# 物理代写|热力学代写thermodynamics代考|Averaged Interaction Energy

## 物理代写|热力学代写thermodynamics代考|Averaged Interaction Energy

Equation (12.10) expresses the score $\Delta P$ as an overlap between the gradient $\hat{P}$ and the bath-induced change of system state $\Delta \hat{\varrho}$. In what follows, we express $\Delta P$ in terms of physically pertinent quantities. To this end, we decompose the total Hamiltonian into system (S), bath (B), and interaction (I) parts,
$$\hat{H}(t)=\hat{H}{\mathrm{S}}(t)+\hat{H}{\mathrm{B}}+\hat{H}{\mathrm{I}}$$ and employ the Liouville-von Neumann equation of the total (system plus bath) state in the interaction picture, $$\frac{\partial}{\partial t} \hat{\varrho}{\mathrm{tot}}(t)=-\frac{i}{\hbar}\left[\hat{H}{\mathrm{I}}(t), \hat{\varrho}{\mathrm{tot}}(t)\right]$$
Here, the interaction Hamiltonian is transformed from the Schrödinger picture to the interaction picture by
$$\hat{H}{\mathrm{I}}(t)=\hat{U}{\mathrm{F}}^{\dagger}(t) \hat{H}{\mathrm{I}}^{(\mathrm{S})}(t) \hat{U}{\mathrm{F}}(t)$$
the free-evolution operator $\hat{U}{\mathrm{F}}(t)$ being given by the chronologically ordered expression $$\hat{U}{\mathrm{F}}(t)=\mathrm{T} \exp \left[-\frac{i}{\hbar} \int_0^t d \tau \hat{H}{\mathrm{S}}(\tau)-\frac{i}{\hbar} \hat{H}{\mathrm{B}} t\right]$$
The solution of (12.13) can be written as the Dyson (state) expansion
\begin{aligned} & \hat{\varrho}{\mathrm{tot}}(t)=\hat{\varrho}{\mathrm{tot}}(0)+\frac{-i}{\hbar} \int_0^t d t_1\left[\hat{H}{\mathrm{I}}\left(t_1\right), \hat{\varrho}{\mathrm{tot}}(0)\right] \ & +\left(\frac{-i}{\hbar}\right)^2 \int_0^t d t_1 \int_0^{t_1} d t_2\left[\hat{H}{\mathrm{I}}\left(t_1\right),\left[\hat{H}{\mathrm{I}}\left(t_2\right), \hat{\varrho}_{\mathrm{tot}}(0)\right]\right]+\ldots, \end{aligned}

## 物理代写|热力学代写thermodynamics代考|Spectral Overlap

Equation (12.24) may be rewritten in the form of an overlap of the system and the bath-response spectral matrices. This can be done by expanding the interaction Hamiltonian in a $d$-dimensional Hilbert space as a sum of products of the system and bath operators,
$$\hat{H}{\mathrm{I}}=\hbar \sum{j=1}^{d^2-1} \hat{S}j \otimes \hat{B}_j .$$ If the mean values vanish, $\left\langle\hat{B}_j\right\rangle=0$, then (12.21) is satisfied. Otherwise, we may include these mean values in the system Hamiltonian, $$\hat{B}_j \rightarrow \hat{B}_j^{\prime}=\hat{B}_j-\left\langle\hat{B}_j\right\rangle \hat{I}, \quad \hat{H}{\mathrm{S}}^{\prime}=\hat{H}{\mathrm{S}}+\sum_j\left\langle\hat{B}_j\right\rangle \hat{S}_j$$ We then transform (12.26) to the interaction picture and expand $$\hat{S}_j(t)=\sum_k \epsilon{j k}(t) \hat{S}_k$$ in the basis of operators $\hat{S}j$ that are Hermitian, traceless, and orthonormalized to $\operatorname{Tr}\left(\hat{S}_j \hat{S}_k\right)=d \delta{j k}$. This expansion defines a (real orthogonal) rotation matrix $\boldsymbol{\epsilon}(t)$ in the Hilbert space of the system, with elements
$$\epsilon_{j k}(t)=\left\langle\hat{S}j(t) \hat{S}_k\right\rangle{\mathrm{id}}$$
in the notation of (12.6). These matrix elements are the dynamical correlation functions of the system basis operators at infinite temperature. We similarly define the bath correlation (response) matrix $\boldsymbol{\Phi}(t)$ at a finite temperature with elements
$$\Phi_{j k}(t)=\left\langle\hat{B}j(t) \hat{B}_k\right\rangle{\mathrm{B}}$$
and a Hermitian matrix $\Xi$ with elements
$$\Xi_{k j}=\left\langle\left[\hat{S}j, \hat{P}\right] \hat{S}_k\right\rangle$$ where $\langle\ldots\rangle=\operatorname{Tr}[\hat{\varrho}(0) \ldots]$. This matrix represents the gradient operator $\hat{P}$ [cf. (12.11b)] in the basis of operators $\hat{S}_j$. Using (12.29) and (12.30), we evaluate the bath and (finite-time) system spectra to be \begin{aligned} \boldsymbol{G}(\omega) & =\int{-\infty}^{\infty} d t e^{i \omega t} \boldsymbol{\Phi}(t), \ \boldsymbol{\epsilon}_t(\omega) & =\frac{1}{\sqrt{2 \pi}} \int_0^t d \tau e^{i \omega \tau} \boldsymbol{\epsilon}(\tau) . \end{aligned}

