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## 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:

1. Electrons can only exist in certain stable orbits about the nucleus of an atom.
2. 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.
3. 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.

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

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：

1. 电子只能存在于围绕原子核的某些稳定轨道上。
2. 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。
3. 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

Ω(和)=(ñ 米)

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ñln⁡Ω(ñ在)