## 物理代写|热力学代写thermodynamics代考|SEM202

statistics-lab™ 为您的留学生涯保驾护航 在代写热力学thermodynamics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。我们的专家在代写热力学thermodynamics代写方面经验极为丰富，各种代写热力学thermodynamics相关的作业也就用不着说。

• 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|热力学代写thermodynamics代考|Electron–Phonon Interaction

The electron-phonon interaction can be shown to bear analogy to the optomechanical interaction in Section 4.2.1. In media where the electron-phonon interaction is weak, the deformation-potential method may be applied to long-wavelength phonons. Whereas in an unstrained (cubic) covalent crystal the electron energy band may be assumed spherical,
$$E_0(\boldsymbol{k})=\frac{\hbar^2 k^2}{2 m^},$$ $m^$ being the effective mass of the conduction electron, a small (uniform) static deformation yields for low $k$,
$$E(\boldsymbol{k}) \simeq E_0(\boldsymbol{k})+C_{\mathrm{d}} \delta \mathcal{V},$$
where $\delta \mathcal{V}$ is the dilation (relative volume change) given by the trace of the strain tensor, and
$$C_{\mathrm{d}}=\frac{\partial E(\mathbf{0})}{\partial(\delta \mathcal{V})}=-\frac{2}{3} E_{\mathrm{F}}$$
for a free electron gas, $E_{\mathrm{F}}$ being the Fermi energy.
For long-wavelength acoustic phonons, we have, instead of (4.23),
$$E(\boldsymbol{k}, \boldsymbol{x})=E_0(\boldsymbol{k})+C_{\mathrm{d}} \delta \mathcal{V}(\boldsymbol{x})$$
We then expand the Hamiltonian of the dilation perturbation in phonon operators $(3.26)$
\begin{aligned} \tilde{H}{\mathrm{d}}= & \int d^3 x \hat{\Psi}^{\dagger}(\boldsymbol{x}) C{\mathrm{d}} \delta \mathcal{V}(\boldsymbol{x}) \hat{\Psi}(\boldsymbol{x})=\sum_{\boldsymbol{k}^{\prime}, \boldsymbol{k}} c_{\boldsymbol{k}^{\prime}}^{\dagger} c_{\boldsymbol{k}}\left\langle\boldsymbol{k}^{\prime}\left|C_{\mathrm{d}} \delta \mathcal{V}\right| \boldsymbol{k}\right\rangle \ = & i C_{\mathrm{d}} \sum_{\boldsymbol{k}^{\prime}, \boldsymbol{k}} c_{\boldsymbol{k}^{\prime}}^{\dagger} c_{\boldsymbol{k}} \sum_{\boldsymbol{q}}|\boldsymbol{q}| \sqrt{\frac{\hbar}{2 \rho \omega_q}}\left[a_{\boldsymbol{q}} \int d^3 x u_{\boldsymbol{k}^{\prime}}^* u_{\boldsymbol{k}} e^{i\left(\boldsymbol{k}-\boldsymbol{k}^{\prime}+\boldsymbol{q}\right) \cdot \boldsymbol{x}}\right. \ & \left.-a_{\boldsymbol{q}}^{\dagger} \int d^3 x u_{\boldsymbol{k}^{\prime}}^* u_{\boldsymbol{k}} e^{i\left(\boldsymbol{k}-\boldsymbol{k}^{\prime}-\boldsymbol{q}\right) \cdot x}\right] \end{aligned}

## 物理代写|热力学代写thermodynamics代考|Polaronic Interaction of a Two-Level System with a Phonon Bath

This model consists of a driven two-level system (TLS) whose $\sigma_z$ operator is coupled to a (dephasing) bath, while its $\sigma_x$ operator is coupled to another (energy-exchange) bath. The Hamiltonian is then
$$H=H_{\mathrm{S}}+H_{\mathrm{SB}}+H_{\mathrm{B}}$$
where
\begin{aligned} H_{\mathrm{S}} & =\frac{\hbar \omega_0}{2} \sigma_z+\frac{\hbar \Omega}{2}\left(\sigma_{+} e^{-i \omega_l t}+\sigma_{-} e^{i \omega_l t}\right), \ H_{\mathrm{SB}} & =\sigma_z \otimes \hbar \sum_k\left(g_k a_k^{\dagger}+g_k^* a_k\right)+\sigma_x \otimes \hbar \sum_k\left(\eta_k b_k^{\dagger}+\eta_k^* b_k\right), \ H_{\mathrm{B}} & =\hbar \sum_k \omega_k a_k^{\dagger} a_k+\hbar \sum_k \tilde{\omega}_k b_k^{\dagger} b_k . \end{aligned}
Here $\Omega$ is the Rabi frequency of the driving field, $g_k$ and $\eta_k$ are coupling strengths of the TLS to the mode $k$ of the corresponding bath, and $a_k^{\dagger}, a_k\left(b_k^{\dagger}, b_k\right)$ are the creation and annihilation operators of the mode $k$ of the corresponding bath. Due to the driving, the system Hamiltonian is not diagonal in the $\sigma_z$ basis, thereby allowing energy exchange with the dephasing bath.

The system dynamics can be studied upon applying to (4.35) the polaron transformation $e^{\mathcal{T}}$, where
$$\mathcal{T}=\sigma_Z \otimes \sum_k\left(\alpha_k a_k^{\dagger}-\alpha_k^* a_k\right), \quad \alpha_k=\frac{g_k}{\omega_k} .$$
This transformation shifts the equilibrium position of the dephasing bath oscillators by a factor proportional to the TLS energy. The transformed Hamiltonian has the form
$$\widetilde{H}=e^{\mathcal{T}} H e^{-\mathcal{T}}=\tilde{H}{\mathrm{S}}+\widetilde{H}{\mathrm{SB}}+\widetilde{H}{\mathrm{B}}$$ where $$\begin{gathered} \widetilde{H}{\mathrm{S}}=\frac{\hbar \omega_0}{2} \sigma_z+\frac{\hbar \Omega_{\mathrm{r}}}{2}\left(\sigma_{+} e^{-i \omega_l t}+\sigma_{-} e^{i \omega_l t}\right) \ \widetilde{H}{\mathrm{SB}}=\frac{\hbar \Omega}{2}\left[e^{-i \omega_l t} \sigma{+} \otimes\left(A_{+}-A\right)+e^{i \omega_l t} \sigma_{-} \otimes\left(A_{-}-A\right)\right] \end{gathered}$$ $\begin{gathered}+\left(\sigma_{+} \otimes A_{+}+\sigma_{-} \otimes A_{-}\right) \otimes \hbar \sum_k\left(\eta_k b_k^{\dagger}+\eta_k^* b_k\right), \ \tilde{H}_{\mathrm{B}}=\hbar \sum_k \omega_k a_k^{\dagger} a_k+\hbar \sum_k \tilde{\omega}_k b_k^{\dagger} b_k .\end{gathered}$

