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Thermodynamic uncertainty relation for generalized time-reversal observables

Tetta Indo* and Yoshihiko Hasegawa†

  • Department of Information and Communication Engineering, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan

  • *Contact author: indo@biom.t.u-tokyo.ac.jp
  • †Contact author: hasegawa@biom.t.u-tokyo.ac.jp
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Phys. Rev. E 114, 054119 – Published 9 November, 2026

DOI: https://doi.org/10.1103/nphl-w7gt

Abstract

Time-reversal symmetry plays an essential role in the thermodynamic uncertainty relations, which bound the fluctuations of observables in terms of the associated dissipation. In fact, thermodynamic uncertainty relations are typically derived under the assumption that the observable of interest is antisymmetric under time reversal. This also suggests that existing thermodynamic uncertainty relations are restricted to a limited class of observables. In this paper, we mitigate this restriction by introducing a new class of observables that do not exhibit exact antisymmetry but change sign under time reversal. We call it generalized time reversal and derive a broadly applicable thermodynamic uncertainty relation for observables with this condition. The generalization is achieved by direct statistical arguments on the observable distributions and holds for both deterministic and stochastic dynamics. We demonstrate the derived thermodynamic uncertainty relation with observables subjected to rectifications, showing that the precision of generalized observables remains expressible in terms of the dissipative cost. The result extends the scope of thermodynamic uncertainty relations beyond the reach of previous frameworks relying on the exact antisymmetry.

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References (63)

