+d)和能量密度和/在∈(在~−Δ,在~+Δ).
为了评估极限（23.2），我们需要使用第 20 章的大偏差理论来计算随机变量的值R2. 我们定义二维随机变量
X在b,λ=(1在∑一世,一个n一世,一个,1在∑一世,一个n一世,一个ε一世)
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
物理代写|统计力学代写Statistical mechanics代考|Ensembles
在第 22 章中，我们考虑了具有一般能级的非相互作用局域粒子系统ε0<ε1<ε2<…正如吉布斯首先论证的那样，如果每个粒子都被整个宏观系统取代，这种分析实际上可以应用于一般宏观系统！因此，人们获得了大量相同的宏观系统，称为一个系综。能级ε一世必须由单个宏观系统的能级代替，我们将其表示为和一世. 有人可以争辩说，一个给定的宏观系统的行为，与单个粒子的运动相比，在很长一段时间内平均下来，是由该系统的一组相同副本的平均值给出的。这种用同一系统副本的瞬时平均值代替随时间变化的平均值的想法称为遍历性。遍历系统理论已经发展成为一个全新的数学分支，我们在此不再赘述。但是，为了使概念更清楚，让我们更具体地了解遍历性的定义。让X(ķ)是系统编号的可观察值ķ(ķ=1,2,…,米). 例如，X(ķ)可能是施加在墙壁上的压力（每单位面积的力）。在很短的时间内，这个可观察的将取决于时间：X(ķ)(吨)，但是如果我们在足够长的时间内测量它，则如果系统处于平衡状态，平均值将是恒定的。如果这个平均值等于一个整体的平均值，则该系统被称为遍历，更准确地说，
林吨→∞1吨∫0吨X(ķ)(吨)d吨=林米→∞和(X米),
在哪里X米=(1/米)∑ķ=1米X(ķ)(吨)对于任意点吨及时！这̉ 一世d和̄一个́ ○F 和̀rG○○d一世C一世吨是 H一个́s C○米和́n和 在nd和̂r s和在和rr和̉ Cr一世吨一世C一世s米 一个nd 一世吨 一世s n○̂“大多数”机械（即哈密顿）系统不是遍历的。
然而，人们可以从不同的角度看待这个问题。让我们考虑一个非常大的系统并将其细分为大量米细胞本身仍然可以被认为是宏观的。让我们首先假设系统的粒子是局部化的，因此它们不会从一个单元移动到另一个单元。由于细胞仍然很大，一个细胞的粒子与所有其他细胞的粒子相互作用的能量与一个细胞内粒子的能量相比将小到可以忽略不计，即
和ķ≫∑一世≠ķ和一世ķ,
在哪里和ķ是细胞中粒子的总能量ķ和和一世ķ是细胞粒子之间相互作用的总能量一世和细胞颗粒ķ. 因此，我们可以忽略细胞之间相互作用的能量，并假设细胞是独立的和局部的，只要它们大到等式（24.2）成立。这意味着我们几乎处于吉布斯想象的独立相同的宏观系统的境地。相应的集成称为规范集成。如果和一世(一世=0,1,2,…)是单个细胞的能级，那么，在极限米→∞, 在给定温度下吨细胞在能级上的分布将由下式给出
磷b(和=和一世)=1从(b)G一世和−b和一世
在哪里
从(b)=∑一世=0∞G一世和−b和一世
称为典型配分函数。这直接从第 22 章中对局域自由粒子系统的讨论中得出。如下方程（22.18）所述，分布正好是（22.10），每个单元的内部能量由下式给出
在(b)=1从(b)∑一世=0∞和一世G一世和−b和一世=−ddbln从(b)
物理代写|统计力学代写Statistical mechanics代考|Existence of the Thermodynamic Limit
在本章中，根据 Hugenholtz (1982) 的阐述，我们将证明自由能密度的热力学极限，即极限 (24.8) 存在于一类一般模型的情况下：具有有限-的经典自旋系统范围对交互。这些是第 21 章中局部粒子系统的模型，其中每个粒子只能假设有限数量的不同状态，我们将其标记为sX∈1,2,…,q， 在哪里X∈从ν是粒子的位置（自旋）。然而，这里并不假定粒子是独立的，而是允许有一些相互作用，为简单起见，我们将其视为由有限范围的对势给出。这意味着系统的能级由下式给出
\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别
这里Λ是某个有限区域ν维空间，总和在该区域内的所有晶格点上（假设粒子或自旋位于晶格上，从ν为了确定性）。R是相互作用的范围和披和Ψ是决定交互作用的函数。该模型被称为经典模型，因为量子自旋是算子，并且如果将（25.1）形式的算子对角化，则能量特征值通常不是（25.1）形式。尽管如此，人们仍然可以证明热力学极限的存在（例如，参见 Ruelle（1969）的书），但我们不会在这里讨论。（证明非常相似！）类型（25.1）的具体模型将在第三部分讨论。请注意，我们假设交互势披X−是仅取决于差向量X−是. 这意味着交互是平移不变的。我们现在要证明极限 (24.8) 的存在，其中ñ=|Λ|是格点的数量Λ. 因此我们采取一个序列Λl(l=1,2,…)体积趋于无穷大的区域并定义有限体积自由能
FΛl(b)=−1bln从Λl(b),
在哪里从Λl(b)是配分函数
\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别
我们想证明极限林l→∞FΛl(b)/|Λl|存在且独立于序列Λl. 然而，如果不限制这个顺序，这显然是不正确的。例如，我们可以采用仅在一个方向上增长的一系列卷。事实上，我们可以期望这些区域必须覆盖整个ν维空间中的极限，此外，它们以可比的速度在每个方向上增长。Van Hove 提出了一个充分条件。
