Finite-time thermodynamics: A journey beginning with optimizing heat engines

Xiu-Hua Zhao; Yu-Han Ma

Finite-time thermodynamics: A journey beginning with optimizing heat engines

T0 review · 0 major / 2 minor · reviewed 2026-05-23 · grok-4.3

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Pith's one-line read Finite-time thermodynamics has developed from heat engine optimization into a framework covering cycle constraints, process control, unconventional engines, and experiments.

desk verdict This is a review paper that organizes existing literature on finite-time thermodynamics but adds no new derivations or results. read the letter →

arxiv 2411.03853 v1 pith:4MESMTGQ submitted 2024-11-06 cond-mat.stat-mech physics.app-phphysics.class-phquant-ph

Yu-Han Ma , Xiu-Hua Zhao This is my paper · ORCID

classification cond-mat.stat-mechphysics.app-phphysics.class-phquant-ph

keywords finite-timethermodynamicsheatenginesthermodynamiccyclesoptimalcontrolunconventionalexperimentalprogressoptimization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper traces the origins of finite-time thermodynamics to early efforts at optimizing heat engines that must run in limited time. It then reviews research from the past two decades on the basic limits that finite operation times impose on thermodynamic cycles. The work examines techniques for optimal control and optimization of processes, the performance of nonstandard heat engines, and experimental realizations of these ideas. A reader would care because the results apply directly to the design of real engines and devices that cannot operate at equilibrium.

What carries the argument

Finite-time thermodynamics, the approach that incorporates the effects of limited operation time on thermodynamic cycles and processes.

What would settle it

Discovery of major recent results on finite-time cycle constraints or experimental engine performance that are absent from the review would show the coverage is incomplete.

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Extended reading notes

Core claim

The paper establishes a historical narrative of finite-time thermodynamics beginning with heat engine optimization and surveys recent advances on the fundamental constraints of finite-time cycles, optimal control methods for thermodynamic processes, the operation of unconventional heat engines, and experimental progress in the field.

Load-bearing premise

The selection of literature and topics over the past two decades is representative of the full scope and key advances in the field without major omissions or bias in coverage.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

This paper is a review that traces the start of finite-time thermodynamics from heat engine optimization and then summarizes work from the past two decades on cycle constraints, optimal control, unconventional engines, and experiments. It presents no original math, data, or first-principles claims. The abstract makes that explicit, and the structure follows the standard pattern of collecting and grouping prior results. That is the main thing to know: it is a consolidation exercise, not an advance. What it does reasonably well is lay out the historical thread and separate the topics into clear sections. A reader who wants a single place to see how the field moved from early bounds to later optimization studies and lab work can get that overview without hunting through many papers. The organization itself is straightforward and follows the topics listed in the abstract. The soft spots are the usual ones for any review. Coverage depends entirely on the authors' choices of what counts as key work over twenty years. Without an explicit statement of search criteria or a discussion of why certain lines were left out, it is hard to know if the summary is balanced or if important constraints and experiments were missed. The paper also does not attempt to resolve open questions or introduce a new organizing principle, so its value stays at the level of a literature map rather than a synthesis that changes how people think about the problems. This paper is for someone entering the subfield or needing a quick map of recent activity. A specialist already working on these cycles will not find new technical content to use. It shows clear enough thinking in how it groups the material, with no internal contradictions visible from the abstract. I would bring it to a reading group only if the group is already focused on finite-time thermodynamics and wants to discuss coverage gaps. I would not cite it in my own papers because it contains no new results to build on. A serious editor should send it to peer review so referees can check whether the literature selection is representative and whether the summaries of the cited papers are accurate. Reviews that are thorough can save time for the community even when they do not break new ground.

Referee Report

0 major / 2 minor

Summary. The manuscript is a review paper that summarizes the historical development of finite-time thermodynamics, beginning with the optimization of heat engines, and surveys research over the past two decades. It focuses on fundamental constraints of finite-time thermodynamic cycles, optimal control and optimization of thermodynamic processes, the operation of unconventional heat engines, and experimental progress.

Significance. As a review without new derivations, models, or data, the manuscript's value would lie in providing a consolidated overview and historical framing for the cond-mat.stat-mech community. Credit is due for the explicit scope statement in the abstract and the choice to anchor the narrative in the classic heat-engine optimization problem, which is a standard and falsifiable entry point for the field.

minor comments (2)

[Abstract] The abstract states the four focus areas but does not indicate the approximate number of references or the cutoff date for the 'past two decades' coverage; adding one sentence would improve clarity for readers.
[Title] The title uses 'a journey beginning with' phrasing that is informal for a journal review; a more direct subtitle such as 'A review of constraints, optimization, and experiments' would align better with conventional review titles.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their thorough reading and positive evaluation of the manuscript. The report correctly identifies the paper as a review without new derivations and notes its value in providing a consolidated overview anchored in the classic heat-engine problem. We appreciate the recommendation to accept.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity in review paper

full rationale

This is a review paper that summarizes historical development and existing literature on finite-time thermodynamics without presenting new derivations, models, predictions, or fitted parameters. No load-bearing steps exist that reduce by construction to the paper's own inputs, self-citations, or ansatzes. External citations support the summary and do not form a self-referential chain for original claims. The paper is self-contained as a literature review against external benchmarks.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

This is a review paper whose central contribution is the accuracy and completeness of its literature summary. No new free parameters, axioms, or invented entities are introduced by the authors themselves; any such elements would reside in the reviewed primary literature.

