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Time-dependent trajectory of a one-dimensional Gaussian non-Markovian observable does not reveal its nonequilibrium character

Roland R. Netz

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Phys. Rev. E 112, 014132 – Published 29 July, 2025

DOI: https://doi.org/10.1103/4rd5-7123

Abstract

When analyzing experimental or simulation time-series data, the question arises whether it is possible to tell from the mere observation of the time-dependent trajectory of a one-dimensional observable whether the system is in equilibrium or not. We here consider the nonequilibrium version of the generalized Langevin equation for a Gaussian non-Markovian observable and show that (i) the multipoint joint distribution solely depends on the two-point correlation function and that (ii) for any nonequilibrium process an equilibrium process with uniquely determined parameters can be found that produces the same two-point correlation function. Since the multipoint joint distribution completely characterizes the dynamics of an observable, we conclude that the nonequilibrium character of a system, in contrast to its non-Markovianity, cannot be read off from the one-dimensional trajectory of a Gaussian observable. These findings are relevant for human cancer and algae cells, whose single-cell velocity distributions have been found to be Gaussian within the data accuracy.

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References (98)

  1. I. Prigogine, Etude Thermodynamic des Phenomene Irreversibles (Desoer, Liege, 1947).
  2. I. Prigogine and P. Mazur, Sur l'extension de la thermodynamique aux phenomenes irreversibles lies aux degres de liberte internes, Physica 19, 241 (1953).
  3. J. L. Lebowitz, Stationary nonequilibrium Gibbsian ensembles, Phys. Rev. 114, 1192 (1959).
  4. R. Zwanzig, Ensemble method in the theory of irreversibility, J. Chem. Phys. 33, 1338 (1960).
  5. S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (North-Holland Pub. Co., Amsterdam, 1962).
  6. H. Grabert, P. Hänggi, and P. Talkner, Microdynamics and nonlinear stochastic processes of gross variables, J. Stat. Phys. 22, 537 (1980).
  7. H. Risken, The Fokker-Planck Equation (Springer, Berlin, 1984).
  8. R. Zwanzig, Nonequilibrium Statistical Mechanics (Oxford University Press, Oxford [u.a.], 2001).
  9. C. Jarzynski, Hamiltonian derivation of a detailed fluctuation theorem, J. Stat. Phys. 98, 77 (2000).
  10. T. Hatano and S. I. Sasa, Steady-state thermodynamics of Langevin systems, Phys. Rev. Lett. 86, 3463 (2001).
  11. U. Seifert, Entropy production along a stochastic trajectory and an integral fluctuation theorem, Phys. Rev. Lett. 95, 040602 (2005).
  12. J. Prost, J.-F. Joanny, and J. M. R. Parrondo, Generalized fluctuation-dissipation theorem for steady-state systems, Phys. Rev. Lett. 103, 090601 (2009).
  13. M. Baiesi, C. Maes, and B. Wynants, Fluctuations and response of nonequilibrium states, Phys. Rev. Lett. 103, 010602 (2009).
  14. F. S. Gnesotto, F. Mura, J. Gladrow, and C. P. Broedersz, Broken detailed balance and non-equilibrium dynamics in living systems: A review, Rep. Prog. Phys. 81, 066601 (2018).
  15. S. Nakajima, On quantum theory of transport phenomena steady diffusion, Prog. Theor. Phys. 20, 948 (1958).
  16. R. Zwanzig, Memory effects in irreversible thermodynamics, Phys. Rev. 124, 983 (1961).
