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  • Open Access

Consistent Time Reversal and Reliable and Accurate Inference in the Presence of Memory

Tassilo Schwarz1,2, Anatoly B. Kolomeisky3,4, and Aljaž Godec5,1,*

  • *Contact author: agodec@physik.uni-freiburg.de
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Phys. Rev. Lett. 137, 187401 – Published 29 October, 2026

DOI: https://doi.org/10.1103/gwf7-6kvr

Abstract

Thermodynamic inference from coarse observations remains a key challenge. Memory, in particular correlations between consecutively observed mesostates, blur signatures of irreversibility and must be accounted for in defining physical time reversal, which remains an open problem. We derive an experimentally accessible kth order estimator for the entropy production rate. Using novel measure-theoretic techniques we prove necessary and sufficient conditions for guaranteed lower bounds on the dissipation even in the strongly non-Markovian setting. The proof reveals that estimators saturated in the order unravel the duration of memory that needs to be considered in defining physically consistent time reversal. We show that Markovian estimators in the absence of a timescale separation lead to artifacts that convey no physical meaning. Similarly, estimators not saturated in the order may overestimate the dissipation. The necessity of correctly accounting for memory in thermodynamic inference from strongly non-Markovian observations underscores the still underappreciated challenges and intricacies in defining and understanding irreversibility in presence of memory. Our results will hopefully stimulate experiments systematically considering thermodynamic inference on multiple scales consistently accounting for memory.

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References (96)

