# Equilibrium and Non-Equilibrium Lattice Dynamics of Anharmonic Systems

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## Abstract

**:**

## 1. Introduction

## 2. The Self-Consistent Phonon Theory

#### 2.1. Determination of the State of Thermal Equilibrium at a Given T

#### 2.2. Previous Works/Implementations by Other Groups

- SCP implemented in the code ALAMODE–ANPHON by Tadano et al. in 2015 [26], where anharmonicity up to the fourth-order is extracted, similar to the present approach, and the self-consistent harmonic approximation is implemented.
- SCP is also implemented in the HiPHive package developed by Eriksson et al. [27].
- SCHA (Self-Consistent Harmonic Approximation) method by Ravichandran and Broido in 2018 [28] also calculates the effective phonon dispersion and lifetimes at finite temperatures.

#### 2.3. Summary

## 3. Non-Equilibrium Heat Transport in Anharmonic Systems

#### 3.1. Equations of Motion and Langevin Thermostats

#### 3.2. Entropy Generation Rate

#### 3.3. Heat Current

#### 3.4. Thermal Expansion and Renormalization of the Force Constants

#### 3.5. Correlation Functions $\langle Y{\eta}^{\u2020}\rangle $ and $\langle Y{Y}^{\u2020}\rangle $

#### 3.5.1. Displacement—Noise Correlations $\langle Y{\eta}^{\u2020}\rangle $ within NESCP

#### 3.5.2. Beyond NESCP

#### 3.5.3. Displacement Autocorrelations $\langle Y{Y}^{\u2020}\rangle $

#### 3.5.4. Beyond NESCP

#### 3.6. The Particular Case of Non-Equilibrium Self-Consistent Phonon Approximation: NESCP

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Calculation of Time Averages

## Appendix B. Explicit Form of the Correlation Functions Including the Atomic and Cartesian Indices

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

#### Appendix B.2. Noise Autocorrelation Functions

## Appendix C. Statement and Proof of the Donsker-Furutsu-Novikov (DFN) Relation

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**Figure 1.**Flowcharts of two possible self-consistent procedures using Broyden’s algorithm depending on whether K and C are treated as dependent (

**left**) or independent parameters (

**right**).

**Figure 2.**The ND atom anharmonic structure is connected to two semi-infinite reservoirs in this example (called L and R). After elimination of the reservoirs’ degrees of freedom, the infinite system is replaced by an isolated “cluster” subject to thermostating forces ${\eta}_{L}$ and ${\eta}_{R}$ applied on the boundary, reflecting the equation of motion in Equation (33). As a result of this boundary condition, force constants $\varphi ,\psi ,\chi $ are replaced with effective force constants $\mathsf{\Phi},\mathsf{\Psi},X$ and a harmonic self energy $\sigma $.

**Figure 3.**Feynman diagrams associated with the approximations of C and ${Z}_{\alpha}$: dashed lines represent $\mathbb{G}$, the Green’s function within NESCP; thick solid lines with two arrows facing each other represent the displacement autocorrelation C; and the solid lines ending with a dot are the noise-displacement correlation ${Z}_{\alpha}$.

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Esfarjani, K.; Liang, Y.
Equilibrium and Non-Equilibrium Lattice Dynamics of Anharmonic Systems. *Entropy* **2022**, *24*, 1585.
https://doi.org/10.3390/e24111585

**AMA Style**

Esfarjani K, Liang Y.
Equilibrium and Non-Equilibrium Lattice Dynamics of Anharmonic Systems. *Entropy*. 2022; 24(11):1585.
https://doi.org/10.3390/e24111585

**Chicago/Turabian Style**

Esfarjani, Keivan, and Yuan Liang.
2022. "Equilibrium and Non-Equilibrium Lattice Dynamics of Anharmonic Systems" *Entropy* 24, no. 11: 1585.
https://doi.org/10.3390/e24111585