# 热力学代写

## 物理代写|热力学代写thermodynamics代考|Electron–Phonon Interaction

$$\text { E_O(lboldsymbol }{\mathrm{k}})=\mid f \mathrm{frac}\left{\backslash \mathrm{hbar} \wedge 2 \mathrm{k}^{\wedge} 2\right}\left{2 \mathrm{~m}^{\wedge}\right},$$

$$E(\boldsymbol{k}) \simeq E_0(\boldsymbol{k})+C_{\mathrm{d}} \delta \mathcal{V}$$

$$C_{\mathrm{d}}=\frac{\partial E(\mathbf{0})}{\partial(\delta \mathcal{V})}=-\frac{2}{3} E_{\mathrm{F}}$$

$$E(\boldsymbol{k}, \boldsymbol{x})=E_0(\boldsymbol{k})+C_{\mathrm{d}} \delta \mathcal{V}(\boldsymbol{x})$$

$$\tilde{H} \mathrm{~d}=\int d^3 x \hat{\Psi}^{\dagger}(\boldsymbol{x}) C \mathrm{~d} \delta \mathcal{V}(\boldsymbol{x}) \hat{\Psi}(\boldsymbol{x})=\sum_{\boldsymbol{k}^{\prime}, \boldsymbol{k}} c_{\boldsymbol{k}^{\prime}}^{\dagger} c_{\boldsymbol{k}}\left\langle\boldsymbol{k}^{\prime}\left|C_{\mathrm{d}} \delta \mathcal{V}\right| \boldsymbol{k}\right\rangle=\quad i C_{\mathrm{d}} \sum_{\boldsymbol{k}^{\prime}, \boldsymbol{k}} c_{\boldsymbol{k}^{\prime}}^{\dagger} c_{\boldsymbol{k}} \sum_{\boldsymbol{q}}|\boldsymbol{q}| \sqrt{\frac{\hbar}{2 \rho \omega}}$$

## 物理代写|热力学代写thermodynamics代考|Polaronic Interaction of a Two-Level System with a Phonon Bath

$$H=H_{\mathrm{S}}+H_{\mathrm{SB}}+H_{\mathrm{B}}$$

$$H_{\mathrm{S}}=\frac{\hbar \omega_0}{2} \sigma_z+\frac{\hbar \Omega}{2}\left(\sigma_{+} e^{-i \omega_l t}+\sigma_{-} e^{i \omega_l t}\right), H_{\mathrm{SB}} \quad=\sigma_z \otimes \hbar \sum_k\left(g_k a_k^{\dagger}+g_k^* a_k\right)+\sigma_x \otimes \hbar \sum_k$$

$$\mathcal{T}=\sigma_Z \otimes \sum_k\left(\alpha_k a_k^{\dagger}-\alpha_k^* a_k\right), \quad \alpha_k=\frac{g_k}{\omega_k} .$$

$$\widetilde{H}=e^{\mathcal{T}} H e^{-\mathcal{T}}=\tilde{H} \mathrm{~S}+\widetilde{H} \mathrm{SB}+\widetilde{H} \mathrm{~B}$$

\begin{aligned} & \widetilde{H} \mathrm{~S}=\frac{\hbar \omega_0}{2} \sigma_z+\frac{\hbar \Omega_{\mathrm{r}}}{2}\left(\sigma_{+} e^{-i \omega_l t}+\sigma_{-} e^{i \omega_l t}\right) \widetilde{H} \mathrm{SB}=\frac{\hbar \Omega}{2}\left[e^{-i \omega_l t} \sigma+\otimes\left(A_{+}-A\right)+e^{i \omega_l t} \sigma_{-} \otimes(A\right. \ & +\left(\sigma_{+} \otimes A_{+}+\sigma_{-} \otimes A_{-}\right) \otimes \hbar \sum_k\left(\eta_k b_k^{\dagger}+\eta_k^* b_k\right), \tilde{H}_{\mathrm{B}}=\hbar \sum_k \omega_k a_k^{\dagger} a_k+\hbar \sum_k \tilde{\omega}_k b_k^{\dagger} b_k \end{aligned}

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|热力学代写thermodynamics代考|NEM2201

statistics-lab™ 为您的留学生涯保驾护航 在代写热力学thermodynamics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。我们的专家在代写热力学thermodynamics代写方面经验极为丰富，各种代写热力学thermodynamics相关的作业也就用不着说。

• 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

Here we consider transitions between two electronic states of an atom, $|e\rangle$ and $|g\rangle$, resulting in the emission or absorption of one photon, via the interaction (4.1). The matrix element for single-photon emission that corresponds to the term linear in $\boldsymbol{A}$ in (4.1), expanded as per (3.5), is given by
\begin{aligned} \left\langle g, n_\lambda(\boldsymbol{k})+1\left|H_{\mathrm{I}}\right| e, n_\lambda(\boldsymbol{k})\right\rangle= & -\frac{e}{m}\left(\frac{2 \pi \hbar}{\mathcal{V} \omega_{\boldsymbol{k}}}\right)^{1 / 2}\left[n_\lambda(\boldsymbol{k})+1\right]^{1 / 2} \ & \times\left\langle g\left|\epsilon_{\boldsymbol{k} \lambda}^* \cdot \sum_i e^{-i \boldsymbol{k} \cdot \boldsymbol{r}i} \boldsymbol{p}_i\right| e\right\rangle, \end{aligned} where $n\lambda(\boldsymbol{k})$ is the initial photon number in the mode $(\boldsymbol{k}, \lambda), \boldsymbol{k}$ being the wave vector and $\lambda$ the polarization, $\omega_{\boldsymbol{k}}$ is the photon frequency at a given $\boldsymbol{k}, \mathcal{V}$ is the quantization volume, $\boldsymbol{\epsilon}{\boldsymbol{k} \lambda}$ is a unit polarization vector of the photon, $e$ and $m$ are the electron charge and mass, whereas $\boldsymbol{r}_i$ and $\boldsymbol{p}_i$ are the position and momentum of the atomic electron $i$. The corresponding transition probability per unit time is then $$w\lambda d \Omega_{\mathrm{s}}=\frac{e^2 \omega_{\mathrm{a}} d \Omega_{\mathrm{s}}}{2 \pi m^2 \hbar c^3}\left[n_\lambda(\boldsymbol{k})+1\right]\left|\epsilon_{\boldsymbol{k} \lambda}^* \cdot\left\langle g\left|\sum_i e^{-i \boldsymbol{k} \cdot \boldsymbol{r}i} p_i\right| e\right\rangle\right|^2,$$ where $\omega{\mathrm{a}}=\left(E_e-E_g\right) / \hbar$ and $E_e$ and $E_g$ are the energies of the excited and ground atomic (electronic) states, $\Omega_{\mathrm{s}}$ being the solid angle of the emission. Equation (4.3) can be adapted to the absorption of a photon upon replacing the factor $\left[n_\lambda(\boldsymbol{k})+1\right]$ by $n_\lambda(\boldsymbol{k})$.

