# Quantum Thermodynamics at Strong Coupling: Operator Thermodynamic Functions and Relations

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## Abstract

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## 1. Quantum and Thermodynamics—Why?

#### 1.1. Quantum in “Gravity as Thermodynamics”

#### 1.2. David Bohm: Quantum in Classical Terms

#### 1.3. Quantum as Thermodynamics?

#### Quantum Thermodynamics

#### 1.4. This Work

## 2. Quantum Thermodynamics at Strong Coupling: Background

#### 2.1. New Challenges in Quantum Thermodynamics

#### 2.2. Goal and Findings of Present Work

## 3. Thermodynamic Functions, Hamiltonian of Mean Force

#### 3.1. Traditional (Weak-Coupling) Equilibrium Thermodynamic Relations

#### 3.2. Quantum System in a Heat Bath with Nonvanishing Coupling

#### 3.3. Hamiltonian of Mean Force

## 4. Quantum Formulation of Gelin and Thoss’ Thermodynamics at Strong Coupling

## 5. Quantum Formulation of Seifert’s Thermodynamics at Strong Coupling

## 6. Issues of These Two Approaches: Entropy and Internal Energy

#### 6.1. Entropy

- (1)
- (2)
- It has been shown [49] that if the composite is in a global thermal state, the discrete energy spectrum of the undamped oscillator will become a continuous one with a unique ground level. This supports physics described by the thermodynamic entropy ${\mathcal{S}}_{s}$.
- (3)
- It has been argued [44,45,46] that the entanglement between the system and the bath prevents the von Neumann entropy from approaching zero at zero temperature. Without quantum entanglement between the system and the bath, the lowest energy level of the composite system will be given by the tensor product of the ground state of the unperturbed system and bath, that is, a pure state. In this case, the von Neumann entropy will go to zero as expected, and this is the scenario that occurred in traditional quantum/classical thermodynamics in the vanishing system-bath coupling limit.

#### 6.2. Internal Energy

- It has been discussed [37,38,39,40] that the internal energy defined in Approach II can lead to anomalous behavior of the heat capacity in the low temperature limit. When the system, consisting of a quantum oscillator [40] or a free particle [37,38,39] is coupled to a heat bath modeled by a large number of quantum harmonic oscillators, the heat capacity of the system can become negative if the temperature of the bath is sufficiently low. If the internal energy defined in Approach I is used to compute the heat capacity, then it has been shown that the heat capacity remains positive for all nonzero temperatures but vanishes in the zero bath temperature limit, for a system with one harmonic oscillator [37], or a finite number of coupled harmonic oscillators [29]. This discrepancy may result from the fact that the internal energy defined in Approach II contains contributions from the interaction and the bath Hamiltonian.

## 7. Quantum Formulation of Jarzynski’s Strong Coupling Thermodynamics

#### 7.1. “Bare” Representation

#### 7.2. “Partial Molar” Representation

#### 7.3. Operator Forms of the Thermodynamic Functions

#### 7.3.1. Enthalpy and Energy Operators: Caution

#### 7.3.2. System Enthalpy Operator: Approved

## 8. Conclusions

#### 8.1. Summary

#### 8.2. Issues

#### 8.3. Further Developments

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Handling Operator Products in Quantum Thermodynamics

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**MDPI and ACS Style**

Hsiang, J.-T.; Hu, B.-L.
Quantum Thermodynamics at Strong Coupling: Operator Thermodynamic Functions and Relations. *Entropy* **2018**, *20*, 423.
https://doi.org/10.3390/e20060423

**AMA Style**

Hsiang J-T, Hu B-L.
Quantum Thermodynamics at Strong Coupling: Operator Thermodynamic Functions and Relations. *Entropy*. 2018; 20(6):423.
https://doi.org/10.3390/e20060423

**Chicago/Turabian Style**

Hsiang, Jen-Tsung, and Bei-Lok Hu.
2018. "Quantum Thermodynamics at Strong Coupling: Operator Thermodynamic Functions and Relations" *Entropy* 20, no. 6: 423.
https://doi.org/10.3390/e20060423