# Advances in Engine Efficiency: Nanomaterials, Surface Engineering, and Quantum-Based Propulsion

## Abstract

**:**

## 1. Introduction

- $\Psi (\mathbf{r},t)$: wavefunction of the stock price S at a given instant of time t.
- ℏ: reduced Planck’s constant.
- m: mass of the particle.
- V: external potential.
- $\mathcal{J}\equiv \frac{{J}^{2}}{2I}-(\mathit{\omega}\xb7\mathbf{J})-\Delta F$: a term that might involve thermomechanical and/or informational aspects, with $\mathbf{J}$ denoting the total sum of orbital and spin angular momentum, I the moment of inertia, $\mathit{\omega}$ the instantaneous angular velocity of the system, and $\Delta F$ the free energy function.

#### 1.1. Thermal Engines

#### 1.2. Information-Burned Engine

#### 1.3. Engine Fueled by Entanglement

- 1.
- Isothermal expansion: during this step, the engine is coupled to a hot thermal reservoir at temperature ${T}_{H}$ and expands isothermally while doing work. The unitary transformation associated with this step is ${\widehat{U}}_{1}={e}^{-i{\widehat{H}}_{1}{\tau}_{1}/\hslash}$, where ${\widehat{H}}_{1}$ is the Hamiltonian of the engine during this step.
- 2.
- Adiabatic expansion: during this step, the engine is thermally isolated and expands adiabatically while doing work. The unitary transformation associated with this step is ${\widehat{U}}_{2}={e}^{-i{\widehat{H}}_{2}{\tau}_{2}/\hslash}$, where ${\widehat{H}}_{2}$ is the Hamiltonian of the engine during this step.
- 3.
- Isothermal compression: during this step, the engine is coupled to a cold thermal reservoir at temperature ${T}_{C}$ and isothermally compresses while working on it. The unitary transformation associated with this step is ${\widehat{U}}_{3}={e}^{-i{\widehat{H}}_{3}{\tau}_{3}/\hslash}$, where ${\widehat{H}}_{3}$ is the Hamiltonian of the engine during this step.
- 4.
- Adiabatic compression: during this step, the engine is thermally isolated and compresses adiabatically while work is carried out. The unitary transformation associated with this step is ${\widehat{U}}_{4}={e}^{-i{\widehat{H}}_{4}{\tau}_{4}/\hslash}$, where ${\widehat{H}}_{4}$ is the Hamiltonian of the engine during this step.

#### 1.4. The Stirling Engine

#### 1.4.1. System Parameters and Constants

#### 1.4.2. Equations Used

#### 1.4.3. Results

- Number of energy values: 4;
- Work per cycle: $0.2425994873046875$ J;
- Power: $\mathrm{60,649,871.82617187}$ W;
- Entropy change per cycle: $6.92314106394308\times {10}^{-6}$ J/K.

#### 1.4.4. Quantum Stirling Engine: Entropy Dynamics

#### 1.4.5. Linear Decoherence in Quantum Stirling Cycles

#### 1.5. Thrust Based on the Gradient in the Refractive Index of a Material

#### Self-Propelled EM Device with Metamaterials

^{2}, we can estimate the thrust to be on the order of nano-newtons to micro-newtons. However, this is a very rough estimate, and the actual thrust generated will depend on the specific properties of the metamaterial and the pulsed electromagnetic wave.

## 2. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Evolution of state populations over multiple cycles in the quantum Stirling cycle. The dynamics indicate a preference towards populating the first energy level as cycles progress.

**Figure 2.**Energy dynamics of the quantum Stirling cycles, highlighting the periodic behaviors of each cycle.

**Figure 3.**Variation of system von Neumann entropy throughout the quantum Stirling cycles. The energy and the entropy, both quantities exhibit periodic behaviors, offering insights into the engine’s thermodynamic processes.

**Figure 4.**Decay of quantum coherence over the cycles in the quantum Stirling engine. The linear decrease in coherence over time highlights consistent system–environment interactions leading to decoherence.

B | Von Neumann Entropy |
---|---|

1 | $6.28\times {10}^{-6}$ |

2 | $1.75\times {10}^{-5}$ |

3 | $2.54\times {10}^{-5}$ |

4 | $3.57\times {10}^{-5}$ |

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**MDPI and ACS Style**

Pinheiro, M.J.
Advances in Engine Efficiency: Nanomaterials, Surface Engineering, and Quantum-Based Propulsion. *Magnetochemistry* **2024**, *10*, 17.
https://doi.org/10.3390/magnetochemistry10030017

**AMA Style**

Pinheiro MJ.
Advances in Engine Efficiency: Nanomaterials, Surface Engineering, and Quantum-Based Propulsion. *Magnetochemistry*. 2024; 10(3):17.
https://doi.org/10.3390/magnetochemistry10030017

**Chicago/Turabian Style**

Pinheiro, Mario J.
2024. "Advances in Engine Efficiency: Nanomaterials, Surface Engineering, and Quantum-Based Propulsion" *Magnetochemistry* 10, no. 3: 17.
https://doi.org/10.3390/magnetochemistry10030017