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  • Open Access

Weak-Memory Dynamics in Discrete Time

Hugues Meyer and Kay Brandner

  • School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, United Kingdom

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Phys. Rev. Lett. 136, 220401 – Published 1 June, 2026

DOI: https://doi.org/10.1103/kdhy-66b6

Abstract

Discrete dynamics arise naturally in systems with broken temporal translation symmetry and are typically described by first-order recurrence relations representing classical or quantum Markov chains. When memory effects induced by hidden degrees of freedom are relevant, however, higher-order discrete evolution equations are generally required. Focusing on linear dynamics, we identify a well-delineated weak-memory regime where such equations can, on an intermediate timescale, be systematically reduced to a unique first-order counterpart acting on the same state space. We formulate our results as a mathematical theorem and work out two examples showing how they can be applied to stochastic Floquet dynamics under coarse-grained and quantum collisional models.

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References (85)

  1. M. Bukov, L. D’Alessio, and A. Polkovnikov, Universal high-frequency behavior of periodically driven systems: From dynamical stabilization to Floquet engineering, Adv. Phys. 64, 139 (2015).
  2. A. Eckardt, Colloquium: Atomic quantum gases in periodically driven optical lattices, Rev. Mod. Phys. 89, 011004 (2017).
  3. T. Oka and S. Kitamura, Floquet engineering of quantum materials, Annu. Rev. Condens. Matter Phys. 10, 387 (2019).
  4. M. Moskalets and M. Büttiker, Floquet scattering theory of quantum pumps, Phys. Rev. B 66, 205320 (2002).
  5. S. Ray and A. C. Barato, Stochastic thermodynamics of periodically driven systems: Fluctuation theorem for currents and unification of two classes, Phys. Rev. E 96, 052120 (2017).
  6. P. Menczel and K. Brandner, Limit cycles in periodically driven open quantum systems, J. Phys. A 52, 43LT01 (2019).
  7. T. Mori, Floquet states in open quantum systems, Annu. Rev. Condens. Matter Phys. 14, 35 (2023).
  8. N. A. Sinitsyn and I. Nemenman, Universal geometric theory of mesoscopic stochastic pumps and reversible ratchets, Phys. Rev. Lett. 99, 220408 (2007).
  9. S. Rahav, J. Horowitz, and C. Jarzynski, Directed flow in nonadiabatic stochastic pumps, Phys. Rev. Lett. 101, 140602 (2008).
  10. M. Esposito and J. M. R. Parrondo, Stochastic thermodynamics of hidden pumps, Phys. Rev. E 91, 052114 (2015).
  11. O. Raz, Y. Subaş𝚤, and C. Jarzynski, Mimicking nonequilibrium steady states with time-periodic driving, Phys. Rev. X 6, 021022 (2016).
  12. K. Funo, N. Lambert, F. Nori, and C. Flindt, Shortcuts to adiabatic pumping in classical stochastic systems, Phys. Rev. Lett. 124, 150603 (2020).
  13. A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum entanglement growth under random unitary dynamics, Phys. Rev. X 7, 031016 (2017).
  14. A. Nahum, S. Vijay, and J. Haah, Operator spreading in random unitary circuits, Phys. Rev. X 8, 021014 (2018).
  15. C. W. Von Keyserlingk, T. Rakovszky, F. Pollmann, and S. L. Sondhi, Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws, Phys. Rev. X 8, 021013 (2018).
  16. V. Khemani, A. Vishwanath, and D. A. Huse, Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws, Phys. Rev. X 8, 031057 (2018).
  17. A. Chan, A. De Luca, and J. T. Chalker, Solution of a minimal model for many-body quantum chaos, Phys. Rev. X 8, 041019 (2018).
  18. T. Rakovszky, F. Pollmann, and C. W. Von Keyserlingk, Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation, Phys. Rev. X 8, 031058 (2018).
  19. T. Rakovszky, F. Pollmann, and C. W. Von Keyserlingk, Sub-ballistic growth of Rényi entropies due to diffusion, Phys. Rev. Lett. 122, 250602 (2019).
  20. B. Bertini, P. Kos, and T. Prosen, Entanglement spreading in a minimal model of maximal many-body quantum chaos, Phys. Rev. X 9, 021033 (2019).
  21. K. Klobas, B. Bertini, and L. Piroli, Exact thermalization dynamics in the “rule 54” quantum cellular automaton, Phys. Rev. Lett. 126, 160602 (2021).
