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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。
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。
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|Fundamentals of Statistical Mechanics
As formulated in the introduction, statistical mechanics is the theory of macroscopic systems from a microscopic point of view, that is, it explains the thermodynamics of a large system from the dynamics of its constituent particles. This requires the identification of the macroscopic observables in terms of microscopic quantities. In this part of the book, we formulate this correspondence starting with a simple example: a model of a paramagnetic salt. We have seen in Part I that the thermodynamic behaviour of a macroscopic system is completely determined by the fundamental equation, i.e., the entropy density as a function of the energy density and the specific volume. The latter two quantities are readily identified with the average energy per particle and the average volume per particle, respectively. The main hypothesis of equilibrium statistical mechanics is Boltzmann’s postulate that the entropy density is given by the logarithm of the number of available states of the entire system of $N$ particles, divided by $N$. This is formulated more precisely in chapter 19. The available micro-states are determined by the quantum mechanical Hamiltonian. This latter concept is explained in chapter 18; which is a very elementary introduction to quantum mechanics. It is essentially just a compilation of some basic facts, but it suffices for understanding the main part of this book. (Only chapters $33-35$ require a more complete understanding of quantum mechanics.) The Hamiltonian determines the possible energy values of the system. It is a sum of a kinetic energy term for each particle and a potential energy term due to the forces between the particles. These forces are usually of a fairly short range and are given by a potential energy function of the form shown in figure 13.1. The forces between the particles cause their individual energies to change during each collision, i.e. every time two particles come within the interaction range of each other. The total energy $E=\epsilon_{1}+\cdots+\epsilon_{N}$ is unchanged, however, provided the system is thermodynamically isolated. In equilibrium, the macroscopic properties of the system are independent of time. We shall identify these macroscopic quantities with averages of corresponding microscopic quantities over all the particles. Thus, for instance, the number of particles with energies $\epsilon_{i}$ in a given range will be independent of time when the system is in equilibrium. Remarkably, although the redistribution of energy over the particles before the attainment of equilibrium is due to the interparticle forces, it turns out that, to a first approximation, the equilibrium state reached can often be described reasonably well by assuming that the particles move independently of one another. We shall therefore first consider models for a system of free particles. The simplest such model is that of a paramagnetic salt. It is described in chapter 19 as a system of independent localized spins. In this chapter, the main hypothesis of statistical mechanics is also introduced. Chapter 20 is a mathematical digression about large deviation theory, a special branch of probability theory. This theory is used extensively in the following chapters. It is used in chapter 21 to rederive the thermodynamics of a paramagnet, and in chapter 22 a more general system of independent localized particles is treated in the same way. In chapter 23 , this analysis is extended further to quantum gases. This chapter is rather technical and can be omitted if one accepts the general formalism of equilibrium statistical mechanics outlined in chapter 24 . The derivation of the general formalism in chapter 24 is not rigorous but should make the basic formulas plausible. These formulas are of course essential for the analysis of all models in equilibrium statistical mechanics. In chapters 25 and 26 , a rigorous proof is given of the existence of the free energy density in the thermodynamic limit for classical spin systems (localized particles) and of the large-deviation property for the energy density. These chapters are optional but they put the theory of chapter 24 on a firm footing for these systems.
物理代写|统计力学代写Statistical mechanics代考|Quantum Mechanics
At the beginning of the twentieth century, it became clear that microscopic particles do not obey Newton’s laws of classical mechanics. Indeed, in order to explain the behaviour of black-body radiation Planck introduced the new postulate that the energy comes in small discrete amounts $E$ called quanta proportional to the frequency $\nu$ (see chapter 15$)$ :
$$
E=h \nu \text { where } h=6.626 \cdot 10^{-34} \mathrm{~J} \mathrm{s.}
$$
The new fundamental constant $h$ is now called Planck’s constant. It is a very small quantity, which is the reason that the discreteness of the energy is not apparent in everyday phenomena.Planck’s idea was then used by Einstein to explain the phenomenon of the photoelectric effect. This effect takes place in a photoelectric cell (figure 18.1) which is a vacuum glass bulb with two electrodes, one of which, called the cathode, is illuminated by a light source. The light liberates electrons from the cathode which results in a small current between the electrodes. It was observed experimentally that no current could flow if the wavelength of the light was larger than a certain threshold. Einstein explained this effect by assuming that an electron is liberated by a single light quantum, or photon, and that this can only happen if the energy of the photon is high enough. By Planck’s relation, therefore, electrons can only be liberated if the frequency is high enough. This idea of light quanta was at odds with the established wisdom that light is a wave phenomenon, of course, but no other explanation of the photoelectric effect seemed possible.
The next step in the development of quantum mechanics was taken by Niels Bohr, who introduced postulates about the discreteness of the energy of the bound electrons in an atom similar to the discreteness of the energy of light quanta. This enabled him to explain the appearance of lines in the radiation spectrum of atoms. His postulates were the following:
- Electrons can only exist in certain stable orbits about the nucleus of an atom.
