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arxiv: 2606.28289 · v1 · pith:X2LQUPCWnew · submitted 2026-06-26 · ❄️ cond-mat.stat-mech · physics.bio-ph· physics.chem-ph

Optimal parameterization of nonequilibrium generalized master equations from discrete-time experimental data

Chih-Wei Joshua Liu , J\'er\'emie Klinger , Grant M. Rotskoff This is my paper

Pith reviewed 2026-06-29 01:44 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.bio-phphysics.chem-ph
keywords generalized master equationsmaximum likelihood estimationoptimal transportnonequilibrium dynamicsMarkov state modelskinetic analysisbiomolecular systemsmemory kernels
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0 comments X

The pith

A maximum-likelihood procedure using optimal transport constructs physically feasible discrete-time generalized master equations from nonequilibrium data.

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

The paper develops a statistical method to fit generalized master equations that include memory effects to time-series data from experiments or simulations. These equations avoid the Markov assumption common in state models and work for systems far from equilibrium. By adapting optimal transport techniques, the approach ensures the fitted models respect physical constraints such as probability conservation. This enables extraction of kinetic quantities like relaxation rates and first-passage times from data on ion channels, molecular motors, and protein folding. A sympathetic reader would care because it offers a more accurate way to analyze complex biomolecular dynamics without forcing simplifying assumptions.

Core claim

The authors present a maximum-likelihood-based procedure to parameterize formally exact, physically feasible, discrete-time generalized master equations from experiments and simulations in and out of equilibrium. By adapting algorithms from optimal transport, they construct conditional-maximum-likelihood estimators for both Nakajima-Zwanzig memory kernels and time-convolutionless GME propagators. When applied to experimental and simulated data, these estimators recover key kinetic parameters including relaxation rates, irreversibilities, dwell times, and first-passage times.

What carries the argument

Conditional-maximum-likelihood estimators of Nakajima-Zwanzig memory kernels and time-convolutionless GME propagators, built by adapting optimal transport algorithms to enforce physical constraints in discrete time.

If this is right

  • The method recovers relaxation rates, irreversibilities, dwell times, and first-passage times from data.
  • Discrete-time GMEs serve as a physically and statistically principled alternative to Markov state models for kinetic analyses.
  • The estimators apply to both experimental recordings and simulated trajectories in and out of equilibrium.
  • They satisfy physical constraints for both memory kernels and propagators.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach could extend to other coarse-grained systems where memory arises from hidden degrees of freedom.
  • Optimal transport methods may generally help enforce constraints in statistical inference for dynamical systems.
  • Integration with techniques for defining states from high-dimensional data could broaden applicability.

Load-bearing premise

Adapting optimal transport algorithms produces estimators that satisfy the physical constraints required for generalized master equations applied to discrete-time data.

What would settle it

Observing that the inferred propagators produce negative probabilities or fail to match the empirical transition statistics in validation data would falsify the claim that the estimators are physically feasible and accurate.

Figures

Figures reproduced from arXiv: 2606.28289 by Chih-Wei Joshua Liu, Grant M. Rotskoff, J\'er\'emie Klinger.

Figure 1
Figure 1. Figure 1: FIG. 1. Coarse-grained MJP and parameterization of its discrete-time GMEs. Toy ergodic MJP (a) generates example [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. CFTR. (a) Cystic-fibrosis transmembrane-conductance regulator CFTR [PDB: 6MSM (upper), 8FZQ (lower)] is a [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. BMF1. (a) Bovine mitochondrial F [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. HP35. (a) The Lys24-Nle/Lys29-Nle variant of villin headpiece HP35 (PDB: 2F4K) exhibits [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Convergence in an adversarial toy example. The coarse-grained MJP in (a) generates example macrostate series (b) [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
read the original abstract

