# Theory of Non-Equilibrium Heat Transport in Anharmonic Multiprobe Systems at High Temperatures

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## Abstract

**:**

## 1. Introduction

## 2. Dynamics

## 3. Physical Observables

#### 3.1. Entropy Generation Rate

#### 3.2. Heat Current

## 4. Constraints

## 5. Thermal Expansion

## 6. Force Constant Renormalization

## 7. Displacement-Noise Correlations

## 8. Displacement Autocorrelations

## 9. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

D | Device |

DC | Direct Current |

DFT | Density Functional Theory |

FC | Force Constant |

GF | Green’s Function |

MD | Molecular Dynamics |

NEMD | Non-Equilibrium Molecular Dynamics |

NEMF | Non-Equilibrium Mean-Field |

## Appendix A. Calculation of Time Averages

## Appendix B. Change of Position Variables Due to Thermal Expansion

#### Appendix B.1. Thermal Expansion

#### Appendix B.2. The New Force Constants

#### Appendix B.3. New Equations of Motion

## Appendix C. Explicit Form of the Correlation Functions including the Atomic and Cartesian Indices

#### Appendix C.1. Lead Self-Energies σ α and Escape Rates Γ α

#### Appendix C.2. Noise Autocorrelation Functions

#### Appendix C.3. Noise-Displacement Correlations Z α

**Figure A1.**Feynman diagrams associated with ${Z}_{\alpha}$ up to second order in anharmonic forces. Dashed lines represent the phonon Green’s function $\mathbb{G}$, the circle represents ${\Gamma}_{\alpha}$, the triangle represents the third-order vertex $\overline{\Psi}$ and the square the fourth-order vertex $\chi $. The thick line with opposite arrows represents the displacement autocorrelation $C\left(\omega \right)$. Frequency must be conserved at each vertex.

#### Appendix C.4. Displacement Autocorrelations C(ω)

**Figure A2.**Feynman diagrams associated with $C\left(\omega \right)$ up to the second order in anharmonic forces. Conventions are the same as in Figure A1.

## Appendix D. Statement and Proof of the Novikov–Furutsu–Donsker (NFD) Relation

**Proof**

**of**

**the**

**NFD**

**relation:**

## Appendix E. Heat Current within the Harmonic Approximation

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**Figure 1.**(

**Top**) a general “molecular” multiprobe geometry where the device defined by atoms within the ellipse are connected to 4 reservoirs imposing a temperature or measuring a current. (

**Bottom**) the 2-probe geometry. Atoms in each lead are connected to a harmonic thermostat at fixed temperature ${T}_{L}$ and ${T}_{R}$. In the Buttiker probe, also called the self-consistent reservoirs geometry, to measure the “local” temperature, each layer of the central device may also be weakly connected to a fictitious thermostat at a temperature to be (self-consistently) determined so that the net heat flow from that probe to the device is zero.

**Figure 2.**Feynman diagram associated with $\mathbb{G}$ represented by thick dashed lines. The phonon Green’s function $\mathcal{G}$ is shown with thin solid lines, and the quartic vertex $\chi $ with the solid square. The thick line with opposite arrows represents the displacement autocorrelation $C\left({\omega}_{1}\right)$. Internal frequency ${\omega}_{1}$ is integrated over.

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Esfarjani, K.
Theory of Non-Equilibrium Heat Transport in Anharmonic Multiprobe Systems at High Temperatures. *Entropy* **2021**, *23*, 1630.
https://doi.org/10.3390/e23121630

**AMA Style**

Esfarjani K.
Theory of Non-Equilibrium Heat Transport in Anharmonic Multiprobe Systems at High Temperatures. *Entropy*. 2021; 23(12):1630.
https://doi.org/10.3390/e23121630

**Chicago/Turabian Style**

Esfarjani, Keivan.
2021. "Theory of Non-Equilibrium Heat Transport in Anharmonic Multiprobe Systems at High Temperatures" *Entropy* 23, no. 12: 1630.
https://doi.org/10.3390/e23121630