  1. C. Jarzynski, Nonequilibrium equality for free energy differences, Phys. Rev. Lett. 78, 2690 (1997).
  2. G. E. Crooks, Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences, Phys. Rev. E 60, 2721 (1999).
  3. G. Gallavotti and E. G. D. Cohen, Dynamical ensembles in nonequilibrium statistical mechanics, Phys. Rev. Lett. 74, 2694 (1995).
  4. M. Esposito and C. Van den Broeck, Three detailed fluctuation theorems, Phys. Rev. Lett. 104, 090601 (2010).
  5. U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
  6. A. C. Barato and U. Seifert, Thermodynamic uncertainty relation for biomolecular processes, Phys. Rev. Lett. 114, 158101 (2015).
  7. 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).
  8. J. M. Horowitz and T. R. Gingrich, Thermodynamic uncertainty relations constrain non-equilibrium fluctuations, Nat. Phys. 16, 15 (2020).
  9. K. Brandner, T. Hanazato, and K. Saito, Thermodynamic bounds on precision in ballistic multiterminal transport, Phys. Rev. Lett. 120, 090601 (2018).
  10. G. T. Landi and M. Paternostro, Irreversible entropy production: From classical to quantum, Rev. Mod. Phys. 93, 035008 (2021).
  11. Y. Hasegawa and T. Van Vu, Fluctuation theorem uncertainty relation, Phys. Rev. Lett. 123, 110602 (2019).
  12. R. J. Harris and G. M. Schütz, Fluctuation theorems for stochastic dynamics, J. Stat. Mech. (2007) P07020.
  13. N. Merhav and Y. Kafri, Statistical properties of entropy production derived from fluctuation theorems, J. Stat. Mech. (2010) P12022.
  14. L. Ziyin and M. Ueda, Universal thermodynamic uncertainty relation in nonequilibrium dynamics, Phys. Rev. Res. 5, 013039 (2023).
  15. I. Di Terlizzi and M. Baiesi, Kinetic uncertainty relation, J. Phys. A: Math. Theor. 52, 02LT03 (2019).
  16. V. T. Vo, T. Van Vu, and Y. Hasegawa, Unified thermodynamic–kinetic uncertainty relation, J. Phys. A: Math. Theor. 55, 405004 (2022).
  17. K. Hiura and S.-I. Sasa, Kinetic uncertainty relation on first-passage time for accumulated current, Phys. Rev. E 103, L050103 (2021).
  18. T. Monnai, Kinetic equality for susceptibility and dynamical activity, Phys. Rev. E 110, L062101 (2024).
  19. V. V. Petrov, On lower bounds for tail probabilities, J. Stat. Plann. Inference 137, 2703 (2007).
  20. Y. Hasegawa and T. Nishiyama, Thermodynamic concentration inequalities and trade-off relations, Phys. Rev. Lett. 133, 247101 (2024).
  21. H. E. D. Scovil and E. O. Schulz-DuBois, Three-level masers as heat engines, Phys. Rev. Lett. 2, 262 (1959).
  22. E. Geva and R. Kosloff, Three-level quantum amplifier as a heat engine: A study in finite-time thermodynamics, Phys. Rev. E 49, 3903 (1994).
  23. E. Boukobza and D. J. Tannor, Thermodynamics of bipartite systems: Application to light-matter interactions, Phys. Rev. A 74, 063823 (2006).
  24. R. Alicki, The quantum open system as a model of the heat engine, J. Phys. A: Math. Gen. 12, L103 (1979).
  25. U. Seifert, Entropy production along a stochastic trajectory and an integral fluctuation theorem, Phys. Rev. Lett. 95, 040602 (2005).
  26. R. E. Spinney and I. J. Ford, Fluctuation relations: A pedagogical overview, in Nonequilibrium Statistical Physics of Small Systems: Fluctuation Relations and Beyond, edited by R. Klages, W. Just, and C. Jarzynski (Wiley–VCH, Weinheim, 2013).
  27. A. M. Timpanaro, G. Guarnieri, J. Goold, and G. T. Landi, Thermodynamic uncertainty relations from exchange fluctuation theorems, Phys. Rev. Lett. 123, 090604 (2019).
  28. S. Boucheron, G. Lugosi, and P. Massart, Concentration Inequalities: A Nonasymptotic Theory of Independence (Oxford University Press, Oxford, 2013).
  29. B. K. Ghosh, Probability inequalities related to Markov's theorem, Am. Stat. 56, 186 (2002).
  30. A. Momeni, B. Rahmani, B. Scellier, L. G. Wright, P. L. McMahon, C. C. Wanjura, Y. Li, A. Skalli, N. G. Berloff, T. Onodera, I. Oguz, F. Morichetti, P. del Hougne, M. Le Gallo, A. Sebastian, A. Mirhoseini, C. Zhang, D. Marković, D. Brunner, C. Moser, et al., Training of physical neural networks, Nature (London) 645, 53 (2025).
  31. I. A. D. Williamson, T. W. Hughes, M. Minkov, B. Bartlett, S. Pai, and S. Fan, Reprogrammable electro-optic nonlinear activation functions for optical neural networks, IEEE J. Sel. Top. Quantum Electron. 26, 1 (2020).
  32. G. H. Y. Li, R. Sekine, R. Nehra, R. M. Gray, L. Ledezma, Q. Guo, and A. Marandi, All-optical ultrafast ReLU function for energy-efficient nanophotonic deep learning, Nanophoton. 12, 847 (2023).
  33. Z. Xu, B. Tang, X. Zhang, J. F. Leong, J. Pan, S. Hooda, E. Zamburg, and A. V.-Y. Thean, Reconfigurable nonlinear photonic activation function for photonic neural network based on non-volatile opto-resistive RAM switch, Light: Sci. Appl. 11, 288 (2022).