  22. K. Klobas and B. Bertini, Entanglement dynamics in rule 54: Exact results and quasiparticle picture, SciPost Phys. 11, 107 (2021).
  23. K. Klobas and B. Bertini, Exact relaxation to Gibbs and non-equilibrium steady states in the quantum cellular automaton rule 54, SciPost Phys. 11, 106 (2021).
  24. B. Buča, K. Klobas, and T. Prosen, Rule 54: Exactly solvable model of nonequilibrium statistical mechanics, J. Stat. Mech. (2021) P074001.
  25. B. Bertini, C. De Fazio, J. P. Garrahan, and K. Klobas, Exact quench dynamics of the Floquet quantum east model at the deterministic point, Phys. Rev. Lett. 132, 120402 (2024).
  26. S. W. P. Kim, F. Hübner, J. P. Garrahan, and B. Doyon, Circuits as a simple platform for the emergence of hydrodynamics in deterministic chaotic many-body systems, arXiv:2503.08788.
  27. D. Karevski and T. Platini, Quantum nonequilibrium steady states induced by repeated interactions, Phys. Rev. Lett. 102, 207207 (2009).
  28. P. Strasberg, G. Schaller, T. Brandes, and M. Esposito, Quantum and information thermodynamics: A unifying framework based on repeated interactions, Phys. Rev. X 7, 021003 (2017).
  29. P. Strasberg, Repeated interactions and quantum stochastic thermodynamics at strong coupling, Phys. Rev. Lett. 123, 180604 (2019).
  30. S. Seah, S. Nimmrichter, D. Grimmer, J. P. Santos, V. Scarani, and G. T. Landi, Collisional quantum thermometry, Phys. Rev. Lett. 123, 180602 (2019).
  31. F. L. S. Rodrigues, G. De Chiara, M. Paternostro, and G. T. Landi, Thermodynamics of weakly coherent collisional models, Phys. Rev. Lett. 123, 140601 (2019).
  32. S. Pigolotti and A. Vulpiani, Coarse graining of master equations with fast and slow states, J. Chem. Phys. 128, 154114 (2008).
  33. M. Esposito, Stochastic thermodynamics under coarse graining, Phys. Rev. E 85, 041125 (2012).
  34. R. Yvinec, C. Zhuge, J. Lei, and M. C. Mackey, Adiabatic reduction of a model of stochastic gene expression with jump Markov process, J. Math. Biol. 68, 1051 (2014).
  35. G. Hummer and A. Szabo, Optimal dimensionality reduction of multistate kinetic and Markov-state models, J. Phys. Chem. B 119, 9029 (2015).
  36. A. Wachtel, R. Rao, and M. Esposito, Thermodynamically consistent coarse graining of biocatalysts beyond Michaelis–Menten, New J. Phys. 20, 042002 (2018).
  37. A. Lapolla and A. Godec, Manifestations of projection-induced memory: General theory and the tilted single file, Front. Phys. 7, 182 (2019).
  38. P. Strasberg and M. Esposito, Non-Markovianity and negative entropy production rates, Phys. Rev. E 99, 012120 (2019).
  39. D. M. Busiello, D. Gupta, and A. Maritan, Coarse-grained entropy production with multiple reservoirs: Unraveling the role of time scales and detailed balance in biology-inspired systems, Phys. Rev. Res. 2, 043257 (2020).
  40. D. Seiferth, P. Sollich, and S. Klumpp, Coarse graining of biochemical systems described by discrete stochastic dynamics, Phys. Rev. E 102, 062149 (2020).
  41. S. A. M. Loos and S. H. L. Klapp, Irreversibility, heat and information flows induced by non-reciprocal interactions, New J. Phys. 22, 123051 (2020).
  42. A. V. Straube, B. G. Kowalik, R. R. Netz, and F. Höfling, Rapid onset of molecular friction in liquids bridging between the atomistic and hydrodynamic pictures, Commun. Phys. 3, 126 (2020).
  43. C. Ayaz, L. Tepper, F. N. Brünig, J. Kappler, J. O. Daldrop, and R. R. Netz, Non-Markovian modeling of protein folding, Proc. Natl. Acad. Sci. U.S.A. 118, e2023856118 (2021).
  44. D. Hartich and A. Godec, Emergent memory and kinetic hysteresis in strongly driven networks, Phys. Rev. X 11, 041047 (2021).
  45. A. Lapolla and A. Godec, Toolbox for quantifying memory in dynamics along reaction coordinates, Phys. Rev. Res. 3, L022018 (2021).
  46. D. Hartich and A. Godec, Violation of local detailed balance upon lumping despite a clear timescale separation, Phys. Rev. Res. 5, L032017 (2023).
  47. F. Coghi, Current fluctuations of a self-interacting diffusion on a ring, J. Phys. A 58, 015002 (2024).