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Cite this review

Pith. "Pith review of Finite-time thermodynamics: A journey beginning with optimizing heat engines." pith.science (2026). https://pith.science/paper/4MESMTGQ

@misc{pith2026241103853,
  author       = {Pith},
  title        = {Pith review of: Finite-time thermodynamics: A journey beginning with optimizing heat engines},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4MESMTGQ}},
  note         = {Machine review of arXiv:2411.03853}
}

read the original abstract

In this paper, we summarize the historical development of finite-time thermodynamics and review the current state of research over the past two decades in this field, focusing on fundamental constraints of finite-time thermodynamic cycles, optimal control and optimization of thermodynamic processes, the operation of unconventional heat engines, and experimental progress.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Revisiting Endo-reversible Carnot engine: Extending the Yvon engine
cond-mat.stat-mech 2024-12 unverdicted novelty 4.0 of 10

The extended Yvon engine is equivalent to the Curzon-Ahlborn engine as the steady-state and cyclic forms of the endo-reversible Carnot heat engine.

Reference graph

Works this paper leans on

131 extracted references · 131 canonical work pages · cited by 1 Pith paper

[1]

Smil, Energy and Civilization: A History (The MIT Press, Cambridge, 2017)

V. Smil, Energy and Civilization: A History (The MIT Press, Cambridge, 2017)

work page 2017
[2]

Carnot, R´ eﬂexions sur la puissance motrice du feu et sur les machines propres a d´ evelopper cette puissance, Annales scientiﬁques de l’ ´Ecole Normale Sup´ erieure1, 393 (1872)

S. Carnot, R´ eﬂexions sur la puissance motrice du feu et sur les machines propres a d´ evelopper cette puissance, Annales scientiﬁques de l’ ´Ecole Normale Sup´ erieure1, 393 (1872)

work page
[3]

Carnot, Reﬂections on the Motive Power of Heat: From the Original French of N.-L.-S

S. Carnot, Reﬂections on the Motive Power of Heat: From the Original French of N.-L.-S. Carnot (Wiley, New York, 1897)

work page
[4]

H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed. (Wiley, New York, 1991)

work page 1991
[5]

Bejan, Advanced Engineering Thermodynamics , 4th ed

A. Bejan, Advanced Engineering Thermodynamics , 4th ed. (Wiley, Hoboken, 2016)

work page 2016
[6]

Andresen, P

B. Andresen, P. Salamon, and R. S. Berry, Thermo- dynamics in ﬁnite time: Extremals for imperfect heat engines, J. Chem. Phys. 66, 1571 (1977)

work page 1977
[7]

Yvon, Saclay Reactor: Acquired Knowledge by Two Years Experience in Heat Transfer Using Compressed Gas, Tech

J. Yvon, Saclay Reactor: Acquired Knowledge by Two Years Experience in Heat Transfer Using Compressed Gas, Tech. Rep. (CEA Saclay, France, 1955)

work page 1955
[8]

Chambadal, Les centrales nucl´ eaires(Armand Colin, Paris, 1957)

P. Chambadal, Les centrales nucl´ eaires(Armand Colin, Paris, 1957)

work page 1957

Show all 131 references

[9]

I. I. Novikov, Eﬃciency of an atomic power generating installation, The Soviet Journal of Atomic Energy 3, 1269 (1957)

1957
[10]

F. L. Curzon and B. Ahlborn, Eﬃciency of a carnot engine at maximum power output, Am. J. Phys. 43, 22 (1975)

1975
[11]

R. S. Berry, P. Salamon, and B. Andresen, eds., Finite-Time Thermodynamics (MDPI - Multidisci- plinary Digital Publishing Institute, 2022)

2022
[12]

De Vos, Eﬃciency of some heat engines at maximum- power conditions, Am

A. De Vos, Eﬃciency of some heat engines at maximum- power conditions, Am. J. Phys. 53, 570 (1985)

1985
[13]

Sahin, A

B. Sahin, A. Kodal, and H. Yavuz, Eﬃciency of a joule-brayton engine at maximum power density, J. Phys. D: Appl. Phys. 28, 1309 (1995)

1995
[14]

Feldmann, E

T. Feldmann, E. Geva, R. Kosloﬀ, and P. Salamon, Heat engines in ﬁnite time governed by master equa- tions, Am. J. Phys. 64, 485 (1996)