  34. Z. Lin and Z. Bai, Probability Inequalities, 1st ed. (Springer, Berlin, Heidelberg, 2011), pp. vii + 181.
  35. F. Balabdaoui and H. Jankowski, Maximum likelihood estimation of a unimodal probability mass function, Stat. Sin. 26, 1061 (2016).
  36. O. E. Barndorff-Nielsen, Normal inverse Gaussian distributions and stochastic volatility modelling, Scand. J. Stat. 24, 1 (1997).
  37. P. Cattiaux, D. Chafaï, and A. Guillin, Central limit theorems for additive functionals of ergodic Markov diffusions, Lat. Am. J. Probab. Math. Stat. 9, 337 (2012).
  38. M. Mercadier and F. Strobel, A one-sided Vysochanskii-Petunin inequality with financial applications, Eur. J. Oper. Res. 295, 374 (2021).
  39. F. Pukelsheim, The three sigma rule, Am. Stat. 48, 88 (1994).
  40. T. M. Sellke and S. H. Sellke, Chebyshev inequalities for unimodal distributions, Am. Stat. 51, 34 (1997).
  41. K. Y. Tan, M. Partanen, R. E. Lake, J. Govenius, S. Masuda, and M. Möttönen, Quantum-circuit refrigerator, Nat. Commun. 8, 15189 (2017).
  42. A. T. Jones, C. P. Scheller, J. R. Prance, Y. B. Kalyoncu, D. M. Zumbühl, and R. P. Haley, Progress in cooling nanoelectronic devices to ultra-low temperatures, J. Low Temp. Phys. 201, 772 (2020).
  43. M. T. Mitchison, Quantum thermal absorption machines: Refrigerators, engines and clocks, Contemp. Phys. 60, 164 (2019).
  44. A. S. Hegde, P. P. Potts, and G. T. Landi, Time-resolved stochastic dynamics of quantum thermal machines, Phys. Rev. Lett. 134, 150402 (2025).
  45. G. Manzano Paule, Thermodynamics and Synchronization in Open Quantum Systems, Springer Theses (Springer, Cham, 2018).
  46. Y. Hasegawa, Quantum thermodynamic uncertainty relation for continuous measurement, Phys. Rev. Lett. 125, 050601 (2020).
  47. C. Jarzynski and D. K. Wójcik, Classical and quantum fluctuation theorems for heat exchange, Phys. Rev. Lett. 92, 230602 (2004).
  48. K. Saito and Y. Utsumi, Symmetry in full counting statistics, fluctuation theorem, and relations among nonlinear transport coefficients in the presence of a magnetic field, Phys. Rev. B 78, 115429 (2008).
  49. G. T. Landi and D. Karevski, Fluctuations of the heat exchanged between two quantum spin chains, Phys. Rev. E 93, 032122 (2016).
  50. S. Lahiri and A. M. Jayannavar, Derivation of not-so-common fluctuation theorems, Indian J. Phys. 89, 515 (2015).
  51. H. Spohn, Large Scale Dynamics of Interacting Particles (Springer, Berlin, 1991).
  52. L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, Macroscopic fluctuation theory, Rev. Mod. Phys. 87, 593 (2015).
  53. T. Bodineau and B. Derrida, Current fluctuations in nonequilibrium diffusive systems: An additivity principle, Phys. Rev. Lett. 92, 180601 (2004).
  54. L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, Current fluctuations in stochastic lattice gases, Phys. Rev. Lett. 94, 030601 (2005).
  55. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. (Pergamon Press, Oxford, 1987).
  56. J. B. Bell, A. L. Garcia, and S. A. Williams, Numerical methods for the stochastic Landau-Lifshitz Navier-Stokes equations, Phys. Rev. E 76, 016708 (2007).
  57. F. Aguirre, A. Sebastian, M. Le Gallo, W. Song, T. Wang, J. Joshua Yang, W. Lu, M.-F. Chang, D. Ielmini, Y. Yang, A. Mehonic, A. Kenyon, M. A. Villena, J. B. Roldán, Y. Wu, H.-H. Hsu, N. Raghavan, J. Suñé, E. Miranda, A. Eltawil, et al., Hardware implementation of memristor-based artificial neural networks, Nat. Commun. 15, 1974 (2024).
  58. K. He, X. Zhang, S. Ren, and J. Sun, Delving deep into rectifiers: Surpassing human-level performance on ImageNet classification, in Proceedings of the IEEE International Conference on Computer Vision (ICCV) (EEE Computer Society, Los Alamitos, CA, 2015), pp. 1026–1034.
  59. D. Andrieux, P. Gaspard, T. Monnai, and S. Tasaki, The fluctuation theorem for currents in open quantum systems, New J. Phys. 11, 043014 (2009).
  60. M. Esposito, U. Harbola, and S. Mukamel, Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems, Rev. Mod. Phys. 81, 1665 (2009).
  61. For ν>0, the elementary Mills bound follows from 1−Φ(ν)=∫ν∞φ(t)dt<∫ν∞(t/ν)φ(t)dt=φ(ν)/ν.
  62. Á. Baricz, Mills' ratio: Reciprocal convexity and functional inequalities, Acta Univ. Sapientiae Math. 4, 26 (2012).
  63. A. Gasull and F. Utzet, Approximating Mills ratio, J. Math. Anal. Appl. 420, 1832 (2014).

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Phys. Rev. E 114, 054119– Published 9 November, 2026
  • Received 27 May 2026
  • Accepted 23 October 2026
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