The electric dipole approximation is valid provided we can approximate the exponential factors in Eqs. (4.2) and (4.3) by unity: $$e^{-i k \cdot \boldsymbol{r}i} \approx 1 .$$ This holds if the wavelength $2 \pi / k$ of the photon is very large compared to the size $R$ of the atom, as in the case of optical atomic transitions. Then the wave functions of $|e\rangle$ and $|g\rangle$ restrict the values of $\boldsymbol{r}_i$ to $\left|\boldsymbol{k} \cdot \boldsymbol{r}_i\right| \lesssim k R \ll 1$. The equation of motion $$i \hbar \dot{\boldsymbol{r}}_i=\left[\boldsymbol{r}_i, H{\mathrm{a}}\right]$$
where $H_{\mathrm{a}}$ is the atomic Hamiltonian, yields
$$\left\langle g\left|\boldsymbol{p}i\right| e\right\rangle=m\left\langle g\left|\dot{\boldsymbol{r}}_i\right| e\right\rangle=-i m \omega{\mathrm{a}}\left\langle g\left|\boldsymbol{r}i\right| e\right\rangle$$ The electric dipole operator $\boldsymbol{d}=e \sum_i \boldsymbol{r}_i$ in this two-state basis may be represented by $$\boldsymbol{d}=\sum{j, l=e, g}|j\rangle\langle j|\boldsymbol{d}| l\rangle\langle l|=\sum_{j, l=e, g} \boldsymbol{\wp}{j l} \sigma{j l}$$
where $\wp_{j l}=\langle j|d| l\rangle$ is the electric-dipole transition matrix element. The transition operators $\sigma_{j l}=|j\rangle\langle l|$ form the set
\begin{aligned} \sigma_z & =|e\rangle\langle e|-| g\rangle\langle g|, \ \sigma_{+} & =|e\rangle\langle g|, \ \sigma_{-} & =|g\rangle\langle e|, \end{aligned}
where $\sigma_{+}, \sigma_{-}$, and $\sigma_z$ satisfy the spin-1/2 algebra of the Pauli matrices, that is,
$$\begin{gathered} {\left[\sigma_{-}, \sigma_{+}\right]=-\sigma_z,} \ {\left[\sigma_{-}, \sigma_z\right]=2 \sigma_{-} .} \end{gathered}$$

## 物理代写|热力学代写thermodynamics代考|Polaronic System–Bath Interactions

We consider the basic opto-mechanical Hamiltonian that governs an optical cavity mode (denoted by O) that is coupled to a photonic bath and to a mechanical oscillator (denoted by M). The total Hamiltonian then has the form
\begin{aligned} H_{\text {Tot }} & =H_{\mathrm{O}+\mathrm{M}}+\left(O^{\dagger}+O\right) \otimes B ; \ H_{\mathrm{O}+\mathrm{M}} & =\omega_{\mathrm{O}} O^{\dagger} O+\Omega_{\mathrm{M}} M^{\dagger} M+g O^{\dagger} O\left(M+M^{\dagger}\right) . \end{aligned}
Here $O^{\dagger}, O$ and $M^{\dagger}, M$ are the creation and annihilation operators of the cavity mode and the oscillator, respectively; $\omega_{\mathrm{O}}, \Omega_{\mathrm{M}}$ and $g$ are their respective frequencies and coupling rate; and $B$ is the photonic-bath operator (Fig. 4.1).

We transform these operators to the basis of hybridized optical-mechanical modes that diagonalize $H_{\mathrm{O}+\mathrm{M}}$ without changing their frequency. Namely,
\begin{aligned} H_{\mathrm{O}+\mathrm{M}} & =\widetilde{H}{\mathrm{O}}+\widetilde{H}{\mathrm{M}}, \quad \widetilde{H}{\mathrm{O}}=\omega{\mathrm{O}} \tilde{O}^{\dagger} \tilde{O}-\left(g \widetilde{O}^{\dagger} \widetilde{O}\right)^2 \frac{1}{\Omega_{\mathrm{M}}}, \quad \widetilde{H}{\mathrm{M}}=\Omega{\mathrm{M}} \tilde{M}^{\dagger} \tilde{M}, \ \tilde{M} & =M+\frac{g}{\Omega_{\mathrm{M}}} O^{\dagger} O, \quad \widetilde{O}=O e^{\frac{g}{\Omega_{\mathrm{M}}^{\mathrm{M}}}\left(M^{\dagger}-M\right)} . \end{aligned}
The new variables can be expressed in terms of the unitary (“polaron”) transformation $$U=e^{\frac{g}{\frac{g}{2 \mathrm{M}}\left(M^{+}-M\right) O^{\dagger} O}} .$$
as $\tilde{O}=U^{\dagger} O U$ and $\tilde{M}=U^{\dagger} M U$. Then, the interaction between the optical mode and the photonic bath is found to indirectly affect the mechanical oscillator.

We shall restrict ourselves to low excitations of the transformed number operators $\hat{n}{\tilde{O}}=\widetilde{O}^{\dagger} \tilde{O}$ and $\hat{n}{\tilde{M}}=\tilde{M}^{\dagger} \tilde{M}$ and to the weak optomechanical-coupling regime. Namely, we shall assume
$$\left(\frac{g}{\Omega_{\mathrm{M}}}\right)^2\left\langle n_{\tilde{M}}\right\rangle \ll 1, \quad \frac{g^2}{\Omega_{\mathrm{M}}}\left\langle n_{\tilde{O}}\right\rangle^2 t \ll 1,$$
where $\left\langle n_{\tilde{M}}\right\rangle$ and $\langle n \tilde{O}\rangle$ are the mean numbers of quanta in the $\tilde{M}$ and $\widetilde{O}$ degrees of freedom, respectively.

# 热力学代写

$$\left\langle g, n_\lambda(\boldsymbol{k})+1\left|H_{\mathrm{I}}\right| e, n_\lambda(\boldsymbol{k})\right\rangle=-\frac{e}{m}\left(\frac{2 \pi \hbar}{\mathcal{V} \omega_{\boldsymbol{k}}}\right)^{1 / 2}\left[n_\lambda(\boldsymbol{k})+1\right]^{1 / 2} \times\left\langle g\left|\epsilon_{\boldsymbol{k} \lambda}^* \cdot \sum_i e^{-i \boldsymbol{k} \cdot \boldsymbol{r i}} \boldsymbol{p}i\right| e\right\rangle$$ 在哪里 $n \lambda(\boldsymbol{k})$ 是模式中的初始光子数 $(\boldsymbol{k}, \lambda), \boldsymbol{k}$ 是波矢量和 $\lambda$ 极化， $\omega_k$ 是给定的光子频率 $\boldsymbol{k}, \mathcal{V}$ 是量化体 积， $\boldsymbol{\epsilon k} \lambda$ 是光子的单位偏振矢量， $e$ 和 $m$ 是电子电荷和质量，而 $\boldsymbol{r}_i$ 和 $\boldsymbol{p}_i$ 是原子电子的位置和动量 $i$. 对应的 单位时间转移概率为 $$w \lambda d \Omega{\mathrm{s}}=\frac{e^2 \omega_{\mathrm{a}} d \Omega_{\mathrm{s}}}{2 \pi m^2 \hbar c^3}\left[n_\lambda(\boldsymbol{k})+1\right]\left|\epsilon_{\boldsymbol{k} \lambda}^* \cdot\left\langle g\left|\sum_i e^{-i \boldsymbol{k} \cdot \boldsymbol{r} i} p_i\right| e\right\rangle\right|^2$$

$$e^{-i k \cdot r i} \approx 1$$

$$\langle g|\boldsymbol{p} i| e\rangle=m\left\langle g\left|\dot{\boldsymbol{r}}i\right| e\right\rangle=-i m \omega \mathrm{a}\langle g|\boldsymbol{r} i| e\rangle$$ 电偶极算子 $\boldsymbol{d}=e \sum_i \boldsymbol{r}_i$ 在这个两国基础上可以表示为 $$\boldsymbol{d}=\sum j, l=e, g|j\rangle\langle j|\boldsymbol{d}| l\rangle\langle l|=\sum{j, l=e, g} \wp j l \sigma j l$$

$$\sigma_z=|e\rangle\langle e|-| g\rangle\left\langle g\left|, \sigma_{+}=\right| e\right\rangle\left\langle g\left|, \sigma_{-}=\right| g\right\rangle\langle e|,$$

$$\left[\sigma_{-}, \sigma_{+}\right]=-\sigma_z,\left[\sigma_{-}, \sigma_z\right]=2 \sigma_{-}$$