  48. 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).
  49. 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).
  50. X. Zhao, D. Hartich, and A. Godec, Emergence of memory in equilibrium versus nonequilibrium systems, Phys. Rev. Lett. 132, 147101 (2024).
  51. M. Wiśniewski and J. Spiechowicz, Memory-induced current reversal of Brownian motors, Phys. Rev. E 111, 024130 (2025).
  52. D. Lesnicki, R. Vuilleumier, A. Carof, and B. Rotenberg, Molecular hydrodynamics from memory kernels, Phys. Rev. Lett. 116, 147804 (2016).
  53. G. Jung, M. Hanke, and F. Schmid, Iterative reconstruction of memory kernels, J. Chem. Theory Comput. 13, 2481 (2017).
  54. H. Meyer, S. Wolf, G. Stock, and T. Schilling, A numerical procedure to evaluate memory effects in non-equilibrium coarse-grained models, Adv. Theor. Simul. 4, 2000197 (2021).
  55. M. T. Vrugt and R. Wittkowski, Projection operators in statistical mechanics: A pedagogical approach, Eur. J. Phys. 41, 045101 (2020).
  56. V. Klippenstein, M. Tripathy, G. Jung, F. Schmid, and N. F. A. Van Der Vegt, Introducing memory in coarse-grained molecular simulations, J. Phys. Chem. B 125, 4931 (2021).
  57. T. Schilling, Coarse-grained modelling out of equilibrium, Phys. Rep. 972, 1 (2022).
  58. B. Hilder and U. Sharma, Quantitative coarse-graining of Markov chains, SIAM J. Math. Anal. 56, 913 (2024).
  59. K. Brandner, Dynamics of microscale and nanoscale systems in the weak-memory regime, Phys. Rev. Lett. 134, 037101 (2025).
  60. 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).
  61. See Supplemental Material at http://link.aps.org/supplemental/10.1103/kdhy-66b6, which provides details of the calculations leading to the results presented in the main text. It is divided in three sections: (i) Master Theorem, (ii) Application 1: Mesoscopic Charge Pump, (iii) Application 2: Collisional Model. It includes Refs. [62–65].
  62. P. Agarwal, M. Jleli, and B. Samet, Fixed Point Theory in Metric Spaces, 1st ed. (Springer Nature, Singapore, 2018).
  63. R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed. (Cambridge University Press, New York, NY, 2013).
  64. Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, 1st ed., SpringerBriefs in Physics (Springer, Berlin, Heidelberg, 2012).
  65. T. Burton, Volterra Integral and Differential Equations, 2nd ed., Mathematics in Science and Engineering Vol. 202 (Elsevier, Amsterdam, 2005).
  66. W. T. Strunz and T. Yu, Convolutionless non-Markovian master equations and quantum trajectories: Brownian motion, Phys. Rev. A 69, 052115 (2004).
  67. C. H. Fleming and N. I. Cummings, Accuracy of perturbative master equations, Phys. Rev. E 83, 031117 (2011).
  68. R. Hartmann and W. T. Strunz, Accuracy assessment of perturbative master equations: Embracing nonpositivity, Phys. Rev. A 101, 012103 (2020).
  69. C. S. Tello Breuer, T. Becker, and A. Eckardt, Benchmarking quantum master equations beyond ultraweak coupling, Phys. Rev. B 110, 064319 (2024).
  70. Z. Sartipi, R. Gundermann, J. Anders, and P. Saalfrank, Canonically consistent quantum master equation for proton-transfer reactions, arXiv:2603.21865.
  71. U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
  72. G. Benenti, G. Casati, K. Saito, and R. S. Whitney, Fundamental aspects of steady-state conversion of heat to work at the nanoscale, Phys. Rep. 694, 1 (2017).
  73. E. Mozgunov and D. Lidar, Completely positive master equation for arbitrary driving and small level spacing, Quantum 4, 227 (2020).
  74. F. Nathan and M. S. Rudner, Universal Lindblad equation for open quantum systems, Phys. Rev. B 102, 115109 (2020).
  75. A. Schnell, A. Eckardt, and S. Denisov, Is there a Floquet Lindbladian?, Phys. Rev. B 101, 100301 (2020).
  76. R. Hotz and G. Schaller, Coarse-graining master equation for periodically driven systems, Phys. Rev. A 104, 052219 (2021).
  77. T. Mori, Strong Markov dissipation in driven-dissipative quantum systems, J. Stat. Phys. 192, 1 (2025).
  78. F. Ciccarello, S. Lorenzo, V. Giovannetti, and G. M. Palma, Quantum collision models: Open system dynamics from repeated interactions, Phys. Rep. 954, 1 (2022).

