# Detailed Fluctuation Theorems: A Unifying Perspective

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## Abstract

**:**

## 1. Introduction

## 2. Markov Jump Process

#### Example

#### Notation

## 3. General Results

#### 3.1. EP Decomposition at the Ensemble Average Level

#### 3.2. EP Decomposition at the Trajectory Level

#### 3.3. Fluctuation Theorems

#### 3.4. EP Fluctuations

#### 3.5. A Gauge Theory Perspective

## 4. Adiabatic–Nonadiabatic Decomposition

#### Additional FTs

## 5. Cycle–Cocycle Decomposition

#### Example

## 6. Stochastic Thermodynamics

#### Example

## 7. System–Reservoirs Decomposition

## 8. Conservative–Nonconservative Decomposition

#### Example

## 9. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DFT | detailed fluctuation theorem |

IFT | integral fluctuation theorem |

PMF | probability mass function |

EP | entropy production |

ME | master equation |

MGF | moment generating function |

## Appendix A. Moment Generating Function Dynamics and Proofs of the FTs

#### Appendix A.1. MGF Dynamics

#### Appendix A.2. Proof of the DFT

#### Appendix A.3. Proof of the DFT for the Sum of Driving and Nonconservative EP

#### Appendix A.4. Proof of the IFT

## Appendix B. Alternative Proofs of the DFT

#### Appendix B.1. Alternative Proof 1

#### Appendix B.2. Alternative Proof 2

## Appendix C. Adiabatic and Nonadiabatic Contributions

## Appendix D. Proofs of the DFTs for the Adiabatic and Driving EP Contributions

#### Appendix D.1. Proof of the DFT for the Adiabatic Contribution

#### Appendix D.2. Proof of the DFT for the Driving Contribution

## References

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**Figure 2.**Schematic representation of the forward and backward processes related by our detailed fluctuation theorem (DFT).

**Figure 4.**Pictorial representation of a system coupled to several reservoirs. Transitions may involve more than one reservoir and exchange between reservoirs. Work reservoirs are also taken into account.

**Figure 5.**Double coupled quantum dot (QD) in contact with three reservoirs. Transitions related to the first reservoir are depicted using solid lines, while those related to the second and third ones using dashed and dotted lines, respectively. The graphical rule was applied to the network of transitions in Figure 1. (

**a**) Pictorial representation of the system. The upper dot $\mathrm{u}$ is in contact with the first reservoir, while the lower dot $\mathrm{d}$ with the second and third reservoirs. Energy and electrons are exchanged, but the dots cannot host more than one electron. (

**b**) Energy landscape of the dot. When both dots are occupied, 11, a repulsive energy u adds to the occupied dots energies, ${\u03f5}_{\mathrm{u}}$ and ${\u03f5}_{\mathrm{d}}$.

**Table 1.**Summary of the reference potentials, affinities, and conservative EP contributions for the specific references discussed in the text. The nonconservative EP contribution follows from $\langle {\underset{\Sigma}{\dot{}}}_{\mathrm{nc}}\rangle ={A}_{e}^{\mathrm{ref}}\langle {j}^{e}\rangle $, whereas the driving one from $\langle {\underset{\Sigma}{\dot{}}}_{\mathrm{d}}\rangle ={\textstyle {\sum}_{n}}{p}_{n}{\mathrm{d}}_{t}{\psi}_{n}^{\mathrm{ref}}$. Overall, $\langle \underset{\Sigma}{\dot{}}\rangle =\langle {\underset{\Sigma}{\dot{}}}_{\mathrm{nc}}\rangle +\langle {\underset{\Sigma}{\dot{}}}_{\mathrm{c}}\rangle =\langle {\underset{\Sigma}{\dot{}}}_{\mathrm{nc}}\rangle +\langle {\underset{\Sigma}{\dot{}}}_{\mathrm{d}}\rangle -{\mathrm{d}}_{t}\mathcal{D}(p\parallel {p}^{\mathrm{ref}})$, where $\mathcal{D}$ is the relative entropy.