Kinetic analyses of experiments often require coarse-grained descriptions, but complex systems rarely conform to the widely used modeling assumptions of Markovianity and thermodynamic equilibrium. Memory is indeed a general and often inevitable consequence of coarse-graining. Markov state models (MSMs) are a popular choice of coarse-grained description, but require microstate assignments -- which are rarely experimentally tunable -- to macrostates that minimize memory. Generalized master equations (GMEs) circumvent this limitation of MSMs by explicitly capturing memory. However, GMEs are difficult to parameterize and usually formally approximate in the experimentally relevant discrete-time setting. Here we introduce a maximum-likelihood-based procedure to parameterize formally exact, physically feasible, discrete-time generalized master equations from experiments and simulations in and out of equilibrium. By adapting algorithms typically used in optimal transport, we construct physical-constraint-satisfying conditional-maximum-likelihood estimators of both exact Nakajima-Zwanzig memory kernels and time-convolutionless GME propagators in discrete time. Applying these estimators to three examples -- experimental recordings of F\"orster-resonance energy-transfer in an ion channel, experimental nanoparticle tracking of a processive molecular motor, and simulated folding of a benchmark protein domain -- we recover kinetic parameters including relaxation rates, irreversibilities, dwell times, and first-passage times. These results establish discrete-time GMEs as a physically and statistically principled alternative to MSMs for kinetic analyses of experimental and simulated biomolecular systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces a maximum-likelihood procedure, adapted from optimal transport algorithms, to parameterize formally exact and physically feasible discrete-time generalized master equations (both Nakajima-Zwanzig memory kernels and time-convolutionless propagators) from experimental and simulation data in and out of equilibrium. The estimators are constructed to satisfy constraints including positivity, normalization, and detailed balance where applicable; the method is demonstrated on three examples (FRET recordings in an ion channel, nanoparticle tracking of a processive motor, and simulated protein domain folding), recovering quantities such as relaxation rates, irreversibilities, dwell times, and first-passage times.

Significance. If the central construction holds, the work supplies a statistically rigorous and constraint-enforcing alternative to Markov state models for non-Markovian kinetic analysis of biomolecular data. It enables direct use of discrete-time recordings without requiring microstate-to-macrostate assignments that minimize memory, and the explicit use of conditional maximum-likelihood estimators with physical feasibility is a clear strength.

minor comments (3)
  1. [Methods] The manuscript would benefit from a short, self-contained paragraph in the methods section explicitly linking the regularized transport problem to the KKT conditions of the original constrained MLE, even if the full derivation is in the supplement.
  2. [Results] In the three validation examples, the reported kinetic parameters (e.g., first-passage times in the protein-folding case) should include uncertainty estimates derived from the estimator; this is a standard expectation for maximum-likelihood procedures.
  3. [Theory] Notation for the discrete-time projection operators and the memory kernel discretization should be unified across the Nakajima-Zwanzig and time-convolutionless sections to avoid reader confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives conditional-maximum-likelihood estimators for discrete-time Nakajima-Zwanzig kernels and time-convolutionless propagators by recasting the constrained likelihood maximization as a regularized optimal-transport problem whose solution is shown to satisfy the KKT conditions of the original constrained MLE; the discrete-time projection operators remain exact by construction. No equation reduces a claimed prediction or first-principles result to a quantity defined solely by the fitted parameters themselves, no uniqueness theorem is imported from self-citation, and no ansatz is smuggled via prior work. The three validation examples on independent experimental and simulation data supply external checks, confirming the central procedure is not equivalent to its inputs by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on the applicability of maximum-likelihood estimation to the GME model and the feasibility of adapting optimal transport algorithms to enforce physical constraints. No free parameters, ad-hoc axioms, or invented entities are mentioned.

axioms (2)
  • standard math Maximum likelihood estimation yields optimal estimators for the GME parameters under the stated model assumptions.
    Basis for constructing the estimators from data.
  • domain assumption Optimal transport algorithms can be adapted to produce estimators that satisfy physical constraints such as probability conservation.
    Central technical step described in the abstract.

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discussion (0)

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Reference graph

Works this paper leans on

142 extracted references

  1. [1]

    [121] to a single unfolded macrostate

    to a single near-native macrostate, and all states described as partly or fully unfolded by Ref. [121] to a single unfolded macrostate. These macrostates reflect the three implied by two-stage folding inT-jump experiments [120]. Using Alg. 2 with Alg. 3 for information projec- tions, we estimate conditional-maximum-likelihood TCL- GME-DT propagators with ...

