# Classical and Quantum H-Theorem Revisited: Variational Entropy and Relaxation Processes

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## Abstract

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## 1. Introduction

## 2. Classical Scheme

#### 2.1. The Boltzmann Transport Equation

#### 2.2. A Summary of the H-Theorem and the Maxwell—Boltzmann Distribution

#### 2.3. Non-Homogeneous Classical H-Functional

#### 2.3.1. Properties of ${\mathcal{H}}^{\prime}$ for Systems in Equilibrium

#### 2.3.2. Proof of the H-Theorem for Non-Homogeneous Distributions

## 3. Quantum Scheme

#### 3.1. H-Theorem and the Fermi–Dirac and Bose–Einstein Distribution Functions

#### 3.2. Out-of-Equilibrium, Non-Homogeneous Quantum Systems

#### 3.2.1. Properties of $\mathcal{H}$ for Systems in Equilibrium

#### 3.2.2. Proof of the Quantum H-Theorem for Non-Homogeneous Systems

## 4. Quantum—Classical Correspondence

## 5. Relaxation Processes in Degenerated Quantum Gases

## 6. Comments and Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

${H}_{B}$ | The original Boltzmann H-functional |

${f}_{B}=f\left(\overrightarrow{v}\right)$ | The Maxwell–Boltzmann distribution function |

${\mathcal{H}}^{\prime}$ | Our H-functional for a classical dilute gas |

${f}_{M}^{\prime}={f}^{\prime}({\overrightarrow{r}}_{M},\overrightarrow{v},t)$ | The classical distribution function of a cell centered at ${\overrightarrow{r}}_{M}$ |

${H}_{T}$ | The H-functional proposed by Tolman |

$\mathcal{H}$ | Our H-functional for a quantum dilute gas |

${f}_{nM}={f}_{M}({\overrightarrow{r}}_{M},{\u03f5}_{n},t)$ | The quantum distribution function of a cell centered at ${\overrightarrow{r}}_{M}$ |

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**MDPI and ACS Style**

Medel-Portugal, C.; Solano-Altamirano, J.M.; Carrillo-Estrada, J.L.E.
Classical and Quantum H-Theorem Revisited: Variational Entropy and Relaxation Processes. *Entropy* **2021**, *23*, 366.
https://doi.org/10.3390/e23030366

**AMA Style**

Medel-Portugal C, Solano-Altamirano JM, Carrillo-Estrada JLE.
Classical and Quantum H-Theorem Revisited: Variational Entropy and Relaxation Processes. *Entropy*. 2021; 23(3):366.
https://doi.org/10.3390/e23030366

**Chicago/Turabian Style**

Medel-Portugal, Carlos, Juan Manuel Solano-Altamirano, and José Luis E. Carrillo-Estrada.
2021. "Classical and Quantum H-Theorem Revisited: Variational Entropy and Relaxation Processes" *Entropy* 23, no. 3: 366.
https://doi.org/10.3390/e23030366