## 物理代写|热力学代写thermodynamics代考|Polaronic System–Bath Interactions

$$H_{\mathrm{Tot}}=H_{\mathrm{O}+\mathrm{M}}+\left(O^{\dagger}+O\right) \otimes B ; H_{\mathrm{O}+\mathrm{M}}=\omega_{\mathrm{O}} O^{\dagger} O+\Omega_{\mathrm{M}} M^{\dagger} M+g O^{\dagger} O\left(M+M^{\dagger}\right) \text {. }$$

$$H_{\mathrm{O}+\mathrm{M}}=\widetilde{H} \mathrm{O}+\widetilde{H} \mathrm{M}, \quad \widetilde{H} \mathrm{O}=\omega \mathrm{O} \tilde{O}^{\dagger} \tilde{O}-\left(g \widetilde{O}^{\dagger} \widetilde{O}\right)^2 \frac{1}{\Omega_{\mathrm{M}}}, \quad \widetilde{H} \mathrm{M}=\Omega \mathrm{M} \tilde{M}^{\dagger} \tilde{M}, \tilde{M} \quad=M$$

$$U=e^{\frac{g}{\frac{g}{2 M}\left(M^{+}-M\right) o \dagger o}} .$$

$$\left(\frac{g}{\Omega_{\mathrm{M}}}\right)^2\left\langle n_{\tilde{M}}\right\rangle \ll 1, \quad \frac{g^2}{\Omega_{\mathrm{M}}}\left\langle n_{\tilde{O}}\right\rangle^2 t \ll 1$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|热力学代写thermodynamics代考|PHYS2712

statistics-lab™ 为您的留学生涯保驾护航 在代写热力学thermodynamics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。我们的专家在代写热力学thermodynamics代写方面经验极为丰富，各种代写热力学thermodynamics相关的作业也就用不着说。

• 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|热力学代写thermodynamics代考|States and Dynamics

A quantum mechanical state of the system is given by a density operator $\rho(t)$, whose evolution,
$$\rho(t)=U_t \rho(0) U_t^{\dagger},$$
is governed by the unitary propagator
$$U_t:=\exp (-i H t / \hbar)=\sum_n \exp \left(-i E_n t / \hbar\right)|n\rangle\langle n| \text {. }$$
Equations (1.12) and (1.13) yicld, for an arbitrary initial state $\rho(0)$,
$$\rho(t)-\sum_{m, n} \rho_{m n}(0) e^{-i\left(E_m-E_n\right) t / \hbar}|m\rangle\langle n|,$$
where $\sum_{m, n}$ is a summation over all $m, n=0,1,2, \ldots, \rho_{m n}(t):=\langle m|\rho(t)| n\rangle$ being the matrix elements of $\rho(t)$.

The ensemble-averaged occupation probability $p_{E_n}$ of an eigenvalue $E_n$ is given by the expectation value of the projector (1.3) onto the corresponding eigenspace,
$$p_{E_n}:=\operatorname{Tr}\left[P_{E_n} \rho(t)\right]=\sum_{E_m=E_n} \rho_{m m}(t)=\sum_{E_m=E_n} p_m,$$

where the level population $p_n$ is the time-independent expectation value of the observable $|n\rangle\langle n|$,
$$p_n:=\operatorname{Tr}[|n\rangle\langle n| \rho(t)]=\rho_{n n}(t)=\rho_{n n}(0),$$
normalized by
$$1=\operatorname{Tr} \rho(t)=\sum_n \rho_{n n}(t)=\sum_n p_n=\sum_{E_n} p_{E_n} .$$
In what follows, we shall employ the energy basis in which all the non-diagonal elements of $P_{E_n} \rho(0) P_{E_n}$ vanish,
$$\rho_{m n}(0)=0 \quad \text { if } \quad m \neq n \quad \text { and } \quad E_m=E_n .$$

## 物理代写|热力学代写thermodynamics代考|The Problem of Equilibration for Physical Observables

The statistical ensemble $\rho(t)$ is not stationary at short $t$ if $\rho(0)$ is out of equilibrium. Yet, if the right-hand side of (1.14) initially depends on $t$, it cannot approach at large $t$ any time-independent “equilibrium ensemble.” Furthermore, any mixed state $\rho(t)$ returns arbitrarily “near” its initial state $\rho(0)$ at certain times $t$ (analogously, but not identically, to pure-state Poincaré recurrences). In what follows, we examine the apparent contradiction of such recurrences with equilibration.
According to (1.14), there exists at least one $\rho_{m n}(0) \neq 0$ with
$$\omega:=\left(E_n-E_m\right) / \hbar \neq 0 .$$
We consider observables represented by Hermitian operators
$$X=\sum_{m, n} X_{m n}|m\rangle\langle n|, \quad X_{m n}:=\langle m|X| n\rangle,$$
with expectation values
$$\langle X\rangle(t):=\operatorname{Tr}[\rho(t) X]$$
For the observable that represents an interlevel transition,
$$X=\hat{X}+\hat{X}^{\dagger}, \quad \hat{X}:=|m\rangle\langle n| / \rho_{m n}(0)$$
we find from (1.14) that
$$\operatorname{Tr}[\rho(t) X]=2 \cos (\omega t)$$
Thus, the mean value of $X$ in the ensemble $\rho(t)$ exhibits permanent oscillations, allowing us to conclude that quantum mechanics and equilibration are, in general,

incompatible. Nevertheless, as shown below, equilibration can approximately hold true for a restricted class of observables $X$ and initial conditions $\rho(0)$.

A measurement of an observable $X$ may be assumed to yield a finite range of possible outcomes,
$$\Delta_X:=\max {\mathcal{H}}\langle\psi|X| \psi\rangle-\min {\mathcal{H}}\langle\psi|X| \psi\rangle=x_{\max }-x_{\min }$$
where the maximization and minimization are over all normalized vectors in the pertinent Hilbert space $\mathcal{H},|\psi\rangle \in \mathcal{H}$, so that $x_{\max }$ and $x_{\min }$ are the largest and smallest eigenvalues of $X$, respectively.