1996
[15]

L. Chen, C. Wu, F. Sun, and S. Cao, Heat transfer eﬀects on the net work output and eﬃ- ciency characteristics for an air-standard otto cycle, Energy Conv. Manag. 39, 643 (1998)

1998
[16]

S. C. Kaushik and S. Kumar, Finite time thermody- namic analysis of endoreversible stirling heat engine with regenerative losses, Energy 25, 989 (2000) . 5

2000
[17]

Kosloﬀ, A quantum mechanical open system as a model of a heat engine, J

R. Kosloﬀ, A quantum mechanical open system as a model of a heat engine, J. Chem. Phys. 80, 1625 (1984)

1984
[18]

Van den Broeck, Thermodynamic eﬃciency at max- imum power, Phys

C. Van den Broeck, Thermodynamic eﬃciency at max- imum power, Phys. Rev. Lett. 95, 190602 (2005)

2005
[19]

Schmiedl and U

T. Schmiedl and U. Seifert, Eﬃciency at maximum power: An analytically solvable model for stochastic heat engines, Europhys. Lett. 81, 20003 (2007)

2007
[20]

Z. C. Tu, Eﬃciency at maximum power of feynman’s ratchet as a heat engine, J. Phys. A: Math. Theor. 41, 312003 (2008)

2008
[21]

A. E. Allahverdyan, R. S. Johal, and G. Mahler, Work extremum principle: Structure and function of quantum heat engines, Phys. Rev. E 77, 041118 (2008)

2008
[22]

Esposito, R

M. Esposito, R. Kawai, K. Lindenberg, and C. Van den Broeck, Eﬃciency at maximum power of low-dissipation carnot engines, Phys. Rev. Lett. 105, 150603 (2010)

2010
[23]

Izumida and K

Y. Izumida and K. Okuda, Eﬃciency at maximum power of minimally nonlinear irreversible heat engines, Europhys. Lett. 97, 10004 (2012)

2012
[24]

Sheng and Z

S. Sheng and Z. C. Tu, Constitutive relation for nonlinear response and universality of eﬃciency at maximum power for tight-coupling heat engines, Phys. Rev. E 91, 022136 (2015)

2015
[25]

Chen, C.-P

J.-F. Chen, C.-P. Sun, and H. Dong, Achieve higher eﬃciency at maximum power with ﬁnite-time quantum otto cycle, Phys. Rev. E 100, 062140 (2019)

2019
[26]

Shiraishi, K

N. Shiraishi, K. Saito, and H. Tasaki, Universal trade-oﬀ relation between power and eﬃciency for heat engines, Phys. Rev. Lett. 117, 190601 (2016)

2016
[27]

Chen and Z

L. Chen and Z. Yan, The eﬀect of heat-transfer law on performance of a two-heat-source endoreversible cycle, J. Chem. Phys. 90, 3740 (1989)

1989
[28]

Holubec and A

V. Holubec and A. Ryabov, Maximum eﬃciency of low-dissipation heat engines at arbitrary power, J. Stat. Mech.-Theory Exp. 2016, 073204 (2016)

2016
[29]

Y.-H. Ma, D. Xu, H. Dong, and C.-P. Sun, Universal constraint for eﬃciency and power of a low-dissipation heat engine, Phys. Rev. E 98, 042112 (2018)

2018
[30]

Abiuso and M

P. Abiuso and M. Perarnau-Llobet, Opti- mal Cycles for Low-Dissipation Heat Engines, Phys. Rev. Lett. 124, 110606 (2020)

2020
[31]

Brandner and K

K. Brandner and K. Saito, Thermodynamic Geometry of Microscopic Heat Engines, Phys. Rev. Lett. 124, 040602 (2020)

2020
[32]

Pietzonka and U

P. Pietzonka and U. Seifert, Universal Trade-Oﬀ be- tween Power, Eﬃciency, and Constancy in Steady-State Heat Engines, Phys. Rev. Lett. 120, 190602 (2018)

2018
[33]

Holubec and A

V. Holubec and A. Ryabov, Cycling Tames Power Fluctuations near Optimum Eﬃciency, Phys. Rev. Lett. 121, 120601 (2018)

2018
[34]

H. J. D. Miller, M. H. Mohammady, M. Perarnau- Llobet, and G. Guarnieri, Thermodynamic Uncertainty Relation in Slowly Driven Quantum Heat Engines, Phys. Rev. Lett. 126, 210603 (2021)

2021
[35]

Y.-H. Ma, D. Xu, H. Dong, and C.-P. Sun, Optimal op- erating protocol to achieve eﬃciency at maximum power of heat engines, Phys. Rev. E 98, 022133 (2018)

2018
[36]

Y. H. Chen, J.-F. Chen, Z. Fei, and H. T. Quan, Microscopic theory of the Curzon-Ahlborn heat engine based on a Brownian particle, Phys. Rev. E 106, 024105 (2022)

