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Unified geometric formalism for dissipation and its fluctuations in finite-time microscopic heat engines

Gentaro Watanabe1,2,*, Guo-Hua Xu3, and Yuki Minami4

  • 1School of Physics and Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou, Zhejiang 310027, China
  • 2Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, Zhejiang 310027, China
  • 3Universal Biology Institute, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
  • 4Faculty of Engineering, Gifu University, Yanagido, Gifu 501-1193, Japan

  • *Contact author: gentaro@zju.edu.cn
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Phys. Rev. Research 8, 043037 – Published 13 October, 2026

DOI: https://doi.org/10.1103/wsl6-bjb7

Abstract

Microscopic heat engines operate in regimes where thermodynamic quantities fluctuate strongly, making stochastic effects an essential aspect of their performance. However, existing geometric formulations of finite-time thermodynamics primarily characterize average dissipation and do not systematically capture its fluctuations. Here, we develop a unified geometric framework that consistently describes both the mean dissipated availability and its fluctuations. In the linear-response regime, we show that these quantities are governed by metric tensors constructed from equilibrium correlation functions, providing a common geometric structure for dissipation and its fluctuations. This framework yields geometric bounds on both the mean and variance of the dissipated availability, and thereby on the efficiency and its fluctuations. The formalism applies broadly to stochastic systems, including Markov jump processes and overdamped and underdamped Brownian dynamics, establishing a unified geometric description across microscopic heat engines.

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References (115)

