# Quantum Thermodynamics: A Dynamical Viewpoint

## Abstract

**:**

## 1. Introduction

- The 0-law of thermodynamics deals with the partition of the system from the bath.
- I-law: The first law of thermodynamics is a statement of conservation of energy.
- II-law: The second law is a statement on the irreversibility of dynamics: the breakup of time reversal symmetry. An empirical definition: heat will flow spontaneously from a hot source to a cold sink. These statements are translated to quantum definitions of positive entropy generation.
- III-law: We will analyze two formulations. The first is that the entropy of any pure substance in thermodynamic equilibrium approaches zero as the temperature approaches absolute zero. The second formulation is a dynamical one, known as the unattainability principle: it is impossible by any procedure, no matter how idealized, to reduce any assembly to absolute zero temperature in a finite number of operations.

## 2. Quantum Open Systems

## 3. The 0-Law

#### 3.1. System Bath Partition

#### 3.2. Thermal Equilibrium

## 4. The I-Law

#### 4.1. The Dynamical Generator in the Weak System-Bath Coupling Limit

#### 4.2. Thermal Generators for Periodic Driving Fields

- (1)
- The system’s renormalized Hamiltonian is now periodic in time:$${\widehat{\mathrm{H}}}_{S}\left(t\right)={\widehat{\mathrm{H}}}_{S}(t+\tau ),\phantom{\rule{4pt}{0ex}}\widehat{\mathrm{U}}(t,0)\equiv \mathcal{T}exp\left(\right)open="\{"\; close="\}">-\frac{i}{\hslash}{\int}_{0}^{t}{\widehat{\mathrm{H}}}_{S}\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds$$$${\widehat{\mathrm{H}}}_{eff}=\sum _{k}{\u03f5}_{k}|k\rangle \langle k|,\phantom{\rule{4pt}{0ex}}\widehat{\mathrm{U}}(\tau ,0)={e}^{-i\frac{1}{\hslash}{\widehat{\mathrm{H}}}_{eff}\tau}$$
- (2)
- The Fourier decomposition (23) is replaced by a double Fourier decomposition:$$\widehat{\mathrm{U}}{(t,0)}^{\u2020}\phantom{\rule{0.166667em}{0ex}}\widehat{\mathrm{S}}\phantom{\rule{0.166667em}{0ex}}\widehat{\mathrm{U}}(t,0)=\sum _{q\in \mathbf{Z}}\sum _{\left\{\omega \right\}}{e}^{i(\omega +q\Omega )t}{\widehat{\mathrm{S}}}_{\omega q}$$
- (3)
- The generator in the interaction picture is the sum of its Fourier components:$$\mathcal{L}=\sum _{q\in \mathbf{Z}}\sum _{\left\{\omega \right\}}={\mathcal{L}}_{\omega q}$$$${\mathcal{L}}_{\omega q}{\widehat{\rho}}_{S}=\frac{1}{2}\gamma (\omega +q\Omega )\left(\right)open="\{"\; close="\}">([{\widehat{\mathrm{S}}}_{\omega q},{\widehat{\rho}}_{S}{\widehat{\mathrm{S}}}_{\omega q}^{\u2020}]+[{\widehat{\mathrm{S}}}_{\omega q}{\widehat{\rho}}_{S},{\widehat{\mathrm{S}}}_{\omega q}^{\u2020}])+{e}^{-\hslash \beta (\omega +q\Omega )}([{\widehat{\mathrm{S}}}_{\omega q}^{\u2020},{\widehat{\rho}}_{S}{\widehat{\mathrm{S}}}_{\omega q}]+[{\widehat{\mathrm{S}}}_{\omega q}^{\u2020}{\widehat{\rho}}_{S},{\widehat{\mathrm{S}}}_{\omega q}])$$

#### 4.3. Heat Flows and Power for Periodically Driven Open Systems

## 5. The II-Law

#### 5.1. Entropy

#### 5.2. Quantum Networks and Quantum Devices

**Figure 1.**The tricycle on the left and the wire on the right; elementary components in a quantum network. The tricycle combines three energy currents. The tricycle in the figure is connected to three heat baths, demonstrating a heat-driven refrigerator. The wire combines two energy currents. The wire in the figure is connected to a hot and cold bath. The I-law and II-law are indicated.

- No process is possible whose sole result is the transfer of heat from a body of lower temperature to a body of higher temperature.

#### 5.3. Approach to Steady State: Limit Cycle

#### 5.4. The Quantum and Thermodynamic Adiabatic Conditions and Quantum Friction

## 6. The III-Law

- The entropy of any pure substance in thermodynamic equilibrium approaches zero as the temperature approaches zero.

#### 6.1. Harmonic Oscillator Cold Heat Bath

#### 6.2. The Existence of a Ground State

#### 6.3. Ideal Bose/Fermi Gas Cold Heat Bath

**Figure 3.**A demonstration of the III-law shown as the vanishing of the cooling current and the rate of temperature decrease as ${T}_{c}\to 0$. The harmonic bath in 3-d is indicated in blue, and the Bose gas in 3-d in red. The Bose gas cools faster when ${T}_{c}\to 0$, but its rate of temperature decrease is slower than the harmonic bath.

#### 6.4. Thermoelectric Refrigerators

## 7. Conclusions

## Acknowledgments

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Kosloff, R.
Quantum Thermodynamics: A Dynamical Viewpoint. *Entropy* **2013**, *15*, 2100-2128.
https://doi.org/10.3390/e15062100

**AMA Style**

Kosloff R.
Quantum Thermodynamics: A Dynamical Viewpoint. *Entropy*. 2013; 15(6):2100-2128.
https://doi.org/10.3390/e15062100

**Chicago/Turabian Style**

Kosloff, Ronnie.
2013. "Quantum Thermodynamics: A Dynamical Viewpoint" *Entropy* 15, no. 6: 2100-2128.
https://doi.org/10.3390/e15062100