  2. [2]

    kinetic distances

    to project transition-matrix iterates onto the feasi- ble set, suggesting connections between entropic optimal transport—a celebrated use of the Sinkhorn-Knopp al- gorithm [45]—and coarse-grained kinetic modeling. Ap- proaches that interpret flux matrices as transport plans between macrostates may merit further investigation. Previous works minimize “kine...

  3. [3]

    The usual Knight-Ruiz-U¸ car algorithm [112] differs from Alg. 5 in lacking the initial geometric symmetriza- tion ˆΘ←sqrt( ˆΘ⊙ ˆΘT); it only preserves the symmetry of already symmetric input fluxes ˆΘ during projection onto the feasible set defined by Eqs. 47-49. Alg. 5 neverthe- less minimizesD KL(Θ∥ ˆΘ) subject to Eqs. 47-49 and 51 with initial geometr...

  4. [4]

    47-49 equals the projection of ˆΘ onto the set satisfying Eqs

    Thus symmetry-preserving projection of sqrt(ˆΘ⊙ ˆΘT) onto the set satisfying Eqs. 47-49 equals the projection of ˆΘ onto the set satisfying Eqs. 47-49 and 51. We now describe a Levenberg-Marquardt algorithm [114, 115] for projecting local-flux estimates ˆΘ onto the subset satisfying Eq. 54. We note that maximizing Eq. 60 is equivalent to minimizing the lo...

    1927

  5. [5]

    00 U (n) ij 0 1 2 3 4 5 6 7 8 9 Delay nτ 10−1 10−5 10−9 K(n) ij Diagrammatic Analytic 0 1 2 3 4 5 6 7 8 9 10 Lagtime nτ

  6. [6]

    00 G(n) ij Cutoff 102 103 104 105 106 Observation time N τ

  7. [7]

    00 ∥ ˆU (n) − U (n)∥2 n = 1 (0.95 CI) n = 2 (0.95 CI) n = 3 (0.95 CI) 102 103 104 105 106 Observation time N τ

  8. [8]

    00 ∥ ˆK(n) − K(n)∥2 n = 0 (0.95 CI) n = 1 (0.95 CI) n = 2 (0.95 CI) 102 103 104 105 106 Observation time N τ

  9. [9]

    00 ∥ ˆG(n) − G(n)∥2 (a) (b) (c) (d) (e) (f) (g) (h) FIG. 5. Convergence in an adversarial toy example. The coarse-grained MJP in (a) generates example macrostate series (b) and lacks separation of timescales. Transition rates between microstatesµ 3 andµ 4, which are coarse-grained into a single macrostate, equal transition rates among macrostatesµ 1,µ 2, ...

  10. [10]

    Schilling, Coarse-grained modeling out of equilib- rium, Phys

    T. Schilling, Coarse-grained modeling out of equilib- rium, Phys. Rep.972, 1 (2022)

    2022

  11. [11]

    Schilling, Evolution equations for open systems and collective variables: Which equation would you like to solve by molecular dynamics simulation?, J

    T. Schilling, Evolution equations for open systems and collective variables: Which equation would you like to solve by molecular dynamics simulation?, J. Chem. Phys.161, 191501 (2024)

    2024

  12. [12]

    Trubiano and M

    A. Trubiano and M. F. Hagan, Markov state model approach to simulate self-assembly, Phys. Rev. X14, 041063 (2024)

    2024

  13. [13]

    Sartore, F

    S. Sartore, F. Teichmann, and G. Stock, Markov-type state models to describe non-Markovian dynamics, J. Chem. Theory Comput.21, 2757 (2025)

    2025

  14. [14]

    B. E. Husic, K. A. McKiernan, H. K. Wayment-Steele, M. M. Sultan, and V. S. Pande, A minimum variance clustering approach produces robust and interpretable coarse-grained models, J. Chem. Theory Comput.14, 1071 (2018)

    2018

  15. [15]