2022
[37]

Salamon and R

P. Salamon and R. S. Berry, Thermody- namic Length and Dissipated Availability, Phys. Rev. Lett. 51, 1127 (1983)

1983
[38]

G. E. Crooks, Measuring Thermodynamic Length, Phys. Rev. Lett. 99, 100602 (2007)

2007
[39]

D. A. Sivak and G. E. Crooks, Ther- modynamic Metrics and Optimal Paths, Phys. Rev. Lett. 108, 190602 (2012)

2012
[40]

P. R. Zulkowski, D. A. Sivak, G. E. Crooks, and M. R. DeWeese, Geometry of thermodynamic control, Phys. Rev. E 86, 041148 (2012)

2012
[41]

Li and Z

G. Li and Z. Tu, Nonequilibrium work rela- tions meet engineered thermodynamic control: A perspective for nonequilibrium measurements, Europhys. Lett. 142, 61001 (2023)

2023
[42]

Gu´ ery-Odelin, C

D. Gu´ ery-Odelin, C. Jarzynski, C. A. Plata, A. Prados, and E. Trizac, Driving rapidly while remaining in con- trol: Classical shortcuts from Hamiltonian to stochastic dynamics, Rep. Prog. Phys. 86, 035902 (2023)

2023
[43]

Van Vu and Y

T. Van Vu and Y. Hasegawa, Geometrical Bounds of the Irreversibility in Markovian Systems, Phys. Rev. Lett. 126, 010601 (2021)

2021
[44]

Wang and J

Z. Wang and J. Ren, Thermodynamic geometry of nonequilibrium ﬂuctuations in cyclically driven trans- port, Phys. Rev. Lett. 132, 207101 (2024)

2024
[45]

Zhong and M

A. Zhong and M. R. DeWeese, Beyond Lin- ear Response: Equivalence between Thermo- dynamic Geometry and Optimal Transport, Phys. Rev. Lett. 133, 057102 (2024)

2024
[46]

Ma, R.-X

Y.-H. Ma, R.-X. Zhai, J. Chen, C. P. Sun, and H. Dong, Experimental Test of the 1 /τ -Scaling En- tropy Generation in Finite-Time Thermodynamics, Phys. Rev. Lett. 125, 210601 (2020)

2020
[47]

Nakazato and S

M. Nakazato and S. Ito, Geometrical aspects of entropy production in stochastic thermodynamics based on wasserstein distance, Phys. Rev. Res. 3, 043093 (2021)

2021
[48]

Schmiedl and U

T. Schmiedl and U. Seifert, Optimal ﬁnite- time processes in stochastic thermodynamics, Phys. Rev. Lett. 98, 108301 (2007)

2007
[49]

Gomez-Marin, T

A. Gomez-Marin, T. Schmiedl, and U. Seifert, Optimal protocols for minimal work processes in underdamped stochastic thermodynamics, J. Chem. Phys. 129, 024114 (2008)

2008
[50]

Aurell, C

E. Aurell, C. Mej ´ ıa-Monasterio, and P. Muratore- Ginanneschi, Optimal protocols and opti- mal transport in stochastic thermodynamics, Phys. Rev. Lett. 106, 250601 (2011)

2011
[51]

P. R. Zulkowski and M. R. DeWeese, Optimal control of overdamped systems, Phys. Rev. E 92, 032117 (2015)

2015
[52]

M. C. Engel, J. A. Smith, and M. P. Brenner, Optimal control of nonequilibrium systems through automatic diﬀerentiation, Phys. Rev. X 13, 041032 (2023)

2023
[53]

G. Li, H. T. Quan, and Z. C. Tu, Shortcuts to isothermality and nonequilibrium work relations, Phys. Rev. E 96, 012144 (2017)

2017
[54]

Li, J.-F

G. Li, J.-F. Chen, C. P. Sun, and H. Dong, Geodesic Path for the Minimal Energy Cost in Shortcuts to Isothermality, Phys. Rev. Lett. 128, 230603 (2022)

2022
[55]

J. A. C. Albay, P.-Y. Lai, and Y. Jun, Re- alization of ﬁnite-rate isothermal compression and expansion using optical feedback trap, Appl. Phys. Lett. 116, 103706 (2020)

2020
[56]

Chen, Optimizing brownian heat engine with shortcut strategy, Phys

J.-F. Chen, Optimizing brownian heat engine with shortcut strategy, Phys. Rev. E 106, 054108 (2022)

2022
[57]

Zhao, Z.-N

X.-H. Zhao, Z.-N. Gong, and Z. C. Tu, Low-dissipation engines: Microscopic construction via shortcuts to adia- 6 baticity and isothermality, the optimal relation between power and eﬃciency, Phys. Rev. E 106, 064117 (2022)

2022
[58]

R. Dann, A. Tobalina, and R. Kosloﬀ, Short- cut to Equilibration of an Open Quantum System, Phys. Rev. Lett. 122, 250402 (2019)

