# Adapted or Adaptable: How to Manage Entropy Production?

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## Abstract

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## 1. Introduction

## 2. System Description

#### 2.1. Boundary Conditions

#### 2.2. Thermodynamic Device

## 3. Local Energy Conversion

#### 3.1. Presentation

#### 3.2. Entropy Production and Efficiency

## 4. Global Conversion System

#### 4.1. Presentation

#### 4.2. Devices with Zero Resting Point

#### 4.3. Devices with Non Zero Resting Point

#### 4.4. Internal Dissipation Devices

## 5. Entropic Point of View

#### 5.1. Devices with Zero Resting Point

#### 5.2. Devices with Non-Zero Resting Points

#### 5.3. Internal Dissipation Devices

## 6. Adaptable or Adapted?

## 7. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

## References

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**Figure 1.**Schematic view of the generic system, with a resource and a sink, whose potential ${\Pi}_{1}^{R}$ and ${\Pi}_{1}^{S}$ are constant. The coupling of the conversion zone (circle) with the two reservoirs is ensured by the elements ${R}_{+}$ and ${R}_{-}$. As a result, the difference potential ${\Pi}_{1}^{+}-{\Pi}_{1}^{-}$ is less than that between reservoir and sink. Power produced in the conversion zone (circle) is $P=-\alpha \Delta {\Pi}_{1}=\Delta {\Pi}_{2}$. The internal resistance ${R}_{2}=\frac{L}{A{\sigma}_{2}}$ gives rise to a dissipative contribution ${R}_{2}{I}_{2}^{2}$. The ${R}_{L}$ resistance is the output load, and the output power is ${P}_{\mathrm{out}}={R}_{L}{I}_{2}^{2}$.

**Figure 2.**Normalized efficiency $\frac{\eta}{{\eta}_{C}}$ according to reduced current $j=\alpha {J}_{2}/{J}_{1}$ with $\gamma =2$ (red, dot dashed), 4 (green, dots), 8 (blue, loosely dashed), 20 (cyan, dashed), $2\times {10}^{2}$ (magenta, loosely dot dashed) in main figure, and $\gamma =2\times {10}^{2}$ (magenta, loosely dot dashed), $2\times {10}^{3}$ (black, dot dashed), $2\times {10}^{4}$ (red, dot dot dashed) in inset. The grey area corresponds to the receptor mode (resp. generator mode). Note that the figure is symmetrical with respect to the Carnot point (blue star), which is never reached. This singular point defines the reversible configuration, where causality is broken.

**Figure 3.**Representations of the powers ${I}_{E+}$, ${I}_{E-}$, with ${R}_{1}^{-1}=0$, ${R}_{2}=0$ (and $P={P}_{\mathrm{out}}$), ${R}_{+}={R}_{-}=2$, ${\Pi}_{1}^{S}=1$, ${\Pi}_{1}^{R}=30$. (

**a**) shows efficiencies (resp. in red and green) ${\eta}_{+/-}={P}_{\mathrm{out}}/{I}_{{2}_{+/-}}$ in function of ${I}_{2}$, the current of matter. (

**b**) is the efficiency in function of the power produced ${P}_{\mathrm{out}}$. (

**c**) show the power (resp. in red, green and black) ${I}_{E+}$, ${I}_{E-}$ and ${P}_{\mathrm{out}}$ in function of ${I}_{2}$. (

**d**) shows (resp. in red, green and black) $CO{E}_{+/-}={I}_{{E}_{+/-}}/{I}_{2}$ and $CO{E}_{{P}_{\mathrm{out}}}={P}_{\mathrm{out}}/{I}_{2}$ in function of ${I}_{2}$. Dotted lines are $\alpha =0.9$, solid lines are $\alpha =1$, dashed lines are $\alpha =1.1$. In (

**c**) and (

**d**) cyan stars show short circuit situations ${I}_{2sc}$, yellow circles are critical points ${I}_{2cp}$. In (

**b**) vertical and horizontal red lines are respectively maximal efficiency and maximal power.

**Figure 4.**Representations of the powers ${I}_{E+}$, ${I}_{E-}$ and P, with ${R}_{1}^{-1}=0.05$, ${R}_{2}=0$ (and $P={P}_{\mathrm{out}}$), ${R}_{+}={R}_{-}=2$, ${\Pi}_{1}^{S}=1$, ${\Pi}_{1}^{R}=30$. (

**a**) shows efficiencies (resp. in red and green) ${\eta}_{+/-}={P}_{\mathrm{out}}/{I}_{{2}_{+/-}}$ in function of ${I}_{2}$ the current of matter; (