    G. R. Bowman, Improved coarse-graining of Markov state models via explicit consideration of statistical un- certainty, J. Chem. Phys.137, 134111 (2012)

    2012

  16. [16]

    P´ erez-Hern´ andez, F

    G. P´ erez-Hern´ andez, F. Paul, T. Giorgino, G. De Fab- ritiis, and F. No´ e, Identification of slow molecular order parameters for Markov model construction, J. Chem. Phys.139, 015102 (2013)

    2013

  17. [17]

    G. Diez, D. Nagel, and G. Stock, Correlation-based fea- ture selection to identify functional dynamics in pro- teins, J. Chem. Theory Comput.18, 5079 (2022)

    2022

  18. [18]

    J. D. Chodera and F. No´ e, Markov state models of biomolecular conformational dynamics, Curr. Opin. Struct. Biol.25, 135 (2015)

    2015

  19. [19]

    B. E. Husic and V. S. Pande, Markov state models: from an art to a science, J. Am. Chem. Soc.140, 2386 (2018)

    2018

  20. [20]

    No´ e, G

    F. No´ e, G. De Fabritiis, and C. Clementi, Machine learning for protein folding and dynamics, Curr. Opin. Struct. Biol.60, 77 (2020)

    2020

  21. [21]

    No´ e and C

    F. No´ e and C. Clementi, Kinetic distance and kinetic maps from molecular dynamics simulation, J. Chem. Theory Comput.11, 5002 (2015)

    2015

  22. [22]

    K. A. Beauchamp, R. McGibbon, Y.-S. Lin, and V. S. Pande, Simple few-state models reveal hidden complex- ity in protein folding, Proc. Natl. Acad. Sci. U.S.A.109, 17807 (2012)

    2012

  23. [23]

    C. R. Schwantes and V. S. Pande, Improvements in Markov state model construction reveal many non- native interactions in the folding of NTL9, J. Chem. Theory Comput.9, 2000 (2013)

    2000

  24. [24]

    W. C. Swope, J. W. Pitera, and F. Suits, Describing protein folding kinetics by molecular dynamics simula- tions. 1. Theory, The Journal of Physical Chemistry B 108, 6571 (2004)

    2004

  25. [25]

    Prinz, J

    J.-H. Prinz, J. D. Chodera, and F. No´ e, Spectral rate theory for two-state kinetics, Phys. Rev. X4, 011020 (2014)

    2014

  26. [26]

    Knoch and T

    F. Knoch and T. Speck, Non-equilibrium Markov state modeling of periodically driven biomolecules, J. Chem. Phys.150, 054103 (2019)

    2019

  27. [27]

    Prinz, H

    J.-H. Prinz, H. Wu, M. Sarich, B. Keller, M. Senne, M. Held, J. D. Chodera, C. Sch¨ utte, and F. No´ e, Markov models of molecular kinetics: generation and validation, J. Chem. Phys.134, 174105 (2011)

    2011

  28. [28]

    K. A. Dill and D. Shortle, Denatured states of proteins, Annu. Rev. Biochem.60, 795 (1991)

    1991

  29. [29]

    Gonzalez-Ballestero, Tutorial: projector approach to master equations for open quantum systems, Quantum 8, 1454 (2024)

    C. Gonzalez-Ballestero, Tutorial: projector approach to master equations for open quantum systems, Quantum 8, 1454 (2024)

    2024

  30. [30]

    A. J. Dominic III, T. Sayer, S. Cao, T. E. Markland, X. Huang, and A. Montoya-Castillo, Building insightful, memory-enriched models to capture long-time biochem- ical processes from short-time simulations, Proc. Natl. Acad. Sci. U.S.A.120, e2221048120 (2023)

    2023

  31. [31]

    Tokuyama and H

    M. Tokuyama and H. Mori, Statistical-mechanical ap- proach to random frequency modulations and the Gaus- sian memory function, Prog. Theor. Exp. Phys.54, 918 (1975)

    1975

  32. [32]