  1. C. Bustamante, J. Liphardt, and F. Ritort, The nonequilibrium thermodynamics of small systems, Phys. Today 58(7), 43 (2005).
  2. S. Ciliberto, Experiments in stochastic thermodynamics: Short history and perspectives, Phys. Rev. X 7, 021051 (2017).
  3. V. Blickle and C. Bechinger, Realization of a micrometre-sized stochastic heat engine, Nat. Phys. 8, 143 (2012).
  4. P. A. Quinto-Su, A microscopic steam engine implemented in an optical tweezer, Nat. Commun. 5, 5889 (2014).
  5. I. A. Martínez, É. Roldán, L. Dinis, D. Petrov, J. M. R. Parrondo, and R. A. Rica, Brownian Carnot engine, Nat. Phys. 12, 67 (2016).
  6. J. Roßnagel, S. T. Dawkins, K. N. Tolazzi, O. Abah, E. Lutz, F. Schmidt-Kaler, and K. Singer, A single-atom heat engine, Science 352, 325 (2016).
  7. I. A. Martínez, É. Roldán, L. Dinis, and R. A. Rica, Colloidal heat engines: A review, Soft Matter 13, 22 (2017).
  8. A. Argun, J. Soni, L. Dabelow, S. Bo, G. Pesce, R. Eichhorn, and G. Volpe, Experimental realization of a minimal microscopic heat engine, Phys. Rev. E 96, 052106 (2017).
  9. G. Maslennikov, S. Ding, R. Hablützel, J. Gan, A. Roulet, S. Nimmrichter, J. Dai, V. Scarani, and D. Matsukevich, Quantum absorption refrigerator with trapped ions, Nat. Commun. 10, 202 (2019).
  10. D. von Lindenfels, O. Gräb, C. T. Schmiegelow, V. Kaushal, J. Schulz, M. T. Mitchison, J. Goold, F. Schmidt-Kaler, and U. G. Poschinger, Spin heat engine coupled to a harmonic-oscillator flywheel, Phys. Rev. Lett. 123, 080602 (2019).
  11. W. Hou, W. Yao, X. Zhao, K. Rehan, Y. Li, Y. Li, E. Lutz, Y. Lin, and J. Du, Combining energy efficiency and quantum advantage in cyclic machines, Nat. Commun. 16, 5127 (2025).
  12. P. G. Steeneken, K. Le Phan, M. J. Goossens, G. E. J. Koops, G. J. A. M. Brom, C. van der Avoort, and J. T. M. van Beek, Piezoresistive heat engine and refrigerator, Nat. Phys. 7, 354 (2011).
  13. J. P. Pekola, Towards quantum thermodynamics in electronic circuits, Nat. Phys. 11, 118 (2015).
  14. T. Hugel, N. B. Holland, A. Cattani, L. Moroder, M. Seitz, and H. E. Gaub, Single-molecule optomechanical cycle, Science 296, 1103 (2002).
  15. J. Klaers, S. Faelt, A. Imamoglu, and E. Togan, Squeezed thermal reservoirs as a resource for a nanomechanical engine beyond the Carnot limit, Phys. Rev. X 7, 031044 (2017).
  16. U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
  17. K. Sekimoto, Langevin equation and thermodynamics, Prog. Theor. Phys. Suppl. 130, 17 (1998).
  18. K. Sekimoto, Stochastic Energetics, Lecture Notes in Physics (Springer, Berlin, 2010), Vol. 799.
  19. K. Saito, Thermodynamics of Fluctuating Systems—Development of Non-Equilibrium Statistical Mechanics (in Japanese), SGC Library (Saiensu-Sha, Tokyo, 2022), Vol. 182.
  20. N. Shiraishi, An Introduction to Stochastic Thermodynamics—From Basic to Advanced (Springer, Singapore, 2023).
  21. L. Peliti and S. Pigolotti, Stochastic Thermodynamics: An Introduction (Princeton University Press, Princeton, 2021).
  22. U. Seifert, Stochastic Thermodynamics (Cambridge University Press, Cambridge, 2025).
  23. C. Jarzynski, Equalities and inequalities: Irreversibility and the second law of thermodynamics at the nanoscale, Annu. Rev. Condens. Matter Phys. 2, 329 (2011).
  24. C. Van den Broeck and M. Esposito, Ensemble and trajectory thermodynamics: A brief introduction, Physica A 418, 6 (2015).
  25. G. Nicolis and Y. De Decker, Stochastic thermodynamics of Brownian motion, Entropy 19, 434 (2017).
  26. D. Guéry-Odelin, C. Jarzynski, C. A. Plata, A. Prados, and E. Trizac, Driving rapidly while remaining in control: Classical shortcuts from Hamiltonian to stochastic dynamics, Rep. Prog. Phys. 86, 035902 (2023).
  27. S. Krishnamurthy, S. Ghosh, D. Chatterji, R. Ganapathy, and A. K. Sood, A micrometre-sized heat engine operating between bacterial reservoirs, Nat. Phys. 12, 1134 (2016).
  28. P. Pietzonka, É. 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).
  29. É. Fodor and M. E. Cates, Active engines: Thermodynamics moves forward, Europhys. Lett. 134, 10003 (2021).