  17. H. Mori, Transport, collective motion, and Brownian motion, Prog. Theor. Phys. 33, 423 (1965).
  18. K. Kawasaki, Simple derivations of generalized linear and nonlinear Langevin equations, J. Phys. A 6, 1289 (1973).
  19. G. Ciccotti and J.-P. Ryckaert, On the derivation of the generalized Langevin equation for interacting Brownian particles, J. Stat. Phys. 26, 73 (1981).
  20. J. E. Straub, M. Borkovec, and B. J. Berne, Calculation of dynamic friction on intramolecular degrees of freedom, J. Phys. Chem. 91, 4995 (1987).
  21. A. Onuki, Phase Transition Dynamics (Cambridge University Press, Cambridge, 2002).
  22. O. F. Lange and H. Grubmüller, Collective Langevin dynamics of conformational motions in proteins, J. Chem. Phys. 124, 214903 (2006).
  23. T. Kinjo and S. A. Hyodo, Equation of motion for coarse-grained simulation based on microscopic description, Phys. Rev. E 75, 051109 (2007).
  24. E. Darve, J. Solomon, and A. Kia, Computing generalized Langevin equations and generalized Fokker–Planck equations, Proc. Natl. Acad. Sci. 106, 10884 (2009).
  25. C. Hijón, P. Español, E. Vanden-Eijnden, and R. Delgado-Buscalioni, Mori–Zwanzig formalism as a practical computational tool, Faraday Discuss. 144, 301 (2010).
  26. S. Izvekov, Microscopic derivation of particle-based coarse-grained dynamics, J. Chem. Phys. 138, 134106 (2013).
  27. H. S. Lee, S.-H. Ahn, and E. F. Darve, The multi-dimensional generalized Langevin equation for conformational motion of proteins, J. Chem. Phys. 150, 174113 (2019).
  28. C. Ayaz, L. Scalfi, B. A. Dalton, and R. R. Netz, Generalized Langevin equation with a nonlinear potential of mean force and nonlinear memory friction from a hybrid projection scheme, Phys. Rev. E 105, 054138 (2022).
  29. H. Vroylandt, L. Goudenège, P. Monmarché, F. Pietrucci, and B. Rotenberg, Likelihood-based non-Markovian models from molecular dynamics, Proc. Natl. Acad. Sci. 119, e2117586119 (2022).
  30. A. Carof, R. Vuilleumier, and B. Rotenberg, Two algorithms to compute projected correlation functions in molecular dynamics simulations, J. Chem. Phys. 140, 124103 (2014).
  31. G. Jung, M. Hanke, and F. Schmid, Iterative reconstruction of memory kernels, J. Chem. Theory Comput. 13, 2481 (2017).
  32. J. O. Daldrop, B. G. Kowalik, and R. R. Netz, External potential modifies friction of molecular solutes in water, Phys. Rev. X 7, 041065 (2017).
  33. J. O. Daldrop, J. Kappler, F. N. Brünig, and R. R. Netz, Butane dihedral angle dynamics in water is dominated by internal friction, Proc. Natl. Acad. Sci. 115, 5169 (2018).
  34. V. Klippenstein and N. F. A. van der Vegt, Cross-correlation corrected friction in (generalized) Langevin models, J. Chem. Phys. 154, 191102 (2021).
  35. L. Tepper, B. Dalton, and R. R. Netz, Accurate memory kernel extraction from discretized time-series data, J. Chem. Theory Comput. 20, 3061 (2024).
  36. D. Mizuno, C. Tardin, C. F. Schmidt, and F. C. MacKintosh, Nonequilibrium mechanics of active cytoskeletal networks, Science 315, 370 (2007).
  37. J. R. Gomez-Solano, A. Petrosyan, S. Ciliberto, R. Chetrite, and K. Gawedzki, Experimental verification of a modified fluctuation-dissipation relation for a micron-sized particle in a nonequilibrium steady state, Phys. Rev. Lett. 103, 040601 (2009).
  38. J. Mehl, V. Blickle, U. Seifert, and C. Bechinger, Experimental accessibility of generalized fluctuation-dissipation relations for nonequilibrium steady states, Phys. Rev. E 82, 032401 (2010).
