# Minimum Entropy Production Effect on a Quantum Scale

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## Abstract

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## 1. Introduction

## 2. Historical Considerations

#### 2.1. The Quantized Electric Conductance

#### 2.2. The Quantized Thermal Conductance

#### 2.3. Lagrangian Description of Heat Conduction

## 3. The Quantized Behavior of the Conductance of Entropy Current and the Entropy Production

## 4. Examples and Applications

#### 4.1. Entropy Change during a Single Quantum Transfer

#### 4.2. Spin-Lattice Relaxation

## 5. An Additional Consequence of the Least Action Principle

## 6. Summary

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The electric conductance showed as a function of the gate voltage. The channel width, w, can be modulated by the gate voltage. The quantized behavior can be read out directly from the figure. The quantum of electric conductance is $2{e}^{2}/h$ based on theoretical predictions. Inspired by the work of van Wees et al. [1].

**Figure 2.**The propagating plane wave in the direction x and the cross-modes in the direction y in the 2D waveguide.

**Figure 3.**Experimental realization of a Si${}_{3}$N${}_{4}$ thermal waveguide [2]. Physical dimensions: length: $L\sim 1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m; width: $w=200$ nm; layer thickness: $d=60$ nm.

**Figure 5.**The same energy phonons travel in both directions, but the probability of emission is greater from the hotter domain $\overline{)1}$ with ${T}_{1}$ towards the colder one $\overline{)2}$ with ${T}_{2}$ than vice versa due to the greater population of $h\nu $ quanta.

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**MDPI and ACS Style**

Márkus, F.; Gambár, K.
Minimum Entropy Production Effect on a Quantum Scale. *Entropy* **2021**, *23*, 1350.
https://doi.org/10.3390/e23101350

**AMA Style**

Márkus F, Gambár K.
Minimum Entropy Production Effect on a Quantum Scale. *Entropy*. 2021; 23(10):1350.
https://doi.org/10.3390/e23101350

**Chicago/Turabian Style**

Márkus, Ferenc, and Katalin Gambár.
2021. "Minimum Entropy Production Effect on a Quantum Scale" *Entropy* 23, no. 10: 1350.
https://doi.org/10.3390/e23101350