  30. S. Toyabe and M. Sano, Nonequilibrium fluctuations in biological strands, machines, and cells, J. Phys. Soc. Jpn. 84, 102001 (2015).
  31. C. Jarzynski, Nonequilibrium equality for free energy differences, Phys. Rev. Lett. 78, 2690 (1997).
  32. G. E. Crooks, Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences, Phys. Rev. E 60, 2721 (1999).
  33. A. C. Barato and U. Seifert, Thermodynamic uncertainty relation for biomolecular processes, Phys. Rev. Lett. 114, 158101 (2015).
  34. T. R. Gingrich, J. M. Horowitz, N. Perunov, and J. L. England, Dissipation bounds all steady-state current fluctuations, Phys. Rev. Lett. 116, 120601 (2016).
  35. U. Seifert, From stochastic thermodynamics to thermodynamic inference, Annu. Rev. Condens. Matter Phys. 10, 171 (2019).
  36. Y. Hasegawa and T. Van Vu, Fluctuation theorem uncertainty relation, Phys. Rev. Lett. 123, 110602 (2019).
  37. J. M. Horowitz and T. R. Gingrich, Thermodynamic uncertainty relations constrain non-equilibrium fluctuations, Nat. Phys. 16, 15 (2020).
  38. K. Liu, Z. Gong, and M. Ueda, Thermodynamic uncertainty relation for arbitrary initial states, Phys. Rev. Lett. 125, 140602 (2020).
  39. T. Koyuk and U. Seifert, Thermodynamic uncertainty relation for time-dependent driving, Phys. Rev. Lett. 125, 260604 (2020).
  40. T. Van Vu and Y. Hasegawa, Thermodynamic uncertainty relations under arbitrary control protocols, Phys. Rev. Res. 2, 013060 (2020).
  41. A. C. Barato, R. Chetrite, A. Faggionato, and D. Gabrielli, Bounds on current fluctuations in periodically driven systems, New J. Phys. 20, 103023 (2018).
  42. T. Koyuk, U. Seifert, and P. Pietzonka, A generalization of the thermodynamic uncertainty relation to periodically driven systems, J. Phys. A: Math. Theor. 52, 02LT02 (2019).
  43. T. Koyuk and U. Seifert, Operationally accessible bounds on fluctuations and entropy production in periodically driven systems, Phys. Rev. Lett. 122, 230601 (2019).
  44. 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).
  45. Z. Wang and J. Ren, Thermodynamic geometry of nonequilibrium fluctuations in cyclically driven transport, Phys. Rev. Lett. 132, 207101 (2024).
  46. K. Sekimoto, F. Takagi, and T. Hondou, Carnot's cycle for small systems: Irreversibility and cost of operations, Phys. Rev. E 62, 7759 (2000).
  47. C. Van den Broeck, Thermodynamic efficiency at maximum power, Phys. Rev. Lett. 95, 190602 (2005).
  48. T. Schmiedl and U. Seifert, Efficiency at maximum power: An analytically solvable model for stochastic heat engines, Europhys. Lett. 81, 20003 (2008).
  49. 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).
  50. A. Dechant, N. Kiesel, and E. Lutz, All-optical nanomechanical heat engine, Phys. Rev. Lett. 114, 183602 (2015).
  51. K. Brandner, K. Saito, and U. Seifert, Thermodynamics of micro- and nano-systems driven by periodic temperature variations, Phys. Rev. X 5, 031019 (2015).
  52. A. Dechant, N. Kiesel, and E. Lutz, Underdamped stochastic heat engine at maximum efficiency, Europhys. Lett. 119, 50003 (2017).
  53. P. Strasberg, C. W. Wächtler, and G. Schaller, Autonomous implementation of thermodynamic cycles at the nanoscale, Phys. Rev. Lett. 126, 180605 (2021).
  54. J. Hoppenau, M. Niemann, and A. Engel, Carnot process with a single particle, Phys. Rev. E 87, 062127 (2013).
  55. V. Holubec, An exactly solvable model of a stochastic heat engine: Optimization of power, power fluctuations and efficiency, J. Stat. Mech. (2014) P05022.
  56. S. Rana, P. S. Pal, A. Saha, and A. M. Jayannavar, Single-particle stochastic heat engine, Phys. Rev. E 90, 042146 (2014).
  57. L. Cerino, A. Puglisi, and A. Vulpiani, Kinetic model for the finite-time thermodynamics of small heat engines, Phys. Rev. E 91, 032128 (2015).
  58. V. Holubec and A. Ryabov, Work and power fluctuations in a critical heat engine, Phys. Rev. E 96, 030102(R) (2017).
  59. V. Holubec and A. Ryabov, Cycling tames power fluctuations near optimum efficiency, Phys. Rev. Lett. 121, 120601 (2018).
  60. A. Dechant, Multidimensional thermodynamic uncertainty relations, J. Phys. A: Math. Theor. 52, 035001 (2019).
