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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|Getting to Boltzmann: A discussion

We’ve not taken the most direct path to arrive at the Boltzmann distribution. Our derivation has almost been a deductive process, but it’s not the case you can start at “line one” and arrive at Eq. (4.31) purely through deduction. A genuinely new equation in physics can’t be derived from something more fundamental and is justified only a posteriori by the success of the theory based on it. $^{13}$ One might say that the appearance of Eq. (4.19) in our derivation was fortuitous, but the form $\mathrm{e}^{-\beta E} \Omega(E)$ presents itself naturally as a probability density having the partition function $Z$ as its normalizing constant, given that $Z$ presents itself naturally as the Laplace transform of the convolution relation for the density of states, Eq. (4.9). We’ve taken this path to support the claim (made on page 59) that the problem of statistical mechanics reduces to one in the theory of probability. ${ }^{14}$ In this subsection, we review some other ways to arrive at the Boltzmann distribution.

• The Boltzmann transport equation. Perhaps the easiest way is to consider stationary solutions of the Boltzmann transport equation, which models the nonequilibrium phase-space probability density $\rho(p, q, t)$. This approach is outside the intended scope of this book, and the Boltzmann equation is not without its own issues that we can’t explore here. Suffice to say that Eq. (4.31) occurs as the steady-state solution of the Boltzmann equation, appropriate to the state of thermal equilibrium.
• As a postulate. In his formulation of statistical mechanics, Gibbs simply started with the form of Eq. (4.31), known as the Gibbs distribution. He assumed ${ }^{15} P=\mathrm{e}^{\eta}$ (because it’s “… the most simple case conceivable”), where $\eta=(\psi-\epsilon) / \Theta$ is a combination of three functions: $\psi$, related to the normalization, $\mathrm{e}^{\psi / \Theta} \equiv Z^{-1}, \epsilon$ the energy, and $\Theta$ which he termed the modulus of the distribution.[19, p33] The linear dependence of $\eta$ on $\epsilon$ determines an ensemble which Gibbs called canonically distributed. ${ }^{16}$ The form $P=\mathrm{e}^{(\psi-\epsilon) / \Theta}$ was “line one” for Gibbs.

## 物理代写|统计力学代写Statistical mechanics代考|Consistency with thermodynamics

A requirement on statistical mechanics is that it reproduce the laws of thermodynamics, a demand ensure by equating macroscopically measurable quantities with appropriate ensemble averages (Section 2.5). As we now show, the framework we’ve established is consistent with thermodynamics if we identify $\beta=(k T)^{-1}$ and if we modify the partition function with a multiplicative factor.
Internal energy is the energy of adiabatic work and heat is the difference between work and adiabatic work (Section 1.2). Adiabatically isolated systems interact with their surroundings through mechanical means only. For such systems, the internal energy $U$ is the conserved energy of mechanical work done on the system, which is the same as the value of the Hamiltonian $H$. Thus, we equate $U$ with the ensemble average of $H$ :
$$U=\langle H\rangle=\int \rho(p, q) H(p, q) \mathrm{d} \Gamma=-\left(\frac{\partial}{\partial \beta} \ln Z\right)_{V},$$
where the final equality follows from Eq. (4.39) with $\Theta=\beta^{-1}$, and which will be recognized as Eq. (4.20) with $\alpha=\beta$. Equation (4.40) is perhaps the most useful formula in statistical mechanics.

Work entails variations in a system’s extensive external parameters $\left{X_{i}\right}, \delta W=\sum_{i} Y_{i} \delta X_{i}$; Eq. (1.4). Adiabatic work $\delta W$ associated with variations $\delta X_{i}$ is reflected in changes $\delta H$ in the value of
$$\delta W=\langle\delta H\rangle=\sum_{i}\left\langle Y_{i}\right\rangle \delta X_{i}=\sum_{i}\left(\int \frac{\partial H}{\partial X_{i}} \rho \mathrm{d} \Gamma\right) \delta X_{i}=-\frac{1}{\beta} \sum_{i}\left(\frac{\partial}{\partial X_{i}} \ln Z\right) \delta X_{i} .$$
The Hamiltonian must therefore be a function of the external parameters ${ }^{22}$ (as well as the canonical coordinates $(p, q))$, with
$$\left\langle Y_{i}\right\rangle=\int \frac{\partial H}{\partial X_{i}} \rho \mathrm{d} \Gamma=\left\langle\frac{\partial H}{\partial X_{i}}\right\rangle=-\frac{1}{\beta} \frac{\partial}{\partial X_{i}} \ln Z .$$

# 统计力学代考

## 物理代写|统计力学代写Statistical mechanics代考|Getting to Boltzmann: A discussion

• 玻尔兹曼输运方程。也许最简单的方法是考虑玻尔兹曼输运方程的平稳解，该方程模拟非平衡相空间概率密度r(p,q,吨). 这种方法超出了本书的预期范围，玻尔兹曼方程也不是没有我们无法在此探讨的问题。可以说Eq。（4.31）作为玻尔兹曼方程的稳态解出现，适合于热平衡状态。
• 作为一个假设。在他对统计力学的表述中，吉布斯只是从方程式的形式开始。(4.31)，称为吉布斯分布。他假设15磷=和这（因为它是“……可以想象的最简单的情况”），其中这=(p−ε)/钍是三个功能的组合：p，与归一化有关，和p/钍≡从−1,ε能量，和钍他将其称为分布的模数。 [19, p33]这上ε确定 Gibbs 称为规范分布的集合。16表格磷=和(p−ε)/钍是吉布斯的“第一线”。

## 物理代写|统计力学代写Statistical mechanics代考|Consistency with thermodynamics

$$U=\langle H\rangle=\int \rho(p, q) H(p, q) \mathrm{d} \Gamma=-\left(\frac{\partial}{\partial \beta} \ln Z\right){V}$$ 其中最终的等式来自等式。(4.39) 与 $\Theta=\beta^{-1}$ ，并将被识别为等式。(4.20) 与 $\alpha=\beta$. 方程 (4.40) 可能是统计力学中最有用的公式。 工作需要系统广泛的外部参数的变化 \left 的分隔符缺失或无法识别；；方程。(1.4)。绝热工作 $\delta W$ 与变化有关 $\delta X{i}$ 反映在变化中 $\delta H$ 在价值
$$\delta W=\langle\delta H\rangle=\sum_{i}\left\langle Y_{i}\right\rangle \delta X_{i}=\sum_{i}\left(\int \frac{\partial H}{\partial X_{i}} \rho \mathrm{d} \Gamma\right) \delta X_{i}=-\frac{1}{\beta} \sum_{i}\left(\frac{\partial}{\partial X_{i}} \ln Z\right) \delta X_{i}$$

$$\left\langle Y_{i}\right\rangle=\int \frac{\partial H}{\partial X_{i}} \rho \mathrm{d} \Gamma=\left\langle\frac{\partial H}{\partial X_{i}}\right\rangle=-\frac{1}{\beta} \frac{\partial}{\partial X_{i}} \ln Z$$

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