# The Problem of Engines in Statistical Physics

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## Abstract

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## 1. Introduction

## 2. Feedback in Classical Engines

#### 2.1. Rayleigh-Eddington Criterion

#### 2.2. Active Force

#### 2.3. Electrostatic Engines

#### 2.3.1. Franklin Bells

#### 2.3.2. Quincke Rotor

## 3. Fluctuations

#### 3.1. Quantum Thermodynamics

#### 3.2. Stochastic Thermodynamics

## 4. Deterministic and Stochastic Engine Dynamics

#### 4.1. Deterministic Model of Oscillating Engine

#### 4.2. Stochastic Model of Oscillating Engine

#### 4.3. Stochastic Model of Rotating Engine

#### 4.4. The Puzzle of Stationary States

## 5. Discussion and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DC | direct current |

EDL | electrical double layer |

emf | electromotive force |

MME | Markovian master Equation |

NESS | non-equilibrium steady state |

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**Figure 1.**The autonomous operation of this heat engine depends on the valves modulating the rate of heat flow between the working substance and the two baths in accordance with Equation (6) so that $W>0$. This gives a positive feedback between the oscillation of x and the modulation of the heat currents. Image taken from [33].

**Figure 2.**A model of an autonomous “putt-putt” heat engine that can raise water from a lower to an upper reservoir. Image taken from [33].

**Figure 3.**Franklin bells: the self-oscillation of the pendulum is powered by the electrostatic potential difference ${V}_{0}$ between the two metal bells. The active force that drives the pendulum results from a positive feedback between the angular position of the pendulum, $\theta \left(t\right)$, and the rate of change of its charge $q\left(t\right)$.

**Figure 4.**The scheme for incorporating thermal noise into the equations of motion for Franklin bells. The stochastic force $\mathcal{F}\left(t\right)$ is associated, by a fluctuation-dissipation relation, to the mechanical damping exerted on the pendulum’s motion by its medium, as represented by the $\gamma \dot{\theta}$ term in Equation (16). The stochastic currents ${\mathcal{I}}_{1,2}\left(t\right)$ correspond to the Johnson–Nyquist noise associated with the corresponding resistances ${R}_{1,2}$ in Equation (18).

**Figure 5.**The Quincke rotor: the self-rotation of the dielectric cylinder is powered by the electrostatic potential difference ${V}_{0}$ between the parallel plates. The torque results from a positive feedback between the turning of the cylinder, described by the angle $\varphi $ and the change in the charge attached to the cylinder’s surface. This torque can lift a weight m, which acts as the load for the engine. The weight and the string attached to it are assumed to be uncharged.

**Figure 6.**Electron shuttle: (

**a**) A metallic grain of mass m is mounted on a harmonic oscillator with the elastic constant k. The position of the grain, measured with respect to its equilibrium, is denoted by x. The oscillator is placed in a constant electric field $\mathbf{E}$ maintained by a voltage source ${V}_{0}$ connected to two electrodes. (

**b**) Electrons can tunnel between the grain and the electrodes, changing the net charge upon which the field $\mathbf{E}$ acts. The electrodes may be considered as thermodynamic reservoirs L and R. Images adapted from [66].

**Figure 7.**Plots of $\omega \left(t\right)$ for the model of the Quincke rotating engine described by Equation (89). Time t is measured in units of $1/\gamma $. The parameters are $\mu =10$, $\Gamma =1$, $J=1$, and ${\tau}_{\mathrm{load}}=0.01$. The colors represent different amplitudes for the white noise term ${\tau}_{L}\left(t\right)$, with blue corresponding to 0, yellow to 0.1, green to 0.5, and dark orange to 1. Initial conditions are $\omega \left(0\right)=0.1$ and $z\left(0\right)=0$. Plot (

**a**) shows single realizations of the stochastic trajectory, while plot (

**b**) shows the averages over 500 realizations.

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Alicki, R.; Gelbwaser-Klimovsky, D.; Jenkins, A.
The Problem of Engines in Statistical Physics. *Entropy* **2021**, *23*, 1095.
https://doi.org/10.3390/e23081095

**AMA Style**

Alicki R, Gelbwaser-Klimovsky D, Jenkins A.
The Problem of Engines in Statistical Physics. *Entropy*. 2021; 23(8):1095.
https://doi.org/10.3390/e23081095

**Chicago/Turabian Style**

Alicki, Robert, David Gelbwaser-Klimovsky, and Alejandro Jenkins.
2021. "The Problem of Engines in Statistical Physics" *Entropy* 23, no. 8: 1095.
https://doi.org/10.3390/e23081095