    Nakajima, On quantum theory of transport phenom- ena: steady diffusion, Prog

    S. Nakajima, On quantum theory of transport phenom- ena: steady diffusion, Prog. Theor. Exp. Phys.20, 948 (1958)

    1958

  33. [33]

    Zwanzig, Memory effects in irreversible thermody- namics, Phys

    R. Zwanzig, Memory effects in irreversible thermody- namics, Phys. Rev.124, 983 (1961)

    1961

  34. [34]

    Kidon, E

    L. Kidon, E. Y. Wilner, and E. Rabani, Exact calcula- tion of the time convolutionless master equation gener- ator: Application to the nonequilibrium resonant level model, J. Chem. Phys.143, 234110 (2015)

    2015

  35. [35]

    Gu, Diagrammatic representation and nonperturba- tive approximations of the exact time-convolutionless master equation, J

    B. Gu, Diagrammatic representation and nonperturba- tive approximations of the exact time-convolutionless master equation, J. Chem. Phys.160, 204113 (2024)

    2024

  36. [36]

    Shi and E

    Q. Shi and E. Geva, A new approach to calculating the memory kernel of the generalized quantum master equa- tion for an arbitrary system–bath coupling, J. Chem. Phys.119, 12063 (2003)

    2003

  37. [37]

    Vroylandt, L

    H. Vroylandt, L. Gouden` ege, P. Monmarch´ e, F. Pietrucci, and B. Rotenberg, Likelihood-based non-Markovian models from molecular dynamics, Proc. Natl. Acad. Sci. U.S.A.119, e2117586119 (2022)

    2022

  38. [38]

    A. V. Oppenheim and R. W. Schafer,Discrete-Time Signal Processing, 3rd ed. (Prentice Hall Press, Upper Saddle River, New Jersey, United States of America, 2009)

    2009

  39. [39]

    Larsson and V

    S. Larsson and V. Thom´ ee,Partial Differential Equa- tions with Numerical Methods(Springer Berlin, Heidel- berg, Baden-W¨ urttemberg, Germany, 2003)

    2003

  40. [40]

    E. C. Goonetilleke and X. Huang, Targeting bacte- rial RNA polymerase: Harnessing simulations and ma- chine learning to design inhibitors for drug-resistant pathogens, Biochemistry64, 1169 (2025)

    2025

  41. [41]

    Y. Wu, S. Cao, Y. Qiu, and X. Huang, Tutorial on how to build non-Markovian dynamic models from molecu- lar dynamics simulations for studying protein conforma- tional changes, J. Chem. Phys.160, 121501 (2024)

    2024

  42. [42]

    Lorpaiboon, S

    C. Lorpaiboon, S. C. Guo, J. Strahan, J. Weare, and A. R. Dinner, Accurate estimates of dynamical statistics using memory, The Journal of Chemical Physics160, 084108 (2024)

    2024

  43. [43]

    I. C. Unarta, S. Cao, E. C. Goonetilleke, J. Niu, S. H. Gellman, and X. Huang, Submillisecond atomistic molecular dynamics simulations reveal hydrogen bond- driven diffusion of a guest peptide in protein–RNA con- densate, The Journal of Physical Chemistry B128, 2347 (2024)

    2024

  44. [44]

    A. J. Dominic III, S. Cao, A. Montoya-Castillo, and X. Huang, Memory unlocks the future of biomolecular dynamics: Transformative tools to uncover physical in- 24 sights accurately and efficiently, Journal of the Ameri- can Chemical Society145, 9916 (2023)

    2023

  45. [45]

    S. Cao, Y. Qiu, M. L. Kalin, and X. Huang, Integra- tive generalized master equation: A method to study long-timescale biomolecular dynamics via the integrals of memory kernels, The Journal of Chemical Physics 159, 134106 (2023)

    2023

  46. [46]

    Makri, Discrete generalized quantum master equa- tions, J

    N. Makri, Discrete generalized quantum master equa- tions, J. Chem. Theory Comput.21, 5037 (2025)

    2025

  47. [47]

    S. Cao, A. Montoya-Castillo, W. Wang, T. E. Markland, and X. Huang, On the advantages of exploiting memory in Markov state models for biomolecular dynamics, J. Chem. Phys.153, 014105 (2020)