**b**) is the efficiency in function of the power produced ${P}_{\mathrm{out}}$; (

**c**) shows the power (resp. in red, green and black) ${I}_{E+}$, ${I}_{E-}$ and ${P}_{\mathrm{out}}$ in function of ${I}_{2}$; (

**d**) shows (resp. in red, green and black) $CO{E}_{+/-}={I}_{{E}_{+/-}}/{I}_{2}$ and $CO{E}_{{P}_{\mathrm{out}}}=P/{I}_{2}$ in a function of ${I}_{2}$. Dotted lines are $\alpha =0.9$, solid lines are $\alpha =1$, dashed lines are $\alpha =1.1$. In (

**c**) and (

**d**), cyan stars show short circuit situations ${I}_{2sc}$, and yellow circles are critical points ${I}_{2cp}$. In (

**b**), vertical and horizontal red lines are respectively maximal efficiency and maximal power.

**Figure 5.**Different representations of the powers ${I}_{E+}$, ${I}_{E-}$ and P, with ${R}_{1}^{-1}=0.05$, ${R}_{2}=4$, ${R}_{+}={R}_{-}=2$, ${\Pi}_{1}^{S}=1$, ${\Pi}_{1}^{R}=30$. (

**a**) shows efficiencies (resp. in red and green) ${\eta}_{+/-}=P/{I}_{{2}_{+/-}}$ as a function of ${I}_{2}$ the current of matter. (

**b**) is the efficiency in function of the power produced P. (

**c**) shows the power (resp. in red, green and black) ${I}_{E+}$, ${I}_{E-}$ and ${P}_{\mathrm{out}}$ in function of ${I}_{2}$. (

**d**) shows (resp. in red, green and black) $CO{E}_{+/-}={I}_{{E}_{+/-}}/{I}_{2}$ and $CO{E}_{{P}_{\mathrm{out}}}={P}_{\mathrm{out}}/{I}_{2}$ as a function of ${I}_{2}$. Dotted lines are $\alpha =0.9$, solid lines are $\alpha =1$, dashed lines are $\alpha =1.1$. In (

**c**) and (

**d**), cyan stars show short circuit situations ${I}_{2sc}$ and yellow circles are critical points ${I}_{2cp}$. In (

**b**), vertical and horizontal red lines are respectively maximal efficiency and maximal power.

**Figure 6.**Evaluation of the entropy production with the same configuration as in Figure 3, ${R}_{1}^{-1}=0$, ${R}_{2}=0$, ${R}_{+}={R}_{-}=2$, ${\Pi}_{1}^{S}=1$, ${\Pi}_{1}^{R}=30$. (

**a**) shows ${\dot{S}}_{E+}$ and (

**b**) shows ${\dot{S}}_{E-}$, both in function of ${I}_{2}$ the current of matter—the same color and line-style code as in Figure 3.

**Figure 7.**Evaluation of the entropy production with the same configuration as in Figure 4 with ${R}_{1}^{-1}=0.05$, ${R}_{2}=0$, ${R}_{+}={R}_{-}=2$, ${\Pi}_{1}^{S}=1$, ${\Pi}_{1}^{R}=30$. (

**a**) shows ${\dot{S}}_{E+}$ and (

**b**) shows ${\dot{S}}_{E-}$, both in function of ${I}_{2}$ the current of matter—the same color and line-style code as in Figure 3.

**Figure 8.**Plot of the entropy production with the same configuration as in Figure 5 with ${R}_{1}^{-1}=0.05$, ${R}_{2}=4$, ${R}_{+}={R}_{-}=2$, ${\Pi}_{1}^{S}=1$, ${\Pi}_{1}^{R}=30$. (

**a**) shows ${\dot{S}}_{E+}$ and (

**b**) shows ${\dot{S}}_{E-}$, both in function of ${I}_{2}$ the current of matter—the same color and linestyle code as in Figure 3.

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**MDPI and ACS Style**

Goupil, C.; Herbert, E.
Adapted or Adaptable: How to Manage Entropy Production? *Entropy* **2020**, *22*, 29.
https://doi.org/10.3390/e22010029

**AMA Style**

Goupil C, Herbert E.
Adapted or Adaptable: How to Manage Entropy Production? *Entropy*. 2020; 22(1):29.
https://doi.org/10.3390/e22010029

**Chicago/Turabian Style**

Goupil, Christophe, and Eric Herbert.
2020. "Adapted or Adaptable: How to Manage Entropy Production?" *Entropy* 22, no. 1: 29.
https://doi.org/10.3390/e22010029