  1. J. Gladrow, N. Fakhri, F. C. MacKintosh, C. F. Schmidt, and C. P. Broedersz, Broken detailed balance of filament dynamics in active networks, Phys. Rev. Lett. 116, 248301 (2016).
  2. C. Battle, C. P. Broedersz, N. Fakhri, V. F. Geyer, J. Howard, C. F. Schmidt, and F. C. MacKintosh, Broken detailed balance at mesoscopic scales in active biological systems, Science 352, 604 (2016).
  3. C. Dieball and A. Godec, Mathematical, thermodynamical, and experimental necessity for coarse graining empirical densities and currents in continuous space, Phys. Rev. Lett. 129, 140601 (2022).
  4. C. Dieball and A. Godec, Coarse graining empirical densities and currents in continuous-space steady states, Phys. Rev. Res. 4, 033243 (2022).
  5. D. Hérisson and M. Ocio, Fluctuation-dissipation ratio of a spin glass in the aging regime, Phys. Rev. Lett. 88, 257202 (2002).
  6. H. Oukris and N. E. Israeloff, Nanoscale non-equilibrium dynamics and the fluctuation–dissipation relation in an ageing polymer glass, Nat. Phys. 6, 135 (2009).
  7. A. J. Bray, S. N. Majumdar, and G. Schehr, Persistence and first-passage properties in nonequilibrium systems, Adv. Phys. 62, 225 (2013).
  8. S. Burov, R. Metzler, and E. Barkai, Aging and nonergodicity beyond the Khinchin theorem, Proc. Natl. Acad. Sci. U.S.A. 107, 13228 (2010).
  9. J. H. P. Schulz, E. Barkai, and R. Metzler, Aging renewal theory and application to random walks, Phys. Rev. X 4, 011028 (2014).
  10. T. GrandPre, G. Teza, and W. Bialek, Direct estimates of irreversibility from time series, arXiv:2412.19772.
  11. A. K. Sangha and T. Keyes, Proteins fold by subdiffusion of the order parameter, J. Phys. Chem. B 113, 15886 (2009).
  12. S. S. Plotkin and P. G. Wolynes, Non-Markovian configurational diffusion and reaction coordinates for protein folding, Phys. Rev. Lett. 80, 5015 (1998).
  13. W. Min, G. Luo, B. J. Cherayil, S. C. Kou, and X. S. Xie, Observation of a power-law memory kernel for fluctuations within a single protein molecule, Phys. Rev. Lett. 94, 198302 (2005).
  14. D. E. Makarov, Interplay of non-Markov and internal friction effects in the barrier crossing kinetics of biopolymers: Insights from an analytically solvable model, J. Chem. Phys. 138, 014102 (2013).
  15. J. Brujić, R. I. Hermans Z., K. A. Walther, and J. M. Fernandez, Single-molecule force spectroscopy reveals signatures of glassy dynamics in the energy landscape of ubiquitin, Nat. Phys. 2, 282 (2006).
  16. I. Grossman-Haham, G. Rosenblum, T. Namani, and H. Hofmann, Slow domain reconfiguration causes power-law kinetics in a two-state enzyme, Proc. Natl. Acad. Sci. U.S.A. 115, 513 (2018).
  17. I. M. Sokolov, Cyclization of a polymer: First-passage problem for a non-Markovian process, Phys. Rev. Lett. 90, 080601 (2003).
  18. T. Guérin, O. Bénichou, and R. Voituriez, Non-Markovian polymer reaction kinetics, Nat. Chem. 4, 568 (2012).
  19. A. M. Berezhkovskii and D. E. Makarov, Single-molecule test for Markovianity of the dynamics along a reaction coordinate, J. Phys. Chem. Lett. 9, 2190 (2018).
  20. A. Lapolla and A. Godec, Toolbox for quantifying memory in dynamics along reaction coordinates, Phys. Rev. Res. 3, L022018 (2021).
  21. F. Ginot, J. Caspers, M. Krüger, and C. Bechinger, Barrier crossing in a viscoelastic bath, Phys. Rev. Lett. 128, 028001 (2022).
  22. N. Narinder, C. Bechinger, and J. R. Gomez-Solano, Memory-induced transition from a persistent random walk to circular motion for achiral microswimmers, Phys. Rev. Lett. 121, 078003 (2018).
  23. S. Nordholm and R. Zwanzig, A systematic derivation of exact generalized Brownian motion theory, J. Stat. Phys. 13, 347 (1975).
  24. H. Grabert, P. Talkner, P. Hänggi, and H. Thomas, Microdynamics and time-evolution of macroscopic non-Markovian systems. II, Z. Phys. B 29, 273 (1978).
  25. M. Ferrario and P. Grigolini, The non-Markovian relaxation process as a “contraction” of a multidimensional one of Markovian type, J. Math. Phys. (N.Y.) 20, 2567 (1979).
  26. S. Rahav and C. Jarzynski, Fluctuation relations and coarse-graining, J. Stat. Mech. (2007) P09012.
  27. H. Wang and H. Qian, On detailed balance and reversibility of semi-Markov processes and single-molecule enzyme kinetics, J. Math. Phys. (N.Y.) 48, 013303 (2007).
  28. J. Mehl, B. Lander, C. Bechinger, V. Blickle, and U. Seifert, Role of hidden slow degrees of freedom in the fluctuation theorem, Phys. Rev. Lett. 108, 220601 (2012).
  29. M. Esposito, Stochastic thermodynamics under coarse graining, Phys. Rev. E 85, 041125 (2012).
  30. A. Puglisi, S. Pigolotti, L. Rondoni, and A. Vulpiani, Entropy production and coarse graining in Markov processes, J. Stat. Mech. (2010) P05015.
  31. G. Teza and A. L. Stella, Exact coarse graining preserves entropy production out of equilibrium, Phys. Rev. Lett. 125, 110601 (2020).
  32. P. Talkner and P. Hänggi, Colloquium: Statistical mechanics and thermodynamics at strong coupling: Quantum and classical, Rev. Mod. Phys. 92, 041002(C) (2020).
  33. D. Hartich and A. Godec, Emergent memory and kinetic hysteresis in strongly driven networks, Phys. Rev. X 11, 041047 (2021).