  39. I. Theurkauff, C. Cottin-Bizonne, J. Palacci, C. Ybert, and L. Bocquet, Dynamic clustering in active colloidal suspensions with chemical signaling, Phys. Rev. Lett. 108, 268303 (2012).
  40. L. Dinis, P. Martin, J. Barral, J. Prost, and J.-F. Joanny, Fluctuation-response theorem for the active noisy oscillator of the hair-cell bundle, Phys. Rev. Lett. 109, 160602 (2012).
  41. P. Bohec, F. F. Gallet, C. Maes, S. Safaverdi, P. Visco, and F. van Wijland, Probing active forces via a fluctuation-dissipation relation: Application to living cells, Europhys. Lett. 102, 50005 (2013).
  42. M. Guo, A. J. Ehrlicher, M. H. Jensen, M. Renz, J. R. Moore, R. D. Goldman, J. Lippincott-Schwartz, F. C. MacKintosh, and D. A. Weitz, Probing the stochastic, motor-driven properties of the cytoplasm using force spectrum microscopy, Cell 158, 822 (2014).
  43. H. Turlier, D. A. Fedosov, B. Audoly, T. Auth, N. S. Gov, C. Sykes, J.-F. Joanny, G. Gompper, and T. Betz, Equilibrium physics breakdown reveals the active nature of red blood cell flickering, Nat. Phys. 12, 513 (2016).
  44. S. S. Plotkin and P. G. Wolynes, Non-Markovian configurational diffusion and reaction coordinates for protein folding, Phys. Rev. Lett. 80, 5015 (1998).
  45. R. Satija and D. E. Makarov, Generalized Langevin equation as a model for barrier crossing dynamics in biomolecular folding, J. Phys. Chem. B 123, 802 (2019).
  46. 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. 118, e2023856118 (2021).
  47. B. A. Dalton, C. Ayaz, H. Kiefer, A. Klimek, L. Tepper, and R. R. Netz, Fast protein folding is governed by memory-dependent friction, Proc. Natl. Acad. Sci. 120, e2220068120 (2023).
  48. B. Bagchi and D. W. Oxtoby, The effect of frequency dependent friction on isomerization dynamics in solution, J. Chem. Phys. 78, 2735 (1983) .
  49. J. E. Straub, M. Borkovec, and B. J. Berne, Non–Markovian activated rate processes: Comparison of current theories with numerical simulation data, J. Chem. Phys. 84, 1788 (1986).
  50. E. Pollak, H. Grabert, and P. Hänggi, Theory of activated rate processes for arbitrary frequency dependent friction: Solution of the turnover problem, J. Chem. Phys. 91, 4073 (1989).
  51. F. N. Brünig, J. O. Daldrop, and R. R. Netz, Pair-reaction dynamics in water: Competition of memory, potential shape, and inertial effects, J. Phys. Chem. B 126, 10295 (2022).
  52. E. Carlon, H. Orland, T. Sakaue, and C. Vanderzande, Effect of memory and active forces on transition path time distributions, J. Phys. Chem. B 122, 11186 (2018).
  53. B. G. Mitterwallner, C. Schreiber, J. O. Daldrop, J. O. Rädler, and R. R. Netz, Non-Markovian data-driven modeling of single-cell motility, Phys. Rev. E 101, 032408 (2020).
  54. A. Klimek, D. Mondal, S. Block, P. Sharma, and R. R. Netz, Data-driven classification of individual cells by their non-Markovian motion, Biophys. J. 123, 1173 (2024).
  55. M. Tuckerman and B. Berne, Vibrational relaxation in simple fluids: Comparison of theory and simulation, J. Chem. Phys. 98, 7301 (1993).
  56. F. Gottwald, S. D. Ivanov, and O. Kühn, Applicability of the Caldeira–Leggett model to vibrational spectroscopy in solution, J. Phys. Chem. Lett. 6, 2722 (2015).
  57. F. N. Brünig, O. Geburtig, A. von Canal, J. Kappler, and R. R. Netz, Time-dependent friction effects on vibrational infrared frequencies and line shapes of liquid water, J. Phys. Chem. B 126, 1579 (2022).