2019
[59]

Pancotti, M

N. Pancotti, M. Scandi, M. T. Mitchison, and M. Perarnau-Llobet, Speed-Ups to Isothermal- ity: Enhanced Quantum Thermal Machines through Control of the System-Bath Coupling, Phys. Rev. X 10, 031015 (2020)

2020
[60]

Ma, Simple realization of the poly- tropic process with a ﬁnite-sized reservoir, Am

Y.-H. Ma, Simple realization of the poly- tropic process with a ﬁnite-sized reservoir, Am. J. Phys. 91, 555 (2023)

2023
[61]

M. J. Ondrechen, B. Andresen, M. Mozurkewich, and R. S. Berry, Maximum work from a ﬁnite reservoir by sequential Carnot cycles, Am. J. Phys. 49, 681 (1981)

1981
[62]

R. S. Johal and R. Rai, Near-equilibrium uni- versality and bounds on eﬃciency in quasi- static regime with ﬁnite source and sink, Europhys. Lett. 113, 10006 (2016)

2016
[63]

Izumida and K

Y. Izumida and K. Okuda, Work Output and Eﬃ- ciency at Maximum Power of Linear Irreversible Heat Engines Operating with a Finite-Sized Heat Source, Phys. Rev. Lett. 112, 180603 (2014)

2014
[64]

Ma, Eﬀect of Finite-Size Heat Source’s Heat Capacity on the Eﬃciency of Heat Engine, Entropy 22, 1002 (2020)

Y.-H. Ma, Eﬀect of Finite-Size Heat Source’s Heat Capacity on the Eﬃciency of Heat Engine, Entropy 22, 1002 (2020)

2020
[65]

Yuan, Y.-H

H. Yuan, Y.-H. Ma, and C. P. Sun, Optimizing thermodynamic cycles with two ﬁnite-sized reservoirs, Phys. Rev. E 105, L022101 (2022)

2022
[66]

Quan, Maximum eﬃciency of ideal heat engines based on a small system: Correc- tion to the carnot eﬃciency at the nanoscale, Phys

H. Quan, Maximum eﬃciency of ideal heat engines based on a small system: Correc- tion to the carnot eﬃciency at the nanoscale, Phys. Rev. E 89, 062134 (2014)

2014
[67]

Fei and Y.-H

Z. Fei and Y.-H. Ma, Temperature ﬂuctuations in meso- scopic systems, Physical Review E 109, 044101 (2024)

2024
[68]

Niedenzu and G

W. Niedenzu and G. Kurizki, Cooperative many- body enhancement of quantum thermal machine power, New J. Phys. 20, 113038 (2018)

2018
[69]

Vroylandt, M

H. Vroylandt, M. Esposito, and G. Verley, Col- lective eﬀects enhancing power and eﬃciency, Europhys. Lett. 120, 30009 (2018)

2018
[70]

C. L. Latune, I. Sinayskiy, and F. Petruccione, Collective heat capacity for quantum ther- mometry and quantum engine enhancements, New J. Phys. 22, 083049 (2020)

2020
[71]

F. S. Filho, G. A. L. For˜ ao, D. M. Busiello, B. Cleuren, and C. E. Fiore, Powerful ordered collective heat en- gines, Phys. Rev. Res. 5, 043067 (2023)

2023
[72]

Rolandi, P

A. Rolandi, P. Abiuso, and M. Perarnau-Llobet, Col- lective Advantages in Finite-Time Thermodynamics, Phys. Rev. Lett. 131, 210401 (2023)

2023
[73]

Campisi and R

M. Campisi and R. Fazio, The power of a critical heat engine, Nat. Commun. 7, 11895 (2016)

2016
[74]

Holubec and A

V. Holubec and A. Ryabov, Work and power ﬂuctuations in a critical heat engine, Phys. Rev. E 96, 030102 (2017)

2017
[75]

Ma, S.-H

Y.-H. Ma, S.-H. Su, and C.-P. Sun, Quantum ther- modynamic cycle with quantum phase transition, Phys. Rev. E 96, 022143 (2017)

2017
[76]

Liang, Y.-H

S. Liang, Y.-H. Ma, D. M. Busiello, and P. D. L. Rios, A minimal model for carnot eﬃciency at maximum power, arXiv preprint arXiv:2312.02323 10.48550/arXiv.2312.02323 (2023)

2023 doi
[77]

Verley, M

G. Verley, M. Esposito, T. Willaert, and C. Van den Broeck, The unlikely carnot eﬃciency, Nat. commun. 5, 4721 (2014)

2014
[78]

Denzler and E

T. Denzler and E. Lutz, Eﬃciency ﬂuc- tuations of a quantum heat engine, Phys. Rev. Res. 2, 032062 (2020)

2020
[79]