  34. D. Hartich and A. Godec, Violation of local detailed balance upon lumping despite a clear timescale separation, Phys. Rev. Res. 5, L032017 (2023).
  35. J. van der Meer, B. Ertel, and U. Seifert, Thermodynamic inference in partially accessible Markov networks: A unifying perspective from transition-based waiting time distributions, Phys. Rev. X 12, 031025 (2022).
  36. C. Dieball and A. Godec, Perspective: Time irreversibility in systems observed at coarse resolution, J. Chem. Phys. 162, 090901 (2025).
  37. P. E. Harunari, A. Dutta, M. Polettini, and E. Roldán, What to learn from a few visible transitions’ statistics?, Phys. Rev. X 12, 041026 (2022).
  38. A. Godec and D. E. Makarov, Challenges in inferring the directionality of active molecular processes from single-molecule fluorescence resonance energy transfer trajectories, J. Phys. Chem. Lett. 14, 49 (2023).
  39. J. van der Meer, J. Degünther, and U. Seifert, Time-resolved statistics of snippets as general framework for model-free entropy estimators, Phys. Rev. Lett. 130, 257101 (2023).
  40. K. Blom, K. Song, E. Vouga, A. Godec, and D. E. Makarov, Milestoning estimators of dissipation in systems observed at a coarse resolution, Proc. Natl. Acad. Sci. U.S.A. 121, e2318333121 (2024).
  41. J. Degünther, J. van der Meer, and U. Seifert, General theory for localizing the where and when of entropy production meets single-molecule experiments, Proc. Natl. Acad. Sci. U.S.A. 121, e2405371121 (2024).
  42. C. Ratzke, B. Hellenkamp, and T. Hugel, Four-colour FRET reveals directionality in the Hsp90 multicomponent machinery, Nat. Commun. 5, 4192 (2014).
  43. J. L. Lebowitz and H. Spohn, A Gallavotti–Cohen-type symmetry in the large deviation functional for stochastic dynamics, J. Stat. Phys. 95, 333 (1999).
  44. P. Gaspard, Fluctuation theorem for nonequilibrium reactions, J. Chem. Phys. 120, 8898 (2004).
  45. U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
  46. D. Hartich and A. Godec, Comment on “Inferring broken detailed balance in the absence of observable currents”, Nat. Commun. 15, 8678 (2024).
  47. I. A. Martínez, G. Bisker, J. M. Horowitz, and J. M. R. Parrondo, Inferring broken detailed balance in the absence of observable currents, Nat. Commun. 10, 3542 (2019).
  48. X. Zhao, D. Hartich, and A. Godec, Emergence of memory in equilibrium versus nonequilibrium systems, Phys. Rev. Lett. 132, 147101 (2024).
  49. Q. Yu, D. Zhang, and Y. Tu, Inverse power law scaling of energy dissipation rate in nonequilibrium reaction networks, Phys. Rev. Lett. 126, 080601 (2021).
  50. Q. Yu and Y. Tu, State-space renormalization group theory of nonequilibrium reaction networks: Exact solutions for hypercubic lattices in arbitrary dimensions, Phys. Rev. E 105, 044140 (2022).
  51. H. Qian, Phosphorylation energy hypothesis: Open chemical systems and their biological functions, Annu. Rev. Phys. Chem. 58, 113 (2007).
  52. R. D. Astumian, Stochastic conformational pumping: A mechanism for free-energy transduction by molecules, Annu. Rev. Biophys. 40, 289 (2011).
  53. N. Shiraishi, K. Funo, and K. Saito, Speed limit for classical stochastic processes, Phys. Rev. Lett. 121, 070601 (2018).
  54. E. Aurell, C. Mejía-Monasterio, and P. Muratore-Ginanneschi, Optimal protocols and optimal transport in stochastic thermodynamics, Phys. Rev. Lett. 106, 250601 (2011).
  55. S. Ito and A. Dechant, Stochastic time evolution, information geometry, and the Cramér-Rao bound, Phys. Rev. X 10, 021056 (2020).
  56. N. Shiraishi and K. Saito, Information-theoretical bound of the irreversibility in thermal relaxation processes, Phys. Rev. Lett. 123, 110603 (2019).
  57. G. Falasco and M. Esposito, Dissipation-time uncertainty relation, Phys. Rev. Lett. 125, 120604 (2020).
  58. T. Van Vu and K. Saito, Thermodynamic unification of optimal transport: Thermodynamic uncertainty relation, minimum dissipation, and thermodynamic speed limits, Phys. Rev. X 13, 011013 (2023).
  59. C. Dieball and A. Godec, Thermodynamic bounds on generalized transport: From single-molecule to bulk observables, Phys. Rev. Lett. 133, 067101 (2024).
  60. A. B. Kolomeisky and M. E. Fisher, Molecular motors: A theorist’s perspective, Annu. Rev. Phys. Chem. 58, 675 (2007).
  61. U. Seifert, Efficiency of autonomous soft nanomachines at maximum power, Phys. Rev. Lett. 106, 020601 (2011).
  62. L. Oberreiter, U. Seifert, and A. C. Barato, Universal minimal cost of coherent biochemical oscillations, Phys. Rev. E 106, 014106 (2022).
  63. Y. Cao, H. Wang, Q. Ouyang, and Y. Tu, The free-energy cost of accurate biochemical oscillations, Nat. Phys. 11, 772 (2015).
  64. J. J. Hopfield, Kinetic proofreading: A new mechanism for reducing errors in biosynthetic processes requiring high specificity, Proc. Natl. Acad. Sci. U.S.A. 71, 4135 (1974).
  65. J. Ninio, Kinetic amplification of enzyme discrimination, Biochimie 57, 587 (1975).
  66. J. D. Mallory, O. A. Igoshin, and A. B. Kolomeisky, Do we understand the mechanisms used by biological systems to correct their errors?, J. Phys. Chem. B 124, 9289 (2020).
  67. P. Mehta and D. J. Schwab, Energetic costs of cellular computation, Proc. Natl. Acad. Sci. U.S.A. 109, 17978 (2012).