  79. In our discussion of collisional models, we formally treat density matrices as vectors and quantum maps as matrices. For this approach to be consistent, |·| must, when applied to a density matrix, be evaluated as the Hilbert-Schmidt norm; ∥·∥ denotes the operator norm induced by |·| throughout.

  80. G. Jung, M. Hanke, and F. Schmid, Iterative reconstruction of memory kernels, J. Chem. Theory Comput. 13, 2481 (2017).
  81. V. Klippenstein, M. Tripathy, G. Jung, F. Schmid, and N. F. A. Van Der Vegt, Introducing memory in coarse-grained molecular simulations, J. Phys. Chem. B 125, 4931 (2021).
  82. H. Vroylandt, L. Goudenège, P. Monmarché, F. Pietrucci, and B. Rotenberg, Likelihood-based non-Markovian models from molecular dynamics, Proc. Natl. Acad. Sci. U.S.A. 119, e2117586119 (2022).
  83. L. Tepper, B. Dalton, and R. R. Netz, Accurate memory kernel extraction from discretized time-series data, J. Chem. Theory Comput. 20, 3061 (2024).
  84. B. A. Dalton, A. Klimek, H. Kiefer, F. N. Brünig, H. Colinet, L. Tepper, A. Abbasi, and R. R. Netz, Memory and friction: From the nanoscale to the macroscale, Annu. Rev. Phys. Chem. 76, 431 (2025).
  85. S. Satore, F. Teichmann, and G. Stock, Markov-type state models to describe non-Markovian dynamics, J. Chem. Theory Comput. 21, 2757 (2025).

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Phys. Rev. Lett. 136, 220401– Published 1 June, 2026
  • Received 5 November 2025
  • Accepted 20 April 2026
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