- If an electron jumps from an orbit of higher energy $E_{2}$ to one of lower energy $E_{1}$, it emits a photon with frequency $\nu$ given by the relation $h \nu=$ $E_{2}-E_{1}$. Conversely, an electron in the lower orbit can absorb a photon with frequency given by this relation and jump to the higher orbit.
- The angular momentum $\vec{l}$ of an electron in one of its stable orbits is always an integral multiple of $\hbar=h / 2 \pi$.
物理代写|统计力学代写Statistical mechanics代考|Non-Interacting Localized Spins
Let us now consider a simple model of a macroscopic system consisting of a large number $N$ of identical particles with ‘spin- $\frac{1}{2}$ ‘, i.e. $S=\frac{1}{2} \hbar$, pinned to fixed positions and immersed in a magnetic field $H$. We shall see that with the help of one simple postulate we can derive the whole thermodynamies of this system. As explained in the previous chapter, the external field $H$ causes the energy levels $\epsilon_{i}$ of each particle to split up into two levels $\epsilon_{i} \pm \epsilon$, where $\epsilon=\mu_{0} \mu H$. (See equation (18.14); $\mu$ is the magnetic moment of the particle, $\mu_{0}=4 \pi \cdot 10^{-7} \mathrm{H} \mathrm{m}{ }^{-1}$ is an absolute constant.) Since all particles are pinned to fixed positions, we can disregard excited states and consider only the ground state of each particle with energy $\epsilon_{0}$. Thus each particle has effectively only two available energy levels: $\epsilon_{0}-\epsilon$ and $\epsilon_{0}+\epsilon$. (If the particles are fixed then it costs a lot of energy to move them, which means that the excited states are much higher than $\epsilon_{0}+\epsilon_{\text {. }}$ ) For simplicity we shall put $\epsilon_{0}=0$ in the following. This does not affect the final result. Since there are $N$ particles in all, this means that there are $2^{N}$ microstates available to the system. If the system is isolated it cannot exchange energy with the surroundings so that the total energy $E$ is fixed. This means that only those microstates are actually accessible to the system for which
$$
M \epsilon-(N-M) \epsilon=E,
$$
where $M$ is the number of particles in the upper energy level. The total number of microstates for given $M(0 \leqslant M \leqslant N)$ is
$$
\Omega(E)=\left(\begin{array}{l}
N \
M
\end{array}\right)
$$
The basic postulate of statistical mechanics is Boltzmann’s postulate:
SM. The entropy density of a localized system in equilibrium with given energy per particle $u$ is
$$
s(u)=\lim {N \rightarrow \infty} \frac{k{\mathrm{B}}}{N} \ln \Omega(N u)
$$
where $\Omega\left(F_{j}\right)$ is the numher of mirrostates acressihle to the system at energy $E$
Remember that the fundamental function $s(u, v)$ determines the thermodynamics of a simple system completely. Here the volume plays no role because all the particles are fixed to their positions. To evaluate the limit (19.3) we need Stirling’s formula for $\ln N$ ! which we shall first derive from Laplace’s principle of steepest descent.
统计力学代考
物理代写|统计力学代写Statistical mechanics代考|Fundamentals of Statistical Mechanics
正如引言中所阐述的，统计力学是从微观角度对宏观系统的理论，即从构成大系统的粒子的动力学来解释大系统的热力学。这需要用微观量来识别宏观可观察物。在本书的这一部分，我们从一个简单的例子开始制定这种对应关系：顺磁性盐的模型。我们在第一部分已经看到，宏观系统的热力学行为完全由基本方程决定，即熵密度是能量密度和比体积的函数。后两个量很容易分别用每个粒子的平均能量和每个粒子的平均体积来识别。ñ粒子，除以ñ. 第 19 章对此进行了更准确的表述。可用的微状态由量子力学哈密顿量决定。第 18 章解释了后一个概念；这是对量子力学的一个非常初级的介绍。它本质上只是一些基本事实的汇编，但足以理解本书的主要部分。（只有章节33−35需要对量子力学有更完整的理解。）哈密顿量决定了系统可能的能量值。它是每个粒子的动能项和由于粒子之间的力而产生的势能项的总和。这些力通常具有相当短的范围，并由图 13.1 所示形式的势能函数给出。粒子之间的力导致它们各自的能量在每次碰撞期间发生变化，即每次两个粒子进入彼此的相互作用范围内。总能量和=ε1+⋯+εñ然而，如果系统是热力学隔离的，则不会改变。在平衡状态下，系统的宏观性质与时间无关。我们将用所有粒子上相应微观量的平均值来识别这些宏观量。因此，例如，具有能量的粒子数ε一世当系统处于平衡状态时，在给定范围内将与时间无关。值得注意的是，尽管在达到平衡之前能量在粒子上的重新分配是由于粒子间的力，但事实证明，对于第一个近似值，达到的平衡状态通常可以通过假设粒子独立移动来合理地描述另一个。因此，我们将首先考虑自由粒子系统的模型。最简单的模型是顺磁性盐模型。它在第 19 章中被描述为一个独立的局部自旋系统。本章还介绍了统计力学的主要假设。第 20 章是关于大偏差理论（概率论的一个特殊分支）的数学题外话。该理论在以下章节中被广泛使用。它在第 21 章中用于重新推导顺磁体的热力学，在第 22 章中以相同的方式处理了一个更一般的独立局部粒子系统。在第 23 章中，这种分析进一步扩展到了量子气体。这一章是相当技术性的，如果人们接受第 24 章中概述的平衡统计力学的一般形式，则可以省略。第 24 章中一般形式的推导并不严格，但应该使基本公式合理。这些公式对于分析平衡统计力学中的所有模型当然是必不可少的。在第 25 章和第 26 章中，对经典自旋系统（局域粒子）的热力学极限中自由能密度的存在和能量密度的大偏差性质给出了严格的证明。
物理代写|统计力学代写Statistical mechanics代考|Quantum Mechanics
二十世纪初，很明显，微观粒子不遵守牛顿经典力学定律。事实上，为了解释黑体辐射的行为，普朗克引入了新的假设，即能量来自少量离散量和称为与频率成比例的量子ν（见第 15 章) :
和=Hν 在哪里 H=6.626⋅10−34 Ĵs.