# 热力学代写

## 物理代写|热力学代写thermodynamics代考|States and Dynamics

$$\rho(t)=U_t \rho(0) U_t^{\dagger}$$

$$U_t:=\exp (-i H t / \hbar)=\sum_n \exp \left(-i E_n t / \hbar\right)|n\rangle\langle n|$$

$$\rho(t)-\sum_{m, n} \rho_{m n}(0) e^{-i\left(E_m-E_n\right) t / \hbar}|m\rangle\langle n|$$

$$p_{E_n}:=\operatorname{Tr}\left[P_{E_n} \rho(t)\right]=\sum_{E_m=E_n} \rho_{m m}(t)=\sum_{E_m=E_n} p_m$$

$$p_n:=\operatorname{Tr}[|n\rangle\langle n| \rho(t)]=\rho_{n n}(t)=\rho_{n n}(0)$$
$$1=\operatorname{Tr} \rho(t)=\sum_n \rho_{n n}(t)=\sum_n p_n=\sum_{E_n} p_{E_n}$$

$$\rho_{m n}(0)=0 \quad \text { if } \quad m \neq n \quad \text { and } \quad E_m=E_n .$$

## 物理代写|热力学代写thermodynamics代考|The Problem of Equilibration for Physical Observables

$$\omega:=\left(E_n-E_m\right) / \hbar \neq 0$$

$$X=\sum_{m, n} X_{m n}|m\rangle\langle n|, \quad X_{m n}:=\langle m|X| n\rangle,$$

$$\langle X\rangle(t):=\operatorname{Tr}[\rho(t) X]$$

$$X=\hat{X}+\hat{X}^{\dagger}, \quad \hat{X}:=|m\rangle\langle n| / \rho_{m n}(0)$$

$$\operatorname{Tr}[\rho(t) X]=2 \cos (\omega t)$$

$$\Delta_X:=\max \mathcal{H}\langle\psi|X| \psi\rangle-\min \mathcal{H}\langle\psi|X| \psi\rangle=x_{\max }-x_{\min }$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|热力学代写thermodynamics代考|MECH3024

statistics-lab™ 为您的留学生涯保驾护航 在代写热力学thermodynamics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。我们的专家在代写热力学thermodynamics代写方面经验极为丰富，各种代写热力学thermodynamics相关的作业也就用不着说。

• 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|热力学代写thermodynamics代考|From Quantum Dynamics to Thermodynamics

In an isolated system with a large but countable number of degrees of freedom (DOF), $1 \ll f<\infty$, governed by an autonomous Hamiltonian $H$, the spectrum is discrete (quantized). Its (typically infinitely many) eigenstates $|n\rangle(n=0,1, \ldots)$ possess eigenvalues $E_n$, ordered as
$$E_0 \leq E_1 \leq E_2 \leq \ldots,$$
with a bounded ground state energy, $E_0>-\infty$. The Hamiltonian can then be written as
$$H:=\sum_n E_n|n\rangle\langle n| .$$
In the presence of energy degeneracy, we may use the projectors onto subspaces of degenerate energies $E_m=E_n$,
$$P_{E_n}:=\sum_{E_m=E_n}|m\rangle\langle m|,$$
to rewrite the Hamiltonian (1.2) as
$$H=\sum_{E_n} E_n P_{E_n},$$
where $\sum_{E_n}$ is a summation over all mutually different $E_n$ values.

## 物理代写|热力学代写thermodynamics代考|Thermodynamic Variables

The number of energy levels below any given $E$ is
$$N(E):=\sum_n \theta\left(E-E_n\right)$$
the Heaviside (step) function $\theta(x)$ being equal to 1 for $x>0$ and 0 for $x \leq 0$. The entropy associated with this number of levels is defined as
$$\mathcal{S}(E):=k_{\mathrm{B}} \ln N(E)$$
where $k_{\mathrm{B}}$ is Boltzmann’s constant. Commonly, this entropy is an extensive quantity, since it scales for a system with $f$ DOF as
$$\mathcal{S}(E) / k_{\mathrm{B}}=O(f)$$
Equation (1.5) implies that for macroscopic $f=O\left(10^{23}\right)$, the level density is staggering even on extremely small energy scales. Hence, the step function $\theta(x)$ in (1.5) may be assumed to be washed out. The level number $N(E)$ then becomes a smooth function of $E$, whose well-defined derivative represents the density of states
$$\Omega(E)=\sum_n \delta\left(E-E_n\right)$$ the delta-function $\delta(x)=\theta^{\prime}(x)$ being also assumed to be washed out over many energy levels.

The coarse-grained entropy defined by (1.6) leads to the definition of temperature, which applies whether the system is at equilibrium or not:
$$T(E):=1 / S^{\prime}(E)$$
In accordance with Nernst’s third law of thermodynamics, the entropy and temperature converge to zero as the energy approaches the ground-state value, $E \rightarrow E_0$. For macroscopic values of $E-E_0$, the dependence of $S$ on $E$ is logarithmic. It then follows from (1.7) and (1.9) that
$$k_{\mathrm{B}} T(E)=O\left(\frac{E-E_0}{f}\right),$$
so that, for any macroscopic energy change $\Delta E$,
$$T(E+\Delta E)=T(E)\left[1+O\left(\frac{\Delta E}{E-E_0}\right)\right]$$
All these relations may fail at extremely low temperatures, which are beyond our consideration here (but cf. references in this chapter).

# 热力学代写

## 物理代写|热力学代写thermodynamics代考|From Quantum Dynamics to Thermodynamics

$$H:=\sum_n E_n|n\rangle\langle n| .$$

$$P_{E_n}:=\sum_{E_m=E_n}|m\rangle\langle m|,$$

$$H=\sum_{E_n} E_n P_{E_n}$$

## 物理代写|热力学代写thermodynamics代考|Thermodynamic Variables

$$N(E):=\sum_n \theta\left(E-E_n\right)$$

$$\mathcal{S}(E):=k_{\mathrm{B}} \ln N(E)$$

$$\mathcal{S}(E) / k_{\mathrm{B}}=O(f)$$

$$\Omega(E)=\sum_n \delta\left(E-E_n\right)$$

$$T(E):=1 / S^{\prime}(E)$$

$$k_{\mathrm{B}} T(E)=O\left(\frac{E-E_0}{f}\right)$$

$$T(E+\Delta E)=T(E)\left[1+O\left(\frac{\Delta E}{E-E_0}\right)\right]$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|热力学代写thermodynamics代考|NEM2201

statistics-lab™ 为您的留学生涯保驾护航 在代写热力学thermodynamics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。我们的专家在代写热力学thermodynamics代写方面经验极为丰富，各种代写热力学thermodynamics相关的作业也就用不着说。

• 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|热力学代写thermodynamics代考|Temperature in K and C

Historically prior to the choice of the triple-point temperatures, the reference of ice water and steam water at 1 atmosphere pressure was used to determine an empirical temperature scale known as the Celsius scale. The temperature difference between ice and steam was chosen to be 100 , that is, $T_s-T_i=100$. Here, $T_S$ and $T_i$ denote the temperature of the reference steam water and ice water mixture. Thus, for a gas thermometer,
$$\frac{T_s}{T_i}=\frac{100+T_i}{T_i}=\lim {n_i \rightarrow 0} \frac{p_s}{p_i}$$ and solving for $T_i$ gives $$T_i=\frac{100}{\frac{p_s}{p_i}-1}$$ Accurate measurement of $\frac{p_s}{p_i}$ gives its value to be $1.3661$. Thus, $T_i=273.15 \mathrm{~K}$. The temperature of the freezing point of water, which in degree Celsius is $0^{\circ} \mathrm{C}$, is $273.15$ in the Kelvin scale. Hence from Eq. 1.6, we write $$T=273.15 \lim {n_0 \rightarrow 0}\left(\frac{p}{p_i}\right)(\mathrm{K})$$
We therefore shift the scale by $273.15$ to convert ${ }^{\circ} \mathrm{C}$ to $\mathrm{K}$, that is, $T(\mathrm{~K})=\theta^{\circ}(\mathrm{C})+273.15$.