  61. K. Ito, G.-H. Xu, C. Jiang, É. Roldán, R. A. Rica-Alarcón, I. A. Martínez, and G. Watanabe, Universal relations and bounds for fluctuations in quasistatic small heat engines, Commun. Phys. 8, 60 (2025).
  62. S. Saryal, M. Gerry, I. Khait, D. Segal, and B. K. Agarwalla, Universal bounds on fluctuations in continuous thermal machines, Phys. Rev. Lett. 127, 190603 (2021).
  63. S. Mohanta, S. Saryal, and B. K. Agarwalla, Universal bounds on cooling power and cooling efficiency for autonomous absorption refrigerators, Phys. Rev. E 105, 034127 (2022).
  64. V. Holubec and A. Ryabov, Fluctuations in heat engines, J. Phys. A: Math. Theor. 55, 013001 (2022).
  65. G. Verley, M. Esposito, T. Willaert, and C. Van den Broeck, The unlikely Carnot efficiency, Nat. Commun. 5, 4721 (2014).
  66. G. Verley, T. Willaert, C. Van den Broeck, and M. Esposito, Universal theory of efficiency fluctuations, Phys. Rev. E 90, 052145 (2014).
  67. K. Proesmans and C. Van den Broeck, Stochastic efficiency: Five case studies, New J. Phys. 17, 065004 (2015).
  68. K. Proesmans, C. Driesen, B. Cleuren, and C. Van den Broeck, Efficiency of single-particle engines, Phys. Rev. E 92, 032105 (2015).
  69. J.-M. Park, H.-M. Chun, and J. D. Noh, Efficiency at maximum power and efficiency fluctuations in a linear Brownian heat-engine model, Phys. Rev. E 94, 012127 (2016).
  70. A. Saha, R. Marathe, P. S. Pal, and A. M. Jayannavar, Stochastic heat engine powered by active dissipation, J. Stat. Mech. (2018) 113203.
  71. S. K. Manikandan, L. Dabelow, R. Eichhorn, and S. Krishnamurthy, Efficiency fluctuations in microscopic machines, Phys. Rev. Lett. 122, 140601 (2019).
  72. H. Vroylandt, M. Esposito, and G. Verley, Efficiency fluctuations of stochastic machines undergoing a phase transition, Phys. Rev. Lett. 124, 250603 (2020).
  73. N. A. Sinitsyn, Fluctuation relation for heat engines, J. Phys. A: Math. Theor. 44, 405001 (2011).
  74. P. S. Pal, S. Lahiri, and A. M. Jayannavar, Transient exchange fluctuation theorems for heat using a Hamiltonian framework: Classical and quantum regimes, Phys. Rev. E 95, 042124 (2017).
  75. K. Proesmans and C. Van den Broeck, Discrete-time thermodynamic uncertainty relation, Europhys. Lett. 119, 20001 (2017).
  76. P. Pietzonka and U. Seifert, Universal trade-off between power, efficiency, and constancy in steady-state heat engines, Phys. Rev. Lett. 120, 190602 (2018).
  77. A. M. Timpanaro, G. Guarnieri, J. Goold, and G. T. Landi, Thermodynamic uncertainty relations from exchange fluctuation theorems, Phys. Rev. Lett. 123, 090604 (2019).
  78. T. Kamijima, S. Otsubo, Y. Ashida, and T. Sagawa, Higher-order efficiency bound and its application to nonlinear nanothermoelectrics, Phys. Rev. E 104, 044115 (2021).
  79. G.-H. Xu and G. Watanabe, Correlation-enhanced stability of microscopic cyclic heat engines, Phys. Rev. Res. 4, L032017 (2022).
  80. G.-H. Xu, C. Jiang, Y. Minami, and G. Watanabe, Relation between fluctuations and efficiency at maximum power for small heat engines, Phys. Rev. Res. 4, 043139 (2022).
  81. C. A. Plata, D. Guéry-Odelin, E. Trizac, and A. Prados, Optimal work in a harmonic trap with bounded stiffness, Phys. Rev. E 99, 012140 (2019).
  82. C. A. Plata, D. Guéry-Odelin, E. Trizac, and A. Prados, Building an irreversible Carnot-like heat engine with an overdamped harmonic oscillator, J. Stat. Mech. (2020) 093207.
  83. C. A. Plata, D. Guéry-Odelin, E. Trizac, and A. Prados, Finite-time adiabatic processes: Derivation and speed limit, Phys. Rev. E 101, 032129 (2020).
  84. P. Jop, A. Petrosyan, and S. Ciliberto, Work and dissipation fluctuations near the stochastic resonance of a colloidal particle, Europhys. Lett. 81, 50005 (2008).
  85. I. A. Martínez, É. Roldán, L. Dinis, D. Petrov, and R. A. Rica, Adiabatic processes realized with a trapped Brownian particle, Phys. Rev. Lett. 114, 120601 (2015).
  86. F. Weinhold, Metric geometry of equilibrium thermodynamics, J. Chem. Phys. 63, 2479 (1975).
  87. F. Weinhold, Thermodynamics and geometry, Phys. Today 29(3), 23 (1976).