Fei, J.-F

Z. Fei, J.-F. Chen, and Y.-H. Ma, Eﬃ- ciency statistics of a quantum otto cycle, Phys. Rev. A 105, 022609 (2022)

2022
[80]

M. O. Scully, M. S. Zubairy, G. S. Agarwal, and H. Walther, Extracting Work from a Sin- gle Heat Bath via Vanishing Quantum Coherence, Science 299, 862 (2003)

2003
[81]

H. Quan, Y. Wang, Y.-x. Liu, C. Sun, and F. Nori, Maxwell’s demon assisted thermody- namic cycle in superconducting quantum circuits, Phys. Rev. Lett. 97, 180402 (2006)

2006
[82]

P. A. Camati, J. F. G. Santos, and R. M. Serra, Co- herence eﬀects in the performance of the quantum Otto heat engine, Phys. Rev. A 99, 062103 (2019)

2019
[83]

Y.-H. Ma, C. Liu, and C. Sun, Works with quantum resource of coherence, arXiv preprint arXiv:2110.04550 10.48550/arXiv.2110.04550 (2021)

2021 doi
[84]

Aimet and H

S. Aimet and H. Kwon, Engineering a heat engine purely driven by quantum coherence, Phys. Rev. A 107, 012221 (2023)

2023
[85]

Pietzonka, ´E

P. Pietzonka, ´E. Fodor, C. Lohrmann, M. E. Cates, and U. Seifert, Autonomous engines driven by active matter: Energetics and design principles, Phys. Rev. X 9, 041032 (2019)

2019
[86]

Kumari, P

A. Kumari, P. S. Pal, A. Saha, and S. Lahiri, Stochastic heat engine using an active particle, Phys. Rev. E 101, 032109 (2020)

2020
[87]

J. S. Lee, J.-M. Park, and H. Park, Brow- nian heat engine with active reservoirs, Phys. Rev. E 102, 032116 (2020)

2020
[88]

Fodor and M

´E. Fodor and M. E. Cates, Active en- gines: Thermodynamics moves forward, Europhysics Letters 134, 10003 (2021)

2021
[89]

Y. Wang, E. Lei, Y.-H. Ma, Z. Tu, and G. Li, Thermody- namic geometric control of active matter, arXiv preprint arXiv:2409.09994 10.48550/arXiv.2409.09994 (2024)

2024 doi
[90]

Benenti, K

G. Benenti, K. Saito, and G. Casati, Ther- modynamic Bounds on Eﬃciency for Sys- tems with Broken Time-Reversal Symmetry, Phys. Rev. Lett. 106, 230602 (2011)

2011
[91]

Brandner, K

K. Brandner, K. Saito, and U. Seifert, Strong Bounds on Onsager Coeﬃcients and Eﬃciency for Three- Terminal Thermoelectric Transport in a Magnetic Field, Phys. Rev. Lett. 110, 070603 (2013)

2013
[92]

A. E. Allahverdyan, K. V. Hovhannisyan, A. V. Melkikh, and S. G. Gevorkian, Carnot Cycle at Finite Power: Attainability of Maximal Eﬃciency, Phys. Rev. Lett. 111, 050601 (2013)

2013
[93]

Zhai, F.-M

R.-X. Zhai, F.-M. Cui, Y.-H. Ma, C. P. Sun, and H. Dong, Experimental test of power- eﬃciency trade-oﬀ in a ﬁnite-time Carnot cycle, Phys. Rev. E 107, L042101 (2023)

2023
[94]

Blickle and C

V. Blickle and C. Bechinger, Realization of a micrometre-sized stochastic heat engine, Nat. Phys. 8, 143 (2012)

2012
[95]

Roßnagel, S

J. Roßnagel, S. T. Dawkins, K. N. Tolazzi, O. Abah, E. Lutz, F. Schmidt-Kaler, and K. Singer, A single-atom 7 heat engine, Science 352, 325 (2016)

2016
[96]

I. A. Mart ´ ınez,´E. Rold´ an, L. Dinis, D. Petrov, J. M. R. Parrondo, and R. A. Rica, Brownian Carnot engine, Nat. Phys. 12, 67 (2016)

2016
[97]

Krishnamurthy, S

S. Krishnamurthy, S. Ghosh, D. Chatterji, R. Gana- pathy, and A. K. Sood, A micrometre-sized heat engine operating between bacterial reservoirs, Nat. Phys. 12, 1134 (2016)

2016
[98]

Liphardt, S

J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, and C. Bustamante, Equilibrium Information from Nonequi- librium Measurements in an Experimental Test of Jarzynski’s Equality, Science 296, 1832 (2002)

2002
[99]

G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, Experimental Demonstra- tion of Violations of the Second Law of Thermody- namics for Small Systems and Short Time Scales, Phys. Rev. Lett. 89, 050601 (2002)

2002
[100]