There are various ways or paths by which the equilibrium state of a system could be changed. A process refers to a particular path causing the change. The system may not be in equilibrium at the different instants during the change; hence, the intermediate non-equilibrium states cannot be defined and the path or process cannot be specified. An important property of energy transfer process in thermodynamics is that it has to be quasi-static.

A quasi-static process is one in which the change of state is effected very slowly so that the state of the system, as well as the environment in which the system interacts with, is arbitrarily close to equilibrium at all times during the process. A process therefore goes through a series of equilibrium states. It may be noted that the equilibrium state corresponds to that of an isolated system when a definite invariant state is reached.
Real processes are not quasi-static since changes occur at finite rates. However, if the time scale of the change is long compared to the relaxation time of the system to equilibrate when perturbed, then the real processes can be approximated as being quasi-static. The state of the system and the environment must also be arbitrarily close to each other since finite gradients in the thermodynamic state will result in finite acceleration and non-uniformities in the system and the environment.

Equilibrium thermodynamics does not involve time. When time appears, it is to be understood that the rate is infinitesimally slow for the process to be quasi-static.

## 物理代写|热力学代写thermodynamics代考|Reversible Process

Reversible processes are of importance in thermodynamics. A reversible process is one in which both the system and the environment with which it interacts with are returned to their original states when the direction of the process is reversed. The system follows the same sequence of equilibrium states in the reverse direction.
A reversible process must necessarily be quasi-static, but a quasi-static process may not be reversible, for example, when a dissipative process like friction is present.
It should be noted that a system can always be made to return to its initial state, but a reversible path also requires that the environment is also returned to its original condition. Internal irreversibility is associated with irreversible processes that occur within the system, for example, diffusion of mass and heat when the system approaches overall equilibrium. External irreversibility is associated with the interaction between the system and the environment, for example, heat exchange across a finite temperature difference between the system and the environment. Chemical reactions among the various chemical species within the system also give rise to internal irreversibility. Irreversibility also results when dissipation processes are involved.
Reversible processes seem to be highly restrictive, but they are very important to provide a reference to assess real processes. Again the heat transfer process can be quasi-static when carried out infinitesimally slowly, and further for heat transfer to be reversible, the temperature difference across the boundary must be vanishingly small. Heat transfer across a finite temperature difference is irreversible and in fact constitutes the second law of thermodynamics.

# 热力学代写

## 物理代写|热力学代写thermodynamics代考|Temperature in K and C

$$\frac{T_s}{T_i}=\frac{100+T_i}{T_i}=\lim n_i \rightarrow 0 \frac{p_s}{p_i}$$

$$T_i=\frac{100}{\frac{p_s}{p_i}-1}$$

$$T=273.15 \lim n_0 \rightarrow 0\left(\frac{p}{p_i}\right)(\mathrm{K})$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|热力学代写thermodynamics代考|MECH3720

statistics-lab™ 为您的留学生涯保驾护航 在代写热力学thermodynamics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。我们的专家在代写热力学thermodynamics代写方面经验极为丰富，各种代写热力学thermodynamics相关的作业也就用不着说。

• 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|热力学代写thermodynamics代考|Empirical Temperature

If $X$ is the value of the thermometric property that changes with temperature parameter, for example, height of the mercury column in the capillary tube or pressure in a constant volume gas thermometer, then the ratio of the thermometric property $X$ can be used to define the ratio of the temperature ” $\theta$ “. The temperature obtained in this manner is an empirical temperature, and its value depends on the particular thermometer used. When the thermometer is brought into thermal contact with heat sources $A$ and $B$ and if the thermometer reads $X_A$ and $X_B$, respectively, we say that the empirical temperatures $\theta_A$ and $\theta_B$ of $A$ and $B$ are in the ratio
$$\frac{\theta_A}{\theta_B}=\frac{X_A}{X_B}$$
To obtain the empirical temperature scale, we need to assign a numerical value to some chosen heat source, for example, temperature of steam at atmospheric pressure. It is agreed upon that the triple point of water (equilibrium between ice, water and steam) be used as a standard heat source and assigned a particular value $\theta_{t p^*}$. Thus, we write Eq. $1.2$ as
$$\theta=\theta_{t p} \frac{X}{X_{t p}}$$
where $\theta$ is the temperature of the system to be measured, and $X$ is the value of the thermometric substance when the thermometer is in thermal equilibrium with the system. $\theta_{t p}$ and the corresponding $X_{t p}$ refer to the triple-point temperature and the value of $\mathrm{X}$ when the thermometer is at thermal equilibrium with a system of ice, water and steam at the triple point. From Eq. 1.2, we see that the ratio of the thermometric substance differs for different thermometers and the empirical temperature $\theta$ measured varies for different thermometers used.

## 物理代写|热力学代写thermodynamics代考|Absolute Temperature T

For a gas thermometer, where we use the pressure of a constant volume gas at low pressure to indicate the temperature, we write Eq. $1.3$ as
$$\theta=\theta_{t p}\left(\frac{p}{p_{t p}}\right)$$
It was found experimentally that $\frac{p}{p_{t p}}$ is independent of the type of gas used in the limit the amount of gas in the bulb (number of moles $n_0$ of it) approaching zero, that is, $\lim {n_0 \rightarrow 0}\left(\frac{p}{p{i p}}\right)$ does not depend on the properties of the thermometric fluid. The temperature so obtained is known as the absolute temperature and is given as
$$T=T_{t p} \lim {n_0 \rightarrow 0}\left(\frac{p}{p{t p}}\right)$$
where $T$ denotes the absolute temperature. It is measured in Kelvin (K).

Historically prior to the choice of the triple-point temperatures, the reference of ice water and steam water at 1 atmosphere pressure was used to determine an empirical temperature scale known as the Celsius scale. The temperature difference between ice and steam was chosen to be 100 , that is, $T_s-T_i=100$. Here, $T_S$ and $T_i$ denote the temperature of the reference steam water and ice water mixture. Thus, for a gas thermometer,
$$\frac{T_s}{T_i}=\frac{100+T_i}{T_i}=\lim {n_i \rightarrow 0} \frac{p_s}{p_i}$$ and solving for $T_i$ gives $$T_i=\frac{100}{\frac{p_s}{p_i}-1}$$ Accurate measurement of $\frac{p_s}{p_i}$ gives its value to be $1.3661$. Thus, $T_i=273.15 \mathrm{~K}$. The temperature of the freezing point of water, which in degree Celsius is $0^{\circ} \mathrm{C}$, is $273.15$ in the Kelvin scale. Hence from Eq. 1.6, we write $$T=273.15 \lim {n_0 \rightarrow 0}\left(\frac{p}{p_i}\right)(\mathrm{K})$$
We therefore shift the scale by $273.15$ to convert ${ }^{\circ} \mathrm{C}$ to $\mathrm{K}$, that is, $T(\mathrm{~K})=\theta^{\circ}(\mathrm{C})+273.15$.

