# Stochastic Stirling Engine Operating in Contact with Active Baths

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## Abstract

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## 1. Introduction

## 2. Stirling Cycle between Equilibrium States: A Quick Review

#### 2.1. Modeling the Motion of a Colloidal Particle

#### 2.2. Energetics of the Stirling Cycle

## 3. Engine Operating between Nonequilibrium Baths

#### 3.1. Modified Langevin Equation

#### 3.2. The Energetics Is Not Altered If We Use Iso-T_{act} Steps

## 4. Energetics Using the Diffusion Constant as an Active Temperature

#### 4.1. A Bath with White but Non-Gaussian Statistics

#### 4.2. A Bath with a Persistent Noise

#### 4.3. A Bath Described by a More General Langevin Equation

## 5. Discussion: Back to Experiments

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Active Particle Dynamics

## Appendix B. Is Kurtosis Related to Efficiency?

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**Figure 1.**Schematic diagram of the Stirling cycle in stiffness-position space. Unlike its thermodynamic counterpart, the cycle is run counter-clockwise but is nevertheless an engine cycle.

**Figure 2.**Log of the probability of the colloid’s position as a function of position (for a unit a), in equilibrium with Gaussian statistics (red at $T/k=2$, green at $T/k=4$) or out of equilibrium as given by ${P}_{\mathrm{ss}}$ (blue at $T/k=2$, orange at $T/k=4$).

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**MDPI and ACS Style**

Zakine, R.; Solon, A.; Gingrich, T.; Van Wijland, F.
Stochastic Stirling Engine Operating in Contact with Active Baths. *Entropy* **2017**, *19*, 193.
https://doi.org/10.3390/e19050193

**AMA Style**

Zakine R, Solon A, Gingrich T, Van Wijland F.
Stochastic Stirling Engine Operating in Contact with Active Baths. *Entropy*. 2017; 19(5):193.
https://doi.org/10.3390/e19050193

**Chicago/Turabian Style**

Zakine, Ruben, Alexandre Solon, Todd Gingrich, and Frédéric Van Wijland.
2017. "Stochastic Stirling Engine Operating in Contact with Active Baths" *Entropy* 19, no. 5: 193.
https://doi.org/10.3390/e19050193