# 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?

## References

- Sekimoto, K. Stochastic Energetics; Lecture Notes in Physics; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
- Seifert, U. Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep. Prog. Phys.
**2012**, 75, 126001. [Google Scholar] [CrossRef] [PubMed] - Schmiedl, T.; Seifert, U. Efficiency at maximum power: An analytically solvable model for stochastic heat engines. Eur. Lett.
**2008**, 81, 20003. [Google Scholar] [CrossRef] - Blickle, V.; Bechinger, C. Realization of a micrometre-sized stochastic heat engine. Nat. Phys.
**2011**, 8, 143–146. [Google Scholar] [CrossRef] - Horowitz, J.M.; Parrondo, J.M.R. Thermodynamics: A Stirling effort. Nat. Phys.
**2012**, 8, 108–109. [Google Scholar] [CrossRef] - Martínez, I.A.; Roldán, E.; Dinis, L.; Petrov, D.; Rica, R.A. Adiabatic Processes Realized with a Trapped Brownian Particle. Phys. Rev. Lett.
**2015**, 114, 120601. [Google Scholar] [CrossRef] [PubMed] - Martinez, I.A.; Roldan, E.; Dinis, L.; Petrov, D.; Parrondo, J.M.R.; Rica, R.A. Brownian Carnot engine. Nat. Phys.
**2016**, 12, 67–70. [Google Scholar] [CrossRef] [PubMed] - Krishnamurthy, S.; Ghosh, S.; Chatterji, D.; Ganapathy, R.; Sood, A.K. A micrometre-sized heat engine operating between bacterial reservoirs. Nat. Phys.
**2016**, 12, 1134–1138. [Google Scholar] [CrossRef] - Wu, X.L.; Libchaber, A. Particle Diffusion in a Quasi-Two-Dimensional Bacterial Bath. Phys. Rev. Lett.
**2000**, 84, 3017–3020. [Google Scholar] [CrossRef] [PubMed] - Kanazawa, K.; Sagawa, T.; Hayakawa, H. Stochastic Energetics for Non-Gaussian Processes. Phys. Rev. Lett.
**2012**, 108, 210601. [Google Scholar] [CrossRef] [PubMed] - Kanazawa, K.; Sagawa, T.; Hayakawa, H. Heat conduction induced by non-Gaussian athermal fluctuations. Phys. Rev. E
**2013**, 87, 052124. [Google Scholar] [CrossRef] [PubMed] - Kubo, R. The fluctuation-dissipation theorem. Rep. Prog. Phys.
**1966**, 29, 255. [Google Scholar] [CrossRef] - Feynman, R.P.; Vernon, F.L. The theory of a general quantum system interacting with a linear dissipative system. Ann. Phys.
**1963**, 24, 118–173. [Google Scholar] [CrossRef] - Solon, A.; Cates, M.; Tailleur, J. Active brownian particles and run-and-tumble particles: A comparative study. Eur. Phys. J. Spec. Top.
**2015**, 224, 1231–1262. [Google Scholar] [CrossRef] - Szamel, G. Self-propelled particle in an external potential: Existence of an effective temperature. Phys. Rev. E
**2014**, 90, 012111. [Google Scholar] [CrossRef] [PubMed] - Van Kampen, N. Stochastic Processes in Physics and Chemistry; Elsevier: Amsterdam, The Netherlands, 1992; Volume 1. [Google Scholar]
- Pawula, R. Generalizations and extensions of the Fokker–Planck–Kolmogorov equations. IEEE Trans. Inf. Theory
**1967**, 13, 33–41. [Google Scholar] [CrossRef] - Popescu, D.M.; Lipan, O. A Kramers-Moyal Approach to the Analysis of Third-Order Noise with Applications in Option Valuation. PLoS ONE
**2015**, 10, e0116752. [Google Scholar] [CrossRef] [PubMed] - Fodor, E.H.; Hayakawa, J.T.; van Wijland, F. What is the role of non Gaussian noise in assemblies of self-propelled active particles? in preparation.
- Fodor, É.; Nardini, C.; Cates, M.E.; Tailleur, J.; Visco, P.; van Wijland, F. How far from equilibrium is active matter? Phys. Rev. Lett.
**2016**, 117, 038103. [Google Scholar] [CrossRef] [PubMed] - Berthier, L.; Kurchan, J. Non-equilibrium glass transitions in driven and active matter. Nat. Phys.
**2013**, 9, 310–314. [Google Scholar] [CrossRef] - Nikola, N.; Solon, A.P.; Kafri, Y.; Kardar, M.; Tailleur, J.; Voituriez, R. Active Particles with Soft and Curved Walls: Equation of State, Ratchets, and Instabilities. Phys. Rev. Lett.
**2016**, 117, 098001. [Google Scholar] [CrossRef] [PubMed] - Tailleur, J.; Cates, M.E. Sedimentation, trapping, and rectification of dilute bacteria. Eur. Lett.
**2009**, 86, 60002. [Google Scholar] [CrossRef]

**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