E. H. Trepagnier, C. Jarzynski, F. Ritort, G. E. Crooks, C. J. Bustamante, and J. Liphardt, Experimental test of hatano and sasa’s nonequilibrium steady-state equality, Proc. Natl. Acad. Sci. U.S.A. 101, 15038 (2004)

2004
[101]

Collin, F

D. Collin, F. Ritort, C. Jarzynski, S. B. Smith, I. Tinoco, and C. Bustamante, Veriﬁcation of the crooks ﬂuctuation theorem and recovery of rna folding free en- ergies, Nature 437, 231 (2005)

2005
[102]

An, J.-N

S. An, J.-N. Zhang, M. Um, D. Lv, Y. Lu, J. Zhang, Z.-Q. Yin, H. T. Quan, and K. Kim, Experimental test of the quantum Jarzynski equality with a trapped-ion system, Nat. Phys. 11, 193 (2015)

2015
[103]

Ciliberto, Experiments in stochastic ther- modynamics: Short history and perspectives, Phys

S. Ciliberto, Experiments in stochastic ther- modynamics: Short history and perspectives, Phys. Rev. X 7, 021051 (2017)

2017
[104]

Y.-H. Ma, C. Sun, and H. Dong, Consistency of optimizing ﬁnite-time carnot engines with the low- dissipation model in the two-level atomic heat engine, Commun. Theor. Phys. 73, 125101 (2021)

2021
[105]

Cavina, P

V. Cavina, P. A. Erdman, P. Abiuso, L. Tolomeo, and V. Giovannetti, Maximum-power heat en- gines and refrigerators in the fast-driving regime, Phys. Rev. A 104, 032226 (2021)

2021
[106]

Wang and D

D. Wang and D. Xu, Nonadiabatic evolution and ther- modynamics of a time-dependent open quantum system, Phys. Rev. A 104, 032201 (2021)

2021
[107]

B´ erut, A

A. B´ erut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, and E. Lutz, Experimental veriﬁ- cation of Landauer’s principle linking information and thermodynamics, Nature 483, 187 (2012)

2012
[108]

P. R. Zulkowski and M. R. DeWeese, Op- timal ﬁnite-time erasure of a classical bit, Phys. Rev. E 89, 052140 (2014)

2014
[109]

Proesmans, J

K. Proesmans, J. Ehrich, and J. Bech- hoefer, Finite-Time Landauer Principle, Phys. Rev. Lett. 125, 100602 (2020)

2020
[110]

Y.-Z. Zhen, D. Egloﬀ, K. Modi, and O. Dahlsten, Uni- versal Bound on Energy Cost of Bit Reset in Finite Time, Phys. Rev. Lett. 127, 190602 (2021)

2021
[111]

Ma, J.-F

Y.-H. Ma, J.-F. Chen, C. P. Sun, and H. Dong, Mini- mal energy cost to initialize a bit with tolerable error, Phys. Rev. E 106, 034112 (2022)

2022
[112]

J. M. R. Parrondo, J. M. Horowitz, and T. Sagawa, Thermodynamics of information, Nat. Phys. 11, 131 (2015)

2015
[113]

Tasnim, N

F. Tasnim, N. Freitas, and D. H. Wolpert, En- tropy production in communication channels, Phys. Rev. E 110, 034101 (2024)

2024
[114]

Zhou, Y.-H

T.-J. Zhou, Y.-H. Ma, and C. Sun, Finite-time optimization of a quantum szilard heat engine, Phys. Rev. Res. 6, 043001 (2024)

2024
[115]

J. V. Koski, V. F. Maisi, J. P. Pekola, and D. V. Averin, Experimental realization of a Szilard engine with a single electron, Proc. Natl. Acad. Sci. U.S.A. 111, 13786 (2014)

2014
[116]

Kowalczyk, I

B. Kowalczyk, I. Lagzi, and B. A. Grzybowski, Nanoseparations: Strategies for size and/or shape-selective puriﬁcation of nanoparticles, Curr. Opin. Colloid Interface Sci. 16, 135 (2011)

2011
[117]

Nasiri, A

R. Nasiri, A. Shamloo, S. Ahadian, L. Amirifar, J. Akbari, M. J. Goudie, K. Lee, N. Ashammakhi, M. R. Dokmeci, D. Di Carlo, and A. Khademhosseini, Microﬂuidic-based approaches in targeted cell/particle separation based on physical properties: Fundamentals and applications, Sma...