# 热力学代写

## 物理代写|热力学代写thermodynamics代考|Empirical Temperature

$$\frac{\theta_A}{\theta_B}=\frac{X_A}{X_B}$$

$$\theta=\theta_{t p} \frac{X}{X_{t p}}$$

## 物理代写|热力学代写thermodynamics代考|Absolute Temperature T

$$\theta=\theta_{t p}\left(\frac{p}{p_{t p}}\right)$$

$$T=T_{t p} \lim n_0 \rightarrow 0\left(\frac{p}{p t p}\right)$$

$$\frac{T_s}{T_i}=\frac{100+T_i}{T_i}=\lim n_i \rightarrow 0 \frac{p_s}{p_i}$$

$$T_i=\frac{100}{\frac{p_s}{p_i}-1}$$

$$T=273.15 \lim n_0 \rightarrow 0\left(\frac{p}{p_i}\right)(\mathrm{K})$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|热力学代写thermodynamics代考|MEC302

statistics-lab™ 为您的留学生涯保驾护航 在代写热力学thermodynamics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。我们的专家在代写热力学thermodynamics代写方面经验极为丰富，各种代写热力学thermodynamics相关的作业也就用不着说。

• 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|热力学代写thermodynamics代考|MASS, MOLECULAR MASS AND MOLES IN A SYSTEM

The mass of a system is the number of molecules $N$ contained in the system multiplied by the mass of each of the molecules in it. Since the mass of a molecule is very small, it is measured in terms of the mass of a standard particle that is chosen to have a mass one-twelfth the mass of an isotope of carbon ${ }^{12}$. The mass of the standard particle $m_0$, known as the atomic mass unit (a.m.u.), is $1.661 \times 10^{-24} \mathrm{~g}$.

The mass of a molecule of a substance is therefore expressed in units of the standard atomic mass unit $m_0$, namely, mass of the molecule $m$ divided by $m_0$, that is, $M=\frac{m}{m_0} . M$ is called the molecular mass.

As an example, the molecular mass of a hydrogen molecule is given as $M_{H_2}=\frac{m_{H_2}}{m_0}$, where $m_{H_2}$ is the mass of the hydrogen molecule and $m_0$ is the mass of the standard particle. It is also spoken of as molecular weight since almost all experiments are carried out in the vicinity of the Earth’s surface where the gravitational constant is the same. We will use the words molecular mass and molecular weight without differentiating between them.

The number of molecules in a macroscopic system is, in general, very large, and we therefore measure it in the unit of mole. A mole is defined as the number of standard particles $N_0$ in $1 \mathrm{~g}$ of it, that is, $N_0=\frac{1}{m_0}=\frac{1}{1.661 \times 10^{-24}}=6.023 \times 10^{23}, N_0$ is called Avogadro number.
The number of the moles of a substance comprising of $N$ molecules is $n=\frac{N}{N_0}$. We can write the molecular mass as
$$M=\frac{m}{m_0}=\frac{m}{m_0} \frac{N_0}{N_0}=m N_0$$
since $m_0 N_0=1 \mathrm{~g}$.
The molecular mass $M$ therefore equals $m N_0$ in unit of grams and is the mass of 1 mole of the substance in grams. Thus, 1 mole of hydrogen has a mass equal to $2 \mathrm{~g}$, and 1 mole of nitrogen is $28 \mathrm{~g}$ and so on. Similarly, the number of moles $n$ of a substance of mass $m \mathrm{~g}$ is $\mathrm{m} / \mathrm{M}$.

## 物理代写|热力学代写thermodynamics代考|INTENSIVE VARIABLES DEFINING A SYSTEM

Energy $U$, volume $V$ and mass $m$ or equivalently the moles $n$, which define a system, are based on the extent of a system. The energy per unit mass $u=\frac{U}{m}$ and the specific volume $v=\frac{V}{m}$ are independent of the extent and are intensive variables. In the following, we define the intensive variables pressure and temperature for defining a simple system. These are independent of its extent and are the so-called intensive variables.

The pressure $p$ is the force per unit area and acts normal to the surface and is independent of the orientation of the surface. The unit of pressure is Newton per square meter (force per unit area) and is called as Pascal (Pa). A standard atmosphere, which is the atmospheric pressure at the standard sea level, is $1.01325 \times 10^5 \mathrm{~Pa}$. Pressure is also denoted in bars and 1 bar $=10^5 \mathrm{~Pa}$.

For a homogeneous system at equilibrium, the pressure is uniformly the same throughout the system. For a system in mechanical equilibrium with its environment, the pressure is the same across the system’s boundary.

Temperature is an intensive variable that has its origin in thermodynamics. It is a measure of the physiological sensation of “hot” and “cold”. The measurement of temperature is based on the fact that two systems brought into thermal contact will eventually reach the same state of “hotness”, that is, a state of thermal equilibrium and will have the same value of temperature. This is the “zeroth” law of thermodynamics that can be stated as follows: If system $\mathrm{A}$ is in thermal equilibrium with system B (such as when brought in contact with each other) and system B is in thermal equilibrium with system $\mathrm{C}$, then systems $\mathrm{A}$ and $\mathrm{C}$ are also in thermal equilibrium.
The zeroth law permits us to choose a test system called a thermometer to compare how “hot” the system of interest is and to determine its temperature.

The substances used in the thermometer should have a property that changes significantly with temperature and can be measured precisely. Most substances change their volume with temperature. Thus, the volume change can be calibrated to indicate the temperature change. A typical thermometric substance is a liquid (e.g., mercury, alcohol) contained in a small thin-walled glass bulb, which connects to a fine-bore capillary tube. The height of the liquid column in the capillary tube can then be calibrated to provide a scale to read the temperature. The property should also change linearly with temperature for easy measurements.

The change of the electrical resistance with temperature or the voltage from a thermocouple can also serve as a thermometer. The different measured parameters of the various thermometers provide the different temperature scales. Practical considerations require a thermometer to be sufficiently small so that it produces negligible effect on the system whose temperature is measured.

In a gas thermometer, a small volume of gas containing $n_0$ moles is enclosed in a bulb and either the volume change at constant pressure or pressure change at fixed volume can be used to measure the temperature changes. A constant volume gas thermometer is preferable since the pressure change with temperature can be accurately measured using a manometer. Use of gas at low pressures appears to provide a thermometric substance independent of the type of gas used.

# 热力学代写

## 物理代写|热力学代写thermodynamics代考|MASS, MOLECULAR MASS AND MOLES IN A SYSTEM

$$M=\frac{m}{m_0}=\frac{m}{m_0} \frac{N_0}{N_0}=m N_0$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|热力学代写thermodynamics代考|AMME2262

statistics-lab™ 为您的留学生涯保驾护航 在代写热力学thermodynamics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。我们的专家在代写热力学thermodynamics代写方面经验极为丰富，各种代写热力学thermodynamics相关的作业也就用不着说。

• 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|热力学代写thermodynamics代考|Grand Canonical Free Energy/Potential

