assignmentutor-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 revisit the entropy [cf. (1.6)] and temperature [cf. (1.9)] defined in Section $1.1$ for the case of Hamiltonian (2.3). For $\epsilon=0$, the difference with the definitions in Section $1.1$ is that the single index $n$ is now replaced by the double indices $\mathrm{nm}$ [see (2.14) and (2.15)].

For $\epsilon=0$ in (2.3), the indices ‘ $S$ ‘ and ‘ $B$ ‘ label the separate, isolated system and bath. The following relations hold between the quantities pertaining to the separate system and bath and those of the system-plus-bath compound (2.3) for $\epsilon=0$ :
$$\begin{gathered} E_{\mathrm{S}}(E):=E-E_{\mathrm{B}}(E) \ \mathcal{S}(E)=\mathcal{S}{\mathrm{B}}\left(E{\mathrm{B}}(E)\right)+\mathcal{S}{\mathrm{S}}\left(E{\mathrm{S}}(E)\right) \ T(E)=T_{\mathrm{B}}\left(E_{\mathrm{B}}(E)\right)=T_{\mathrm{S}}\left(E_{\mathrm{S}}(E)\right) \end{gathered}$$
These relations are asymptotically exact in the thermodynamic limit $f_{\mathrm{B}} \rightarrow \infty$.
In other words, in the limit of zero coupling the entropies of the system-plusbath complex are additive, provided the sum $\mathcal{S}{\mathrm{B}}\left(E^{\prime}\right)+\mathcal{S}{\mathrm{S}}\left(E-E^{\prime}\right)$ is maximized. The consequence is that all three temperatures in Eq. (2.23) are identical, that is, the equilibrium condition (or zeroth law of thermodynamics).

For low-dimensional systems it is appropriate to resort to the von Neumann (VN) entropy
$$\mathcal{S}{\mathrm{VN}}=-k{\mathrm{B}} \operatorname{Tr}(\rho \ln \rho),$$
which is an invariant of the density operator $\rho$. The $\mathrm{VN}$ entropy is nonnegative and bounded from below by the Shannon entropy
$$\mathcal{S}{\mathrm{SH}}=-k{\mathrm{B}} \sum_{n} p_{n} \ln p_{n} .$$
The VN entropy is maximized for a fixed mean energy $E=\operatorname{Tr}(\rho H)$ in the Gibbs state.

## 物理代写|热力学代写thermodynamics代考|Thermal Equilibrium and Correlation Functions

The description of quantum baths and their interactions with smaller quantum systems cannot be accomplished by detailed solutions of their Schrödinger or Liouville equations on account of their huge dimensionalities, up to infinity in the thermodynamic limit. The alternative in this limit is to resort to Kubo’s multi-time correlation (or Green) functions of the pertinent observables in the equilibrium state. In this book we shall only consider two-time correlations (autocorrelation functions) of the same observable, $\epsilon \tilde{H}{\mathrm{SB}} \sim \hat{S} \cdot \hat{B}$, where $\hat{S}$ and $\hat{B}$ are operators pertaining to the system and the bath, respectively. These functions are $$\Phi(t, 0)=\epsilon^{2} \operatorname{Tr}\left[\rho \tilde{H}{\mathrm{SB}}(t) \tilde{H}{\mathrm{SB}}(0)\right],$$ where $\tilde{H}{\mathrm{SB}}(t)$ is the interaction-picture form of $\tilde{H}_{\mathrm{SB}}$ and $\rho$ is the density matrix of the equilibrium state of system $\mathrm{S}$.

Hereafter, we shall simplify the notation to $\Phi(t)$. Assuming that the Hamiltonian spectrum is discrete, $\Phi(t)$ is a quasi-periodic function that after sufficient time returns arbitrarily close to its initial value, thus satisfying the quantum analog of classical Poincaré recurrences.

However, if the description of $\rho$ as a Gibbs state at temperature $T$ is valid, the functions $\Phi(t)$ can be extended by analytic continuation to the complex domain $(t \rightarrow z)$, where they satisfy the following Kubo-Martin-Schwinger (KMS) condition at any time,
$$\Phi(-t)=\Phi(t-i \hbar \beta),$$
with $\beta=1 /\left(k_{\mathrm{B}} T\right)$ being the inverse temperature. As shown in Chapter 4 , for typical bosonic or fermionic quantum baths, the KMS condition implies the decay of $\Phi(t)$ to zero as $t \rightarrow \infty$ (i.e., to the loss of recurrences and irreversibility). Explicitly, for a bosonic or fermionic bath,
$$\Phi(t) \propto \int_{-\infty}^{+\infty} d \omega e^{-i \omega t}{[1-(\mp) n(\omega)] \theta(\omega)+n(|\omega|) \theta(-\omega)}$$

# 热力学代写

$$E_{\mathrm{S}}(E):=E-E_{\mathrm{B}}(E) \mathcal{S}(E)=\mathcal{S B}(E \mathrm{~B}(E))+\mathcal{S S}(E \mathrm{~S}(E)) T(E)=T_{\mathrm{B}}\left(E_{\mathrm{B}}(E)\right)=T_{\mathrm{S}}\left(E_{\mathrm{S}}(E)\right)$$

$$\mathcal{S V N}=-k \mathrm{~B} \operatorname{Tr}(\rho \ln \rho),$$

$$\mathcal{S S H}=-k \mathrm{~B} \sum_{n} p_{n} \ln p_{n} .$$

## 物理代写|热力学代写thermodynamics代考|Thermal Equilibrium and Correlation Functions

$$\Phi(t, 0)=\epsilon^{2} \operatorname{Tr}[\rho \tilde{H} \operatorname{SB}(t) \tilde{H} \operatorname{SB}(0)],$$

$$\Phi(t) \propto \int_{-\infty}^{+\infty} d \omega e^{-i \omega t}[1-(\mp) n(\omega)] \theta(\omega)+n(|\omega|) \theta(-\omega)$$

## 有限元方法代写

assignmentutor™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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师