新的基本常数H现在称为普朗克常数。这是一个非常小的量，这就是能量的离散性在日常现象中并不明显的原因。爱因斯坦随后用普朗克的想法来解释光电效应现象。这种效应发生在一个光电电池（图 18.1）中，它是一个带有两个电极的真空玻璃灯泡，其中一个电极称为阴极，由光源照亮。光从阴极释放电子，从而在电极之间产生小电流。实验观察到，如果光的波长大于某个阈值，则没有电流可以流动。爱因斯坦通过假设单个光量子或光子释放电子来解释这种效应，并且只有当光子的能量足够高时才会发生这种情况。因此，根据普朗克的关系，只有当频率足够高时才能释放电子。当然，光量子的这种想法与光是一种波动现象的既定观点是不一致的，但似乎没有其他对光电效应的解释是可能的。
Niels Bohr 迈出了量子力学发展的下一步，他引入了关于原子中束缚电子能量离散性的假设，类似于光量子能量的离散性。这使他能够解释原子辐射光谱中线条的出现。他的假设如下：
- 电子只能存在于围绕原子核的某些稳定轨道上。
- 如果一个电子从更高能量的轨道上跳跃和2低能量之一和1, 它发射一个具有频率的光子ν由关系给出Hν= 和2−和1. 相反，较低轨道中的电子可以吸收具有该关系给定频率的光子并跳到较高轨道。
- 角动量l→一个电子在它的一个稳定轨道上总是是整数倍ℏ=H/2圆周率.
物理代写|统计力学代写Statistical mechanics代考|Non-Interacting Localized Spins
现在让我们考虑一个由大量数组成的宏观系统的简单模型ñ具有“自旋”的相同粒子12’， IE小号=12ℏ，固定在固定位置并浸入磁场中H. 我们将看到，借助一个简单的假设，我们可以推导出这个系统的整个热力学。如前一章所述，外部场H导致能量水平ε一世每个粒子分为两个级别ε一世±ε， 在哪里ε=μ0μH. （见方程（18.14）；μ是粒子的磁矩，μ0=4圆周率⋅10−7H米−1是一个绝对常数。）由于所有粒子都固定在固定位置，我们可以忽略激发态，只考虑每个具有能量的粒子的基态ε0. 因此，每个粒子实际上只有两个可用的能级：ε0−ε和ε0+ε. （如果粒子是固定的，那么移动它们会消耗大量能量，这意味着激发态要远高于ε0+ε. ) 为简单起见，我们将把ε0=0在下面的。这不影响最终结果。既然有ñ总而言之，这意味着有2ñ系统可用的微观状态。如果系统是孤立的，它就不能与周围环境进行能量交换，因此总能量和是固定的。这意味着只有那些微状态才能被系统实际访问
米ε−(ñ−米)ε=和,
在哪里米是上能级的粒子数。给定的微状态总数米(0⩽米⩽ñ)是
Ω(和)=(ñ 米)
统计力学的基本假设是玻尔兹曼假设：
SM。与给定的每个粒子能量平衡的局域系统的熵密度在是
s(在)=林ñ→∞ķ乙ñlnΩ(ñ在)
在哪里Ω(Fj)是能源系统的镜像状态数和
请记住，基本功能s(在,在)完全决定了一个简单系统的热力学。在这里，体积不起作用，因为所有粒子都固定在它们的位置上。为了评估极限（19.3），我们需要斯特林公式lnñ！我们将首先从拉普拉斯的最速下降原理推导出来。