# 热力学代考

## 物理代写|热力学代写thermodynamics代考|Averaged Interaction Energy

$$\hat{H}(t)=\hat{H} \mathrm{~S}(t)+\hat{H} \mathrm{~B}+\hat{H} \mathrm{I}$$

$$\frac{\partial}{\partial t} \hat{\varrho} \operatorname{tot}(t)=-\frac{i}{\hbar}[\hat{H} \mathrm{I}(t), \hat{\varrho} \operatorname{tot}(t)]$$

$$\hat{H} \mathrm{I}(t)=\hat{U} \mathrm{~F}^{\dagger}(t) \hat{H} \mathrm{I}^{(\mathrm{S})}(t) \hat{U} \mathrm{~F}(t)$$

$$\hat{U} \mathrm{~F}(t)=\mathrm{T} \exp \left[-\frac{i}{\hbar} \int_0^t d \tau \hat{H} \mathrm{~S}(\tau)-\frac{i}{\hbar} \hat{H} \mathrm{~B} t\right]$$
(12.13) 的解可以写成 Dyson (state) 展开式

## 物理代写|热力学代写thermodynamics代考|Spectral Overlap

$$\hat{H} \mathrm{I}=\hbar \sum j=1^{d^2-1} \hat{S} j \otimes \hat{B}j$$ 如果均值消失， $\left\langle\hat{B}_j\right\rangle=0$, 那么 (12.21) 是满足的。否则，我们可以将这些平均值包含 在系统哈密顿量中， $$\hat{B}_j \rightarrow \hat{B}_j^{\prime}=\hat{B}_j-\left\langle\hat{B}_j\right\rangle \hat{I}, \quad \hat{H} \mathrm{~S}^{\prime}=\hat{H} \mathrm{~S}+\sum_j\left\langle\hat{B}_j\right\rangle \hat{S}_j$$ 然后我们将 (12.26) 转换为交互图片并展开 $$\hat{S}_j(t)=\sum_k \epsilon j k(t) \hat{S}_k$$ 在运营商的基础上 $\hat{S} j$ 是 Hermitian 的、无迹的和正交归一化的 $\operatorname{Tr}\left(\hat{S}_j \hat{S}_k\right)=d \delta j k$. 此 扩展定义了一个 (实正交) 旋转矩阵 $\epsilon(t)$ 在系统的希尔伯特空间中，有元素 $$\epsilon{j k}(t)=\left\langle\hat{S} j(t) \hat{S}k\right\rangle \mathrm{id}$$ 在 (12.6) 的符号中。这些矩阵元素是无限温度下系统基算子的动态相关函数。我们类 似地定义 bath 相关 (响应) 矩阵 $\boldsymbol{\Phi}(t)$ 在具有元素的有限温度下 $$\Phi{j k}(t)=\left\langle\hat{B} j(t) \hat{B}k\right\rangle \mathrm{B}$$ 和厄米特矩阵 $\Xi$ 与元素 $$\Xi{k j}=\left\langle[\hat{S} j, \hat{P}] \hat{S}_k\right\rangle$$

$$\boldsymbol{G}(\omega)=\int-\infty^{\infty} d t e^{i \omega t} \boldsymbol{\Phi}(t), \boldsymbol{\epsilon}_t(\omega)=\frac{1}{\sqrt{2 \pi}} \int_0^t d \tau e^{i \omega \tau} \boldsymbol{\epsilon}(\tau)$$

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