  58. E. Herrera-Delgado, J. Briscoe, and P. Sollich, Tractable nonlinear memory functions as a tool to capture and explain dynamical behaviors, Phys. Rev. Res. 2, 043069 (2020).
  59. A. J. Chorin, O. H. Hald, and R. Kupferman, Optimal prediction and the Mori–Zwanzig representation of irreversible processes, Proc. Natl. Acad. Sci. 97, 2968 (2000).
  60. B. Robertson, Equations of motion in nonequilibrium statistical mechanics, Phys. Rev. 144, 151 (1966).
  61. S. Nordholm and R. Zwanzig, A systematic derivation of exact generalized Brownian motion theory, J. Stat. Phys. 13, 347 (1975).
  62. R. H. Picard and C. R. Willis, Time-dependent projection-operator approach to master equations for coupled systems. II. Systems with correlations, Phys. Rev. A 16, 1625 (1977).
  63. C. Uchiyama and F. Shibata, Unified projection operator formalism in nonequilibrium statistical mechanics, Phys. Rev. E 60, 2636 (1999).
  64. T. Koide, Derivation of transport equations using the time-dependent projection operator method, Prog. Theor. Phys. 107, 525 (2002).
  65. A. Latz, Non-equilibrium projection-operator for a quenched thermostatted system, J. Stat. Phys. 109, 607 (2002).
  66. H. Meyer, T. Voigtmann, and T. Schilling, On the non-stationary generalized Langevin equation, J. Chem. Phys. 147, 214110 (2017).
  67. B. Cui and A. Zaccone, Generalized Langevin equation and fluctuation-dissipation theorem for particle-bath systems in external oscillating fields, Phys. Rev. E 97, 060102(R) (2018).
  68. M. te Vrugt and R. Wittkowski, Mori-Zwanzig projection operator formalism for far-from-equilibrium systems with time-dependent Hamiltonians, Phys. Rev. E 99, 062118 (2019).
  69. R. R. Netz, Derivation of the non-equilibrium generalized Langevin equation from a time-dependent Hamiltonian, Phys. Rev. E 110, 014123 (2024).
  70. L. Lavacchi, J. O. Daldrop, and R. R. Netz, Non-Arrhenius barrier crossing dynamics of non-equilibrium non-Markovian systems, Europhys. Lett. 139, 51001 (2022).
  71. J. Shea, G. Jung, and F. Schmid, Passive probe particle in an active bath: Can we tell it is out of equilibrium? Soft Matter 18, 6965 (2022).
  72. D. Lucente, A. Baldassarri, A. Puglisi, A. Vulpiani, and M. Viale, Inference of time irreversibility from incomplete information: Linear systems and its pitfalls, Phys. Rev. Res. 4, 043103 (2022).
  73. G. Knotz and M. Krüger, Mean back relaxation for position and densities, Phys. Rev. E 110, 044137 (2024).
  74. K. Engbring, D. Boriskovsky, Y. Roichman, and B. Lindner, A nonlinear fluctuation-dissipation test for Markovian systems, Phys. Rev. X 13, 021034 (2023).
  75. R. Rodríguez-García, I. Lopez-Montero, M. Mell, G. Egea, and N. S. Gov, Direct cytoskeleton forces cause membrane softening in red blood cells, Biophys. J. 108, 2794 (2015).
  76. J. R. Medeiros and S. M. Duarte Queiros, Thermostatistics of a damped bimodal particle, Phys. Rev. E 92, 062145 (2015).
  77. A. Militaru, M. Innerbichler, M. Frimmer, F. Tebbenjohanns, L. Novotny, and C. Dellago, Escape dynamics of active particles in multistable potentials, Nat. Commun. 12, 2446 (2021).
  78. G. Paneru, T. Tlusty, and H. K. Pak, Bona fide stochastic resonance under nonGaussian active fluctuations, Soft Matter 19, 1356 (2023).
  79. T. M. Muenker, G. Knotz, M. Krüger, and T. Betz, Accessing activity