  88. G. Ruppeiner, Thermodynamics: A Riemannian geometric model, Phys. Rev. A 20, 1608 (1979).
  89. G. Ruppeiner, Riemannian geometry in thermodynamic fluctuation theory, Rev. Mod. Phys. 67, 605 (1995).
  90. B. Andresen, Current trends in finite-time thermodynamics, Angew. Chem. Int. Ed. 50, 2690 (2011).
  91. P. Salamon and R. S. Berry, Thermodynamic length and dissipated availability, Phys. Rev. Lett. 51, 1127 (1983).
  92. B. Andresen, P. Salamon, and R. S. Berry, Thermodynamics in finite time, Phys. Today 37(9), 62 (1984).
  93. J. Nulton, P. Salamon, B. Andresen, and Q. Anmin, Quasistatic processes as step equilibrations, J. Chem. Phys. 83, 334 (1985).
  94. P. Salamon, J. Nulton, and E. Ihrig, On the relation between entropy and energy versions of thermodynamic length, J. Chem. Phys. 80, 436 (1984).
  95. R. Gilmore, Length and curvature in the geometry of thermodynamics, Phys. Rev. A 30, 1994 (1984).
  96. F. Schlögl, Thermodynamic metric and stochastic measures, Z. Phys. B 59, 449 (1985).
  97. D. Brody and N. Rivier, Geometrical aspects of statistical mechanics, Phys. Rev. E 51, 1006 (1995).
  98. G. E. Crooks, Measuring thermodynamic length, Phys. Rev. Lett. 99, 100602 (2007).
  99. P. R. Zulkowski, D. A. Sivak, G. E. Crooks, and M. R. DeWeese, Geometry of thermodynamic control, Phys. Rev. E 86, 041148 (2012).
  100. T. V. Vu and Y. Hasegawa, Geometrical bounds of the irreversibility in Markovian systems, Phys. Rev. Lett. 126, 010601 (2021).
  101. J. Eglinton and K. Brandner, Geometric bounds on the power of adiabatic thermal machines, Phys. Rev. E 105, L052102 (2022).
  102. A. G. Frim and M. R. DeWeese, Geometric bound on the efficiency of irreversible thermodynamic cycles, Phys. Rev. Lett. 128, 230601 (2022).
  103. J. R. Sawchuk and D. A. Sivak, Global thermodynamic manifold for conservative control of stochastic systems, Phys. Rev. Res. 8, 013004 (2026).
  104. D. A. Sivak and G. E. Crooks, Thermodynamic metrics and optimal paths, Phys. Rev. Lett. 108, 190602 (2012).
  105. K. Brandner and K. Saito, Thermodynamic geometry of microscopic heat engines, Phys. Rev. Lett. 124, 040602 (2020).
  106. P. Abiuso and M. Perarnau-Llobet, Optimal cycles for low-dissipation heat engines, Phys. Rev. Lett. 124, 110606 (2020).
  107. A. G. Frim and M. R. DeWeese, Optimal finite-time Brownian Carnot engine, Phys. Rev. E 105, L052103 (2022).
  108. Z. Li and Y. Izumida, Decomposition of metric tensor in thermodynamic geometry in terms of relaxation timescales, Phys. Rev. E 111, 034113 (2025).
  109. G. Watanabe and Y. Minami, Finite-time thermodynamics of fluctuations in microscopic heat engines, Phys. Rev. Res. 4, L012008 (2022).
  110. H. J. D. Miller, M. Scandi, J. Anders, and M. Perarnau-Llobet, Work fluctuations in slow processes: Quantum signatures and optimal control, Phys. Rev. Lett. 123, 230603 (2019).
  111. H. J. D. Miller and M. Mehboudi, Geometry of work fluctuations versus efficiency in microscopic thermal machines, Phys. Rev. Lett. 125, 260602 (2020).
  112. H. Risken, The Fokker–Planck Equation, 2nd ed. (Springer, Berlin, 1988).
  113. The similarity-transformed eigenfunctions of the Hermitian operator are ϕ̃i=eΦ/2ϕi, while the functions eΦϕi correspond to the left eigenfunctions of the Fokker-Planck operator since LFP†=eΦLFPe−Φ.
  114. The additive constant in Φ(x) has no effect on the spectrum of the Fokker-Planck operator. We keep it here to remain consistent with the conventional similarity transformation used in standard treatments ( see, e.g., Ref. [112]).
  115. The larger scatter of the experimental data points in Fig. 4 at smaller 1/τ may be due to the smaller number of samples. In the experiment of Ref. [5], cycles were repeated for a fixed duration of 50 s at each value of τ, so the number of samples decreases as τ increases.

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Phys. Rev. Research 8, 043037– Published 13 October, 2026
  • Received 6 April 2026
  • Accepted 21 September 2026
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