  68. See Code at SchwarzTass/consistent-time-reversal for an efficient implementation of the estimator. https://github.com/schwarzTass/Consistent-Time-Reversal.

  69. See Supplemental Material at http://link.aps.org/supplemental/10.1103/gwf7-6kvr for details on the models and the underlying calculations and mathematical proofs of the claims, which includes Refs. [70–75].
  70. C. Godrèche and J. M. Luck, Statistics of the occupation time of renewal processes, J. Stat. Phys. 104, 489 (2001).
  71. K. J. Falconer, Fractals: A Very Short Introduction, 2 (Oxford University Press, Oxford, 2013), 10.1093/actrade/9780199675982.003.0002.
  72. C. Song, S. Havlin, and H. A. Makse, Self-similarity of complex networks, Nature 433, 392 (2005).
  73. B. Krön, Growth of self-similar graphs, J. Graph Theory 45, 224 (2004).
  74. L. Malozemov and A. Teplyaev, Pure point spectrum of the Laplacians on fractal graphs, J. Funct. Anal. 129, 390 (1995).
  75. L. Malozemov and A. Teplyaev, Self-similarity, operators and dynamics, Math. Phys. Anal. Geom. 6, 201 (2003).
  76. W. Feller, An Introduction to Probability Theory and Its Applications, 3rd ed. (John Wiley & Sons, Inc, New York, 1968).
  77. L. Cocconi, G. Salbreux, and G. Pruessner, Scaling of entropy production under coarse graining in active disordered media, Phys. Rev. E 105, L042601 (2022).
  78. X. Zhao, D. E. Makarov, and A. Godec, Towards Markov-state holography, New J. Phys. 27, 125002 (2025).

  79. That is, the sequence of states of the embedded Markov chain.

  80. É. Roldán and J. M. R. Parrondo, Estimating dissipation from single stationary trajectories, Phys. Rev. Lett. 105, 150607 (2010).
  81. Q. Yu and P. E. Harunari, Dissipation at limited resolutions: Power law and detection of hidden dissipative scales, J. Stat. Mech. 2024, 103201 (2024).
  82. B. Ertel and U. Seifert, Estimator of entropy production for partially accessible Markov networks based on the observation of blurred transitions, Phys. Rev. E 109, 054109 (2024).
  83. J. Schnakenberg, Network theory of microscopic and macroscopic behavior of master equation systems, Rev. Mod. Phys. 48, 571 (1976).
  84. K. Brandner, Dynamics of microscale and nanoscale systems in the weak-memory regime: A mathematical framework beyond the Markov approximation, Phys. Rev. E 111, 014137 (2025).
  85. K. Brandner, Dynamics of microscale and nanoscale systems in the weak-memory regime, Phys. Rev. Lett. 134, 037101 (2025).
  86. R. F. Fox, Correlation time expansion for non-Markovian, Gaussian, stochastic processes, Phys. Lett. 94A, 281 (1983).
  87. R. F. Fox, Functional-calculus approach to stochastic differential equations, Phys. Rev. A 33, 467 (1986).
  88. P. J. Foster, J. Bae, B. Lemma, J. Zheng, W. Ireland, P. Chandrakar, R. Boros, Z. Dogic, D. J. Needleman, and J. J. Vlassak, Dissipation and energy propagation across scales in an active cytoskeletal material, Proc. Natl. Acad. Sci. U.S.A. 120, e2207662120 (2023).
  89. L. Vollmar, R. Bebon, J. Schimpf, B. Flietel, S. Celiksoy, C. Sönnichsen, A. Godec, and T. Hugel, Model-free inference of memory in conformational dynamics of a multi-domain protein, J. Phys. A 57, 365001 (2024).
  90. M. Schörner, S. R. Beyer, J. Southall, R. J. Cogdell, and J. Köhler, Conformational memory of a protein revealed by single-molecule spectroscopy, J. Phys. Chem. B 119, 13964 (2015).
  91. K. Song, R. Park, A. Das, D. E. Makarov, and E. Vouga, Non-Markov models of single-molecule dynamics from information-theoretical analysis of trajectories, J. Chem. Phys. 159, 064104 (2023).
  92. V. Salnikov, M. T. Schaub, and R. Lambiotte, Using higher-order Markov models to reveal flow-based communities in networks, Sci. Rep. 6, 23194 (2016).
  93. G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems, 1st ed. (Wiley, New York, 1977).
  94. H. Qian, S. Saffarian, and E. L. Elson, Concentration fluctuations in a mesoscopic oscillating chemical reaction system, Proc. Natl. Acad. Sci. U.S.A. 99, 10376 (2002).
  95. J. H. Fritz, B. Nguyen, and U. Seifert, Stochastic thermodynamics of chemical reactions coupled to finite reservoirs: A case study for the Brusselator, J. Chem. Phys. 152, 235101 (2020).

  96. That is, W is a lumping that refines the partition U.

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Phys. Rev. Lett. 137, 187401– Published 29 October, 2026
  • Received 14 October 2024
  • Revised 15 December 2025
  • Accepted 25 September 2026
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