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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.

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

∑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.

s~(在~,ρ)=林Δ,d→0林在→∞ķ乙在ln⁡ΩΔ,d(在~,ρ)

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)

\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|Existence of the Thermodynamic Limit

\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别

FΛl(b)=−1bln⁡从Λl(b),

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