    2020

  48. [48]

    Sayer and A

    T. Sayer and A. Montoya-Castillo, Compact and com- plete description of non-Markovian dynamics, J. Chem. Phys.158, 014105 (2023)

    2023

  49. [49]

    Mori, Transport, collective motion, and Brownian motion, Prog

    H. Mori, Transport, collective motion, and Brownian motion, Prog. Theor. Exp. Phys.33, 423 (1965)

    1965

  50. [50]

    Zwanzig, Ensemble method in the theory of irre- versibility, J

    R. Zwanzig, Ensemble method in the theory of irre- versibility, J. Chem. Phys.33, 1338 (1960)

    1960

  51. [51]

    A. S. Nemirovski and D. B. Yudin,Slozhnost’ zadach i effecktivnost’ metodov optimizacii(Nauka, Moscow, Russian Soviet Federative Socialist Republic, Union of Soviet Socialist Republics, 1979)

    1979

  52. [52]

    L. M. Bregman, A relaxation method of finding a com- mon point of convex sets and its application to the solu- tion of problems in convex programming, U.S.S.R. Com- put. Math. Math. Phys.7, 200 (1967)

    1967

  53. [53]

    Knopp and R

    P. Knopp and R. Sinkhorn, Concerning nonnegative ma- trices and doubly stochastic matrices, Pacific J. Math 21, 343 (1967)

    1967

  54. [54]

    M. Cuturi, Sinkhorn distances: lightspeed computation of optimal transport, inProceedings of the 27th Inter- national Conference on Neural Information Processing Systems - Volume 2, NIPS’13 (Curran Associates Inc., Red Hook, New York, United States of America, 2013) pp. 2292–2300

    2013

  55. [55]

    Meng and D

    X.-L. Meng and D. B. Rubin, Maximum likelihood esti- mation via the ECM algorithm: A general framework, Biometrika80, 267 (1993)

    1993

  56. [56]

    F¨ orster, Zwischenmolekulare Energiewanderung und Fluoreszenz, Ann

    T. F¨ orster, Zwischenmolekulare Energiewanderung und Fluoreszenz, Ann. Phys. (Berl.)437, 55 (1948)

    1948

  57. [57]

    Levring, D

    J. Levring, D. S. Terry, Z. Kilic, G. Fitzgerald, S. C. Blanchard, and J. Chen, CFTR function, pathology, and pharmacology at single-molecule resolution, Nature 616, 606 (2023)

    2023

  58. [58]

    Kobayashi, H

    R. Kobayashi, H. Ueno, C.-B. Li, and H. Noji, Rotary catalysis of bovine mitochondrial F 1-ATPase studied by single-molecule experiments, Proc. Natl. Acad. Sci. U.S.A.117, 1447 (2020)

    2020

  59. [59]

    Piana, K

    S. Piana, K. Lindorff-Larsen, and D. E. Shaw, Protein folding kinetics and thermodynamics from atomistic simulation, Proc. Natl. Acad. Sci. U.S.A.109, 17845 (2012)

    2012

  60. [60]

    Hummer and A

    G. Hummer and A. Szabo, Optimal dimensionality re- duction of multistate kinetic and Markov-state models, J. Phys. Chem. B119, 9029 (2014)

    2014

  61. [61]

    Inferring bro- ken detailed balance in the absence of observable cur- rents

    D. Hartich and A. Godec, Comment on “Inferring bro- ken detailed balance in the absence of observable cur- rents”, Nat. Commun.15, 8678 (2024)

    2024

  62. [62]

    Inferring broken de- tailed balance in the absence of observable currents

    G. Bisker, I. A. Mart´ ınez, J. M. Horowitz, and J. M. R. Parrondo, Reply to: Comment on “Inferring broken de- tailed balance in the absence of observable currents”, Nat. Commun.15, 8679 (2024)

    2024

  63. [63]

    Stella and S

    L. Stella and S. Melchionna, Equilibration and sampling in molecular dynamics simulations of biomolecules, J. Chem. Phys.109, 10115 (1998)