2020
[118]

Reimann, Brownian motors: Noisy transport far from equilibrium, Phys

P. Reimann, Brownian motors: Noisy transport far from equilibrium, Phys. Rep. 361, 57 (2002)

2002
[119]

S/suppress lapik, J

A. S/suppress lapik, J. /suppress Luczka, P. H¨ anggi, and J. Spiechow- icz, Tunable Mass Separation via Negative Mobility, Phys. Rev. Lett. 122, 070602 (2019)

2019
[120]

Herman, J

A. Herman, J. W. Ager, S. Ardo, and G. Segev, Ratchet-Based Ion Pumps for Selective Ion Separations, PRX Energy 2, 023001 (2023)

2023
[121]

X.-H. Zhao, Z. C. Tu, and Y.-H. Ma, Engineering ratchet-based particle separation via extended shortcuts to isothermality, Phys. Rev. E 110, 034105 (2024)

2024
[122]

Jarzynski, Nonequilibrium Equality for Free Energy Diﬀerences, Phys

C. Jarzynski, Nonequilibrium Equality for Free Energy Diﬀerences, Phys. Rev. Lett. 78, 2690 (1997)

1997
[123]

G. E. Crooks, Entropy production ﬂuctuation theorem and the nonequilibrium work relation for free energy diﬀerences, Phys. Rev. E 60, 2721 (1999)

1999
[124]

C. D. Christ, A. E. Mark, and W. F. van Gunsteren, Basic ingredients of free energy calculations: A review, J. Comput. Chem. 31, 1569 (2010)

2010
[125]

Li and Z

G. Li and Z. C. Tu, Equilibrium free-energy diﬀerences from a linear nonequilibrium equality, Phys. Rev. E 103, 032146 (2021)

2021
[126]

Whitelam, Free-energy estimates from nonequilib- rium trajectories under varying-temperature protocols, Phys

S. Whitelam, Free-energy estimates from nonequilib- rium trajectories under varying-temperature protocols, Phys. Rev. E 110, 014142 (2024)

2024
[127]

Zhong, B

A. Zhong, B. Kuznets-Speck, and M. R. DeWeese, Time-asymmetric ﬂuctuation theorem and eﬃcient free- energy estimation, Phys. Rev. E 110, 034121 (2024)

2024
[128]

G. Lan, P. Sartori, S. Neumann, V. Sourjik, and Y. Tu, The energy–speed–accuracy trade-oﬀ in sensory adap- tation, Nat. Phys. 8, 422 (2012)

2012
[129]

Sartori and S

P. Sartori and S. Pigolotti, Thermodynamics of Error Correction, Phys. Rev. X 5, 041039 (2015)

2015
[130]

Fang and J

X. Fang and J. Wang, Nonequilibrium Ther- modynamics in Cell Biology: Extending Equi- librium Formalism to Cover Living Systems, Ann. Rev. Biophys. 49, 227 (2020)

2020
[131]

X. Yang, M. Heinemann, J. Howard, G. Huber, S. Iyer- Biswas, G. Le Treut, M. Lynch, K. L. Montooth, D. J. Needleman, S. Pigolotti, J. Rodenfels, P. Ron- ceray, S. Shankar, I. Tavassoly, S. Thutupalli, D. V. Titov, J. Wang, and P. J. Foster, Physical bioenerget- ics: Energy ﬂux...

2021

Pith tools

Reviewed May 23, 2026 · model on record in the stance chip above.

[1] [1]

Smil, Energy and Civilization: A History (The MIT Press, Cambridge, 2017)

V. Smil, Energy and Civilization: A History (The MIT Press, Cambridge, 2017)

work page 2017

[2] [2]

Carnot, R´ eﬂexions sur la puissance motrice du feu et sur les machines propres a d´ evelopper cette puissance, Annales scientiﬁques de l’ ´Ecole Normale Sup´ erieure1, 393 (1872)

S. Carnot, R´ eﬂexions sur la puissance motrice du feu et sur les machines propres a d´ evelopper cette puissance, Annales scientiﬁques de l’ ´Ecole Normale Sup´ erieure1, 393 (1872)

work page

[3] [3]

Carnot, Reﬂections on the Motive Power of Heat: From the Original French of N.-L.-S

S. Carnot, Reﬂections on the Motive Power of Heat: From the Original French of N.-L.-S. Carnot (Wiley, New York, 1897)

work page

[4] [4]

H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed. (Wiley, New York, 1991)

work page 1991

[5] [5]

Bejan, Advanced Engineering Thermodynamics , 4th ed

A. Bejan, Advanced Engineering Thermodynamics , 4th ed. (Wiley, Hoboken, 2016)

work page 2016

[6] [6]

Andresen, P

B. Andresen, P. Salamon, and R. S. Berry, Thermo- dynamics in ﬁnite time: Extremals for imperfect heat engines, J. Chem. Phys. 66, 1571 (1977)

work page 1977

[7] [7]

Yvon, Saclay Reactor: Acquired Knowledge by Two Years Experience in Heat Transfer Using Compressed Gas, Tech

J. Yvon, Saclay Reactor: Acquired Knowledge by Two Years Experience in Heat Transfer Using Compressed Gas, Tech. Rep. (CEA Saclay, France, 1955)

work page 1955

[8] [8]

Chambadal, Les centrales nucl´ eaires(Armand Colin, Paris, 1957)

P. Chambadal, Les centrales nucl´ eaires(Armand Colin, Paris, 1957)

work page 1957