# Landauer’s Principle in a Quantum Szilard Engine without Maxwell’s Demon

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## Abstract

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## 1. Introduction

## 2. A Szilard Engine without Maxwell’s Demon?

## 3. Thermodynamics of a Demonless Quantum Szilard Engine under Confinement Effects

#### 3.1. Step I: Creating Superposition by Insertion

#### 3.2. Step II: Localization by Quantum Measurement

#### 3.3. Step III: Work Extraction by Expansion

#### 3.4. Step IV: Removal

## 4. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Analytical Expressions for Work, Heat and Energy Exchanges

## References

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**Figure 1.**A classical Szilard engine setup without Maxwell’s demon. Schematic shows the step after the partition (piston) symmetrically inserted. Magnetic rods are attached to the both sides of the piston and two solenoids are placed on both ends which are connected to a passive diode bridge. Regardless of the particle’s position, magnetic rods moving inside the solenoids induce an electric current that can be used from the output of the diode bridge.

**Figure 2.**A quantum Szilard engine setup composed of three components, system $\mathcal{S}$, measuring device $\mathcal{D}$ and heat bath $\mathcal{B}$. $\mathcal{S}$ denotes the container with a single particle inside. $\mathcal{D}$ is the device which measures the particle. $\mathcal{B}$ at temperature T is in contact with both $\mathcal{S}$ and $\mathcal{D}$, keeping all processes isothermal. Dotted turquoise lines denote the effective regions that the particle occupies because of the confinement effects. (I→II) Inserting the partition into the container. Symmetric insertion divides the container into two equal compartments and creates an entangled state of the particle’s position. (II→III) Performing quantum measurement to localize the particle into one of the compartments. (III→IV) Letting particle to expand the partition and extracting work from the system. (IV→I) Removing the partition from the container at the boundary, which completes the cycle.

**Figure 3.**Simulation of quasistatic isothermal insertion process for a container with sizes ${L}_{x}=20$ nm, ${L}_{y}=10$ nm and at temperature $T=300$ K. d denotes the depth of the partition inserted into the domain. (

**a**) Quantum-mechanical thermal probability density distribution of the particle is non-uniform inside the container due to quantum size effects. Magnitudes of the density distributions are represented by the rainbow color scale, where red and blue colors denote higher and lower density regions, respectively. Partition with zero thickness entering the container in y-direction, $d=1$ nm. (

**b**) Partition enters almost halfway of the container, $d=4$ nm. Although it has zero thickness, confined particle perceives a finite effective thickness ($2\delta $). (

**c**) Partition is at $d=7$ nm depth. (

**d**) Partition separates the container into two equally sized compartments, $d=10$ nm. Particle has equal probability to be at both sides. Variation of (

**e**) Helmholtz free energy, (

**f**) entropy and (

**g**) internal energy with respect to partition’s penetration depth d in nm’s.

**Figure 4.**Work and heat exchanges as well as changes in internal energy during insertion process with respect to (

**a**) domain size L and (

**b**) temperature T. Black/gray, teal/turquoise and purple/pink curves represent insertion work, insertion heat and internal energy change during insertion between $\mathcal{S}$ and $\mathcal{B}$ respectively for (a)/(b).

**Figure 5.**Simulation of quasistatic isothermal expansion process without performing the quantum measurement and localizing the particle. l denotes the lateral position of the partition. (

**a**) Partition symmetrically divides the container into two equally sized compartments. (

**b**) Partition is moved to the position of $l=14$ nm by preserving quantum superposition and without localizing the particle. (

**c**) Partition is at $l=17$ nm. (

**d**) Partition is moved to the right boundary of the container, $l=20$ nm. Initial situation before the insertion is recovered. Variation of (

**e**) Helmholtz free energy, (

**f**) entropy and (

**g**) internal energy with respect to partition’s lateral position l in nm’s.

**Figure 6.**Simulation of quasistatic isothermal expansion process after the measurement and localization of the particle. (

**a**) After the symmetric insertion of the partition, quantum measurement is performed, and the particle is localized at the left compartment. (

**b**) Partition expands to the position of $l=14$ nm, which is caused by the pressure exerted on the partition by the localized particle. (

**c**) Expansion of the partition at $l=17$ nm. (

**d**) Partition expands to the right boundary of the container, $l=20$ nm. Initial situation before the insertion is recovered. Variation of (

**e**) Helmholtz free energy, (

**f**) entropy and (

**g**) internal energy with respect to partition’s lateral position l in nm’s.

**Table 1.**Changes of free energy, entropy and internal energy in the system $\mathcal{S}$, device $\mathcal{D}$ and bath $\mathcal{B}$, for insertion (I), measurement (II), expansion (III) and removal (IV) processes.

I | II | III | IV | ||
---|---|---|---|---|---|

$\mathcal{S}$ | $\Delta F$ | +${W}_{\mathrm{ins}}$ | +${W}_{\mathrm{msr}}$ | $-{W}_{\mathrm{msr}}$$-{W}_{\mathrm{ins}}$ | 0 |

$\Delta S$ | $-{Q}_{\mathrm{ins}}$ | $-{Q}_{\mathrm{msr}}$ | +${Q}_{\mathrm{msr}}$+${Q}_{\mathrm{ins}}$ | 0 | |

$\Delta U$ | +$\Delta {U}_{\mathrm{ins}}$ | 0 | $-\Delta {U}_{\mathrm{ins}}$ | 0 | |

$\mathcal{D}$ | $\Delta F$ | 0 | $-{W}_{\mathrm{msr}}$ | 0 | 0 |

$\Delta S$ | 0 | +${Q}_{\mathrm{msr}}$ | 0 | 0 | |

$\Delta U$ | 0 | 0 | 0 | 0 | |

$\mathcal{B}$ | $\Delta F$ | $-{W}_{\mathrm{ins}}$ | 0 | +${W}_{\mathrm{msr}}$+${W}_{\mathrm{ins}}$ | 0 |

$\Delta S$ | +${Q}_{\mathrm{ins}}$ | 0 | $-{Q}_{\mathrm{msr}}$$-{Q}_{\mathrm{ins}}$ | 0 | |

$\Delta U$ | $-\Delta {U}_{\mathrm{ins}}$ | 0 | +$\Delta {U}_{\mathrm{ins}}$ | 0 |

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**MDPI and ACS Style**

Aydin, A.; Sisman, A.; Kosloff, R. Landauer’s Principle in a Quantum Szilard Engine without Maxwell’s Demon. *Entropy* **2020**, *22*, 294.
https://doi.org/10.3390/e22030294

**AMA Style**

Aydin A, Sisman A, Kosloff R. Landauer’s Principle in a Quantum Szilard Engine without Maxwell’s Demon. *Entropy*. 2020; 22(3):294.
https://doi.org/10.3390/e22030294

**Chicago/Turabian Style**

Aydin, Alhun, Altug Sisman, and Ronnie Kosloff. 2020. "Landauer’s Principle in a Quantum Szilard Engine without Maxwell’s Demon" *Entropy* 22, no. 3: 294.
https://doi.org/10.3390/e22030294