# Nonequilibrium Thermodynamics and Steady State Density Matrix for Quantum Open Systems

## Abstract

**:**

## 1. Introduction

## 2. Non-Equilibrium Steady State

#### 2.1. System and Initial Conditions

#### 2.2. The NE Density Matrix ${\rho}^{\mathrm{NE}}$

#### 2.3. Three Equivalent Expressions for ${\rho}^{\mathrm{NE}}$

## 3. Entropy Production

#### 3.1. Entropy Production Rate

#### 3.2. An Example

## 4. Nonequilibrium Gibbs–von Neumann Entropies

#### 4.1. Which Density Matrix?

#### 4.2. An Example for the Entropy of the Central Region

## 5. Discussion

## Acknowledgments

## Conflicts of Interest

## References and Notes

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**Figure 1.**NE entropy production rate $\Delta {\dot{S}}^{\mathrm{NE}}$ versus the energy level ${\epsilon}_{0}$ for different transport regimes. $\Delta {\dot{S}}^{\mathrm{NE}}$ is always a positive quantity. (

**a**) only temperature differences (${\mu}_{L}={\mu}_{R}=0.2$) ${k}_{B}{T}_{L}=0.1$, ${k}_{B}{T}_{R}=0.05$ (solid line) and ${k}_{B}{T}_{L}=0.1$, ${k}_{B}{T}_{R}=0.3$ (dashed line); (

**b**) both chemical potential and temperature differences (${\mu}_{L}=0.3$, ${\mu}_{R}=0.2$) ${k}_{B}{T}_{L}=0.1$, ${k}_{B}{T}_{R}=0.05$ (solid line) and ${k}_{B}{T}_{L}=0.1$, ${k}_{B}{T}_{R}=0.3$ (dashed line); (

**c**) both temperature and chemical potential differences (${k}_{B}{T}_{L}=0.1$, ${k}_{B}{T}_{R}=0.3$) ${\mu}_{L}=0.3$, ${\mu}_{R}=0.2$ (solid line) and ${\mu}_{L}=0.2$, ${\mu}_{R}=0.3$ (dashed line); (

**d**) comparison with results of Figure 3b in Reference [11]. (${\mu}_{L}=0.05$, ${\mu}_{R}=0.0$) ${k}_{B}{T}_{L}=0.026$, ${k}_{B}{T}_{R}=26/{30.10}^{-3}$, strong coupling ${v}_{L}={v}_{R}=0.2$ (solid line) and weak coupling ${v}_{L}={v}_{R}=0.02$ (dashed line, amplitude rescaled by a factor $\times 100$). The other parameters are ${t}_{L}={t}_{R}=2.0$ and ${v}_{L}={v}_{R}=0.25$ (when not specified otherwise). All parameters are given in dimension of energy in [eV].

**Figure 2.**NE entropy production rate $\Delta {\dot{S}}^{\mathrm{NE}}$ versus the temperature difference ($\Delta T={T}_{L}-{T}_{R}$ and $\Delta \beta ={\beta}_{L}-{\beta}_{R}$) and/or the chemical potential difference ($\Delta \mu ={\mu}_{L}-{\mu}_{R}$). $\Delta {\dot{S}}^{\mathrm{NE}}$ is always a positive quantity, and increases when $\Delta \mu $ or $\Delta T$ ($|\Delta \beta |$) increases. The solid (dashed) lines are for the resonant (off-resonant) transport regime, i.e., ${\epsilon}_{0}\sim (>)\phantom{\rule{4pt}{0ex}}{\mu}^{\mathrm{eq}}$ when $\Delta \mu =0$. (

**a**) system at a unique temperature ${k}_{B}{T}_{L}={k}_{B}{T}_{R}=0.1$; (

**b**) system with a unique chemical potential ${\mu}_{L}={\mu}_{R}=0.2$. In the inset, we also show the dependence of $\Delta {\dot{S}}^{\mathrm{NE}}$ vs. $\Delta \beta $; (

**c**) system with a temperature difference ${k}_{B}{T}_{L}=0.1,{k}_{B}{T}_{R}=0.2$; (

**d**) system with a chemical potential difference ${\mu}_{L}=0.35,{\mu}_{R}=0.05$. The inset shows the $\Delta {\dot{S}}^{\mathrm{NE}}$ vs. $\Delta \beta $. The other parameters are ${t}_{L}={t}_{R}=2.0$ and ${v}_{L}={v}_{R}=0.25$ (given in [eV]). For the dependence on $\Delta \mu $, we take ${\mu}_{L}={\mu}^{\mathrm{eq}}+\Delta \mu /2$ and ${\mu}_{R}={\mu}^{\mathrm{eq}}-\Delta \mu /2$ with ${\mu}^{\mathrm{eq}}=0.2$. For the dependence on $\Delta T$, we take and ${T}_{R}={T}^{0}=0.1$, ${T}_{L}={T}^{0}+\Delta T$ (hence, $\Delta \beta <0$ for $\Delta T>0$).