    1998

  64. [64]

    E. B. Walton and K. J. VanVliet, Equilibration of exper- imentally determined protein structures for molecular dynamics simulation, Phys. Rev. E74, 061901 (2006)

    2006

  65. [65]

    Chapman, On the Brownian displacements and ther- mal diffusion of grains suspended in a non-uniform fluid, Proceedings of the Royal Society of London

    S. Chapman, On the Brownian displacements and ther- mal diffusion of grains suspended in a non-uniform fluid, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character119, 34 (1928)

    1928

  66. [66]

    Kolmogoroff, ¨Uber die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Mathematische Annalen 104, 415 (1931)

    A. Kolmogoroff, ¨Uber die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Mathematische Annalen 104, 415 (1931)

    1931

  67. [67]

    Tokuyama and H

    M. Tokuyama and H. Mori, Statistical-mechanical the- ory of random frequency modulations and generalized Brownian motions, Prog. Theor. Exp. Phys.55, 411 (1976)

    1976

  68. [68]

    Chaturvedi and F

    S. Chaturvedi and F. Shibata, Time-convolutionless projection operator formalism for elimination of fast variables. Applications to Brownian motion, Z. Phys. B35, 297 (1979)

    1979

  69. [69]

    Timm, Time-convolutionless master equation for quantum dots: Perturbative expansion to arbitrary or- der, Phys

    C. Timm, Time-convolutionless master equation for quantum dots: Perturbative expansion to arbitrary or- der, Phys. Rev. B83, 115416 (2011)

    2011

  70. [70]

    C. W. Sinclair and J. Rottler, Influence of solute induced memory on interface migration, Phys. Rev. Mater.9, 123402 (2025)

    2025

  71. [71]

    Bement and J

    P. Bement and J. Rottler, Models for polymer dynam- ics from dimensionality reduction techniques, J. Chem. Phys.163, 104902 (2025)

    2025

  72. [72]

    Strasberg and M

    P. Strasberg and M. Esposito, Non-Markovianity and negative entropy production rates, Phys. Rev. E99, 012120 (2019)

    2019

  73. [73]

    Maldonado-Mundo, P

    D. Maldonado-Mundo, P. ¨Ohberg, B. W. Lovett, and E. Andersson, Investigating the generality of time-local master equations, Phys. Rev. A86, 042107 (2012)

    2012

  74. [74]

    Chru´ sci´ nski and A

    D. Chru´ sci´ nski and A. Kossakowski, Non-Markovian quantum dynamics: Local versus nonlocal, Phys. Rev. Lett.104, 070406 (2010)

    2010

  75. [75]

    S. C. Hou, X. X. Yi, S. X. Yu, and C. H. Oh, Singularity of dynamical maps, Phys. Rev. A86, 012101 (2012)

    2012

  76. [76]

    Nestmann, V

    K. Nestmann, V. Bruch, and M. R. Wegewijs, How quantum evolution with memory is generated in a time- local way, Phys. Rev. X11, 021041 (2021)

    2021

  77. [77]

    E. M. Laine, K. Luoma, and J. Piilo, Local-in-time mas- ter equations with memory effects: applicability and in- terpretation, J. Phys. B45, 154004 (2012)

    2012

  78. [78]

    Brandner, Dynamics of microscale and nanoscale systems in the weak-memory regime: A mathemati- cal framework beyond the Markov approximation, Phys

    K. Brandner, Dynamics of microscale and nanoscale systems in the weak-memory regime: A mathemati- cal framework beyond the Markov approximation, Phys. Rev. E111, 014137 (2025)

    2025

  79. [79]

    Shabani and D

    A. Shabani and D. A. Lidar, Completely positive post- Markovian master equation via a measurement ap- proach, Phys. Rev. A71, 020101 (2005)

    2005

  80. [80]

    Gerry and D

    M. Gerry and D. Segal, Full counting statistics and co- herences: Fluctuation symmetry in heat transport with the unified quantum master equation, Phys. Rev. E107, 054115 (2023)

    2023

Showing first 80 references.