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• Foundations of Data Science 数据科学基础

## 物理代写|热力学代写thermodynamics代考|Additivity of Entropy

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

## 物理代写|热力学代写thermodynamics代考|Additivity of Entropy

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

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

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

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

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

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

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