For many applications, the system is subject to a constant temperature and a constant chemical potential. Therefore, we want to define a new thermodynamic potentialgrand canonical potential (or grand canonical free energy) that includes $T$ and $\mu$ in the variables.
Recall
\begin{aligned} &\mathrm{U}=\mathrm{U}\left(\mathrm{S}, \mathrm{V}, \mathrm{N}1, \ldots, \mathrm{N}{\mathrm{r}}\right)=\mathrm{TS}-\mathrm{PV}+\sum \mu_{\mathrm{i}} \mathrm{N}{\mathrm{i}} \ &\mathrm{dU}=\mathrm{TdS}-\mathrm{PdV}+\sum \mu{\mathrm{i}} \mathrm{dN}{\mathrm{i}} \end{aligned} Apply the partial Legendre transformation to the internal energy function $\mathrm{U}$, $$\psi=\mathrm{Y}-\sum{\mathrm{k}+1}^{\mathrm{n}} \mathrm{P}{\mathrm{i}} \mathrm{X}{\mathrm{i}} \quad \text { and } \quad \mathrm{P}{\mathrm{i}}=\frac{\partial \mathrm{Y}}{\partial \mathrm{X}{\mathrm{i}}}$$
the grand canonical free energy function is defined as the following:
\begin{aligned} \Omega &=\mathrm{U}-\mathrm{TS}-\sum \mu_{\mathrm{i}} \mathrm{N}{\mathrm{i}}=-\mathrm{PV} \ \mathrm{d} \Omega &=\mathrm{dU}-\mathrm{d}(\mathrm{TS})-\mathrm{d}\left(\sum \mu{\mathrm{i}} \mathrm{N}{\mathrm{i}}\right) \ &=-\mathrm{SdT}-\mathrm{PdV}-\sum \mathrm{N}{\mathrm{i}} \mathrm{d} \mu_{\mathrm{i}} \end{aligned}
As seen from the above equation, the grand canonical free energy is a function of temperature $\mathrm{T}$, volume $\mathrm{V}$ and chemical potentials $\mu_{\mathrm{i}}$,
$$\Omega=\Omega\left(\mathrm{T}, \mathrm{V},\left{\mu_{\mathrm{i}}\right}\right)$$
Comparing it with
$$\mathrm{U}=\mathrm{U}\left(\mathrm{S}, \mathrm{V},\left{\mathrm{N}{\mathrm{i}}\right}\right)$$ we see that $\mathrm{T}$ and $\mu{\mathrm{i}}$ in the $\Omega$ function replace $\mathrm{S}$ and $\mathrm{N}{\mathrm{i}}$ in the U function. For a process with constant $\mathrm{T}$ and $\mu{\mathrm{i}}$ (i.e., thermal and chemical equilibrium), the change in the grand canonical free energy is the mechanical work,
$$\mathrm{d} \Omega=-\mathrm{PdV}=\mathrm{dW} .$$

## 物理代写|热力学代写thermodynamics代考|Helmholtz Potential Minimum Principle

Consider a composite system consisting of two subsystems separated by a partition, as illustrated in the figure below. In subsystem A (left side), there is a dilute aqueous solution consisting of water and a salt (e.g., $\mathrm{NaC1}$ ). There is only water in subsystem $B$ (right side). However, the water presents in two phases (i.e., liquid and vapor) in subsystem B. Assume that the total volume and the total mass of the system are constant. The system is surrounded by a thermal reservoir.

The constraints for the combined system (the system and the reservoir) are:
\begin{aligned} &\mathrm{U}+\mathrm{U}^{\mathrm{R}}=\text { constant } \ &\mathrm{N}{\mathrm{i}}=\text { constant }, \quad i=1,2, \ldots r \ &\mathrm{~V}{\text {Total }}-\text { constant } \end{aligned}
The internal constraints are (1) the subsystems are separated by a rigid partition so that the volumes of the subsystems are constant. (2) The partition is semi-permeable, and only water molecules can pass through the partition.
\begin{aligned} &N_{A W}+N_{B W}^L+N_{B W}^V=\text { constant } \ &\mathrm{N}{\text {solute }}=\text { constant, } \ &V_A=\text { constant } \ &V_B=V{B L}+V_{B V}=\mathrm{constant} \end{aligned}
where the subscripts $\mathrm{A}$ and $\mathrm{B}$ stand for the subsystem $\mathrm{A}$ and subsystem $\mathrm{B}$; the subscript W stands for water; and the subscript or superscripts L and V stand for liquid and vapor, respectively.

We would like to find (1) what thermodynamic function should be used as the thermodynamic potential to model this system, and (2) the equilibrium conditions for this system.

# 热力学代写

## 物理代写|热力学代写thermodynamics代考|Grand Canonical Free Energy/Potential

\begin{aligned} &mathrm{U}=\mathrm{U}\left(\mathrm{S}, \mathrm{V}, \mathrm{N}1, \ldots, \mathrm{N}{mathrm{r}}\right)=\mathrm{TS}-\mathrm{PV}+\sum 缪{mathrm{i}}. \mathrm{N}{\mathrm{i}} \ &\mathrm{dU}=\mathrm{TdS}-\mathrm{PdV}+\sum \mu_{\mathrm{i}} \mathrm{dN}{\mathrm{i}} \end{aligned} 对内能函数$\mathrm{U}$ 应用部分 Legendre变换。 $$\psi=\mathrm{Y}-\sum{\mathrm{k}+1}^{\mathrm{n}} \mathrm{P}{\mathrm{i}} \mathrm{X}{\mathrm{i}} \夸张的文字 { 和}。\夸张的是，mathrm{P}{mathrm{i}}=frac{partial /mathrm{Y}}{partial /mathrm{X}{mathrm{i}}。$$

\begin{aligned} \Omega &=mathrm{U}-mathrm{TS}-sum mu_{mathrm{i}}. \mathrm{N}{\mathrm{i}}=-\mathrm{PV} \ \Omega \Omega &=mathrm{dU}-mathrm{d}(\mathrm{TS})-mathrm{d}\left(\sum \mu{mathrm{i}}{mathrm{N}{mathrm{i}}}right) \ &=-\mathrm{SdT}-\mathrm{PdV}-\sum \mathrm{N}{\mathrm{i}} \ǞǞǞǞ \mu{mathrm{i}}。 \end{aligned}

$$\Omega=\Omega\left(\mathrm{T}, \mathrm{V},\left{\mu_{\mathrm{i}}\right}\right)$$

$$\mathrm{U}=\mathrm{U}\left(\mathrm{S}, \mathrm{V},\left{\mathrm{N}{\mathrm{i}}\right}\right)$$ 我们看到$mathrm{T}$和$mu{mathrm{i}}$在$Omega$函数中取代$mathrm{S}$和$mathrm{N}{mathrm{i}}$在U函数中。对于一个具有恒定的$mathrm{T}$和$mu{mathrm{i}}$的过程（即热平衡和化学平衡），大典自由能的变化就是机械功。
$$\䗖䗖䗖 \Omega=-\mathrm{PdV}=\mathrm{dW} .$$

## 物理代写|热力学代写thermodynamics代考|Helmholtz Potential Minimum Principle

$$S=S_1+S_2=text {平衡时的最大值 }$$

$$\mathrm{dS}=\mathrm{d}\left(\mathrm{S}1+\mathrm{S}_2\right)=\mathrm{d} \mathrm{S}_1+\mathrm{d} \1+mathrm{S}_2=0。$$ 因为 $$\mathrm{dS}=\frac{1}{\mathrm{~T}} \mathrm{dU}+\frac{\mathrm{P}}{\mathrm{T}} \V-Sigma V-\Sigma \frac{\mu{\mathrm{k}}}{\mathrm{T}} \mathrm{dN}{\mathrm{k}}$$ 而从约束条件来看，我们有 $$\Ǟn}{mathrm{k}}=0, ǞdV}=0 。$$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。