**Figure 3.**Gibbs–von Neumann NE entropy for the central region ${S}_{C}^{\mathrm{NE}}$ versus the energy level ${\epsilon}_{0}$ for the different transport regimes considered in Figure 1. The Gibbs NE entropy ${S}_{C}^{\mathrm{NE}}$ is always a positive quantity as expected. (

**a**) (${\mu}_{L}={\mu}_{R}=0.2$) ${k}_{B}{T}_{L}=0.1$, ${k}_{B}{T}_{R}=0.05$ (solid line) and ${k}_{B}{T}_{L}=0.1$, ${k}_{B}{T}_{R}=0.3$ (dashed line); (

**b**) both chemical potential and temperature differences (${\mu}_{L}=0.3$, ${\mu}_{R}=0.2$) ${k}_{B}{T}_{L}=0.1$, ${k}_{B}{T}_{R}=0.05$ (solid line) and ${k}_{B}{T}_{L}=0.1$, ${k}_{B}{T}_{R}=0.3$ (dashed line); (

**c**) both temperature and chemical potential differences (${k}_{B}{T}_{L}=0.1$, ${k}_{B}{T}_{R}=0.3$) ${\mu}_{L}=0.3$, ${\mu}_{R}=0.2$ (solid line) and ${\mu}_{L}=0.2$, ${\mu}_{R}=0.3$ (dashed line); (

**d**) comparison with results of Figure 3b in Reference [11]. (${\mu}_{L}=0.05$, ${\mu}_{R}=0.0$) ${k}_{B}{T}_{L}=0.026$, ${k}_{B}{T}_{R}=26/{30.10}^{-3}$, strong coupling ${v}_{L}={v}_{R}=0.2$ (solid line) and weak coupling ${v}_{L}={v}_{R}=0.02$ (da${S}_{C}^{\mathrm{NE}}$shed line). The other parameters are ${t}_{L}={t}_{R}=2.0$ and ${v}_{L}={v}_{R}=0.25$ (when not specified otherwise) and given in [eV].

**Figure 4.**Gibbs–von Neumann NE entropy for the central region ${S}_{C}^{\mathrm{NE}}$ versus the temperature difference ($\Delta T={T}_{L}-{T}_{R}$ and $\Delta \beta ={\beta}_{L}-{\beta}_{R}$) and/or the chemical potential difference ($\Delta \mu ={\mu}_{L}-{\mu}_{R}$). $\Delta {\dot{S}}^{\mathrm{NE}}$ is always a positive quantity, and increases when $\Delta \mu $ or $\Delta T$ ($|\Delta \beta |$) increases. The solid (dashed) lines are for the resonant (off-resonant) transport regime, i.e., ${\epsilon}_{0}\sim (>)\phantom{\rule{4pt}{0ex}}{\mu}^{\mathrm{eq}}$ when $\Delta \mu =0$. (

**a**) system at a unique temperature ${k}_{B}{T}_{L}={k}_{B}{T}_{R}=0.1$; (

**b**) system with a unique chemical potential ${\mu}_{L}={\mu}_{R}=0.2$. In the inset, we also show the dependence of $\Delta {\dot{S}}^{\mathrm{NE}}$ vs. $\Delta \beta $; (

**c**) system with a temperature difference ${k}_{B}{T}_{L}=0.1,{k}_{B}{T}_{R}=0.2$; (

**d**) system with a chemical potential difference ${\mu}_{L}=0.35,{\mu}_{R}=0.05$. The inset shows the $\Delta {\dot{S}}^{\mathrm{NE}}$ vs. $\Delta \beta $. The other parameters are ${t}_{L}={t}_{R}=2.0$ and ${v}_{L}={v}_{R}=0.25$ (given in [eV]). For the dependence on $\Delta \mu $, we take ${\mu}_{L}={\mu}^{\mathrm{eq}}+\Delta \mu /2$ and ${\mu}_{R}={\mu}^{\mathrm{eq}}-\Delta \mu /2$ with ${\mu}^{\mathrm{eq}}=0.2$. For the dependence on $\Delta T$, we take ${T}_{R}={T}^{0}=0.1$, ${T}_{L}={T}^{0}+\Delta T$ (hence, $\Delta \beta <0$ for $\Delta T>0$).

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Ness, H.
Nonequilibrium Thermodynamics and Steady State Density Matrix for Quantum Open Systems. *Entropy* **2017**, *19*, 158.
https://doi.org/10.3390/e19040158

**AMA Style**

Ness H.
Nonequilibrium Thermodynamics and Steady State Density Matrix for Quantum Open Systems. *Entropy*. 2017; 19(4):158.
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**Chicago/Turabian Style**

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2017. "Nonequilibrium Thermodynamics and Steady State Density Matrix for Quantum Open Systems" *Entropy* 19, no. 4: 158.
https://doi.org/10.3390/e19040158