Decomposition | ${\mathit{\psi}}_{\mathit{n}}^{\mathrm{ref}}$ | ${\mathit{A}}_{\mathit{e}}^{\mathrm{ref}}$ | $\langle {\underset{\Sigma}{\dot{}}}_{\mathrm{c}}\rangle $ |
---|---|---|---|

adiabatic-nonadiabatic | $-\mathrm{ln}{p}_{n}^{\mathrm{ss}}$ | $\mathrm{ln}\frac{{w}_{e}{p}_{o(e)}^{\mathrm{ss}}}{{w}_{-e}{p}_{o(-e)}^{\mathrm{ss}}}$ | $-\langle {j}^{e}\rangle {D}_{e}^{n}\mathrm{ln}\left\{{p}_{n}/{p}_{n}^{\mathrm{ss}}\right\}$ |

cycle–cocycle | $-\mathrm{ln}\left\{{\Pi}_{e\in {\mathcal{T}}_{n}}{w}_{e}-Z\right\}$ | $\{\begin{array}{cc}0,\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}e\in \mathcal{T},\hfill \\ {\mathcal{A}}_{e},\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}e\in {\mathcal{T}}^{*}\hfill \end{array}$ | ${\Sigma}_{e\in \mathcal{T}}\langle {j}_{e}\rangle {A}_{e}$ |

system–reservoir | ${S}_{\mathrm{mc}}-{S}_{n}$ | $\delta {S}_{e}^{\mathrm{r}}=-{f}_{y}\delta {X}_{e}^{y}$ | $[{S}_{n}-\mathrm{ln}{p}_{n}]{D}_{e}^{n}\langle {j}^{e}\rangle $ |

conservative–nonconservative | ${\mathsf{\Phi}}_{\mathrm{gg}}-[{S}_{n}-{F}_{y}{L}_{n}^{\lambda}]$ | ${\mathcal{F}}_{{y}_{\mathrm{f}}}\delta {X}_{e}^{{y}_{\mathrm{f}}}$ | $[{S}_{n}-{F}_{\lambda}{L}_{n}^{\lambda}-\mathrm{ln}{p}_{n}]{D}_{e}^{n}\langle {j}^{e}\rangle $ |

**Table 2.**Examples of system quantity–intensive field conjugated pairs in the entropy representation. ${\beta}_{r}:=1/{T}_{r}$ denotes the inverse temperature of the reservoir. Since charges are carried by particles, the conjugated pair $({Q}_{n},-{\beta}_{r}{V}_{r})$ is usually embedded in $({N}_{n},-{\beta}_{r}{\mu}_{r})$.

System Quantity ${\mathit{X}}^{\mathit{\kappa}}$ | Intensive Field ${\mathit{f}}_{(\mathit{\kappa},\mathit{r})}$ |
---|---|

energy, ${E}_{n}$ | inverse temperature, ${\beta}_{r}$ |

particles number, ${N}_{n}$ | chemical potential, $-{\beta}_{r}{\mu}_{r}$ |

charge, ${Q}_{n}$ | electric potential, $-{\beta}_{r}{V}_{r}$ |

displacement, ${X}_{n}$ | generic force, $-{\beta}_{r}{k}_{r}$ |

angle, ${\theta}_{n}$ | torque, $-{\beta}_{r}{\tau}_{r}$ |

Index | Label for | Number |
---|---|---|

n | state | ${\mathsf{N}}_{\mathrm{n}}$ |

e | transition | ${\mathsf{N}}_{\mathrm{e}}$ |

$\kappa $ | system quantity | ${\mathsf{N}}_{\kappa}$ |

r | reservoir | ${\mathsf{N}}_{\mathrm{r}}$ |

$y\equiv (\kappa ,r)$ | conserved quantity ${X}^{\kappa}$ from reservoir r | ${\mathsf{N}}_{\mathrm{y}}$ |

$\lambda $ | conservation law and conserved quantity | ${\mathsf{N}}_{\lambda}$ |

${y}_{\mathrm{p}}$ | “potential” y | ${\mathsf{N}}_{\lambda}$ |

${y}_{\mathrm{f}}$ | “force” y | ${\mathsf{N}}_{\mathrm{y}}-{\mathsf{N}}_{\lambda}$ |

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Rao, R.; Esposito, M.
Detailed Fluctuation Theorems: A Unifying Perspective. *Entropy* **2018**, *20*, 635.
https://doi.org/10.3390/e20090635

**AMA Style**

Rao R, Esposito M.
Detailed Fluctuation Theorems: A Unifying Perspective. *Entropy*. 2018; 20(9):635.
https://doi.org/10.3390/e20090635

**Chicago/Turabian Style**

Rao, Riccardo, and Massimiliano Esposito.
2018. "Detailed Fluctuation Theorems: A Unifying Perspective" *Entropy* 20, no. 9: 635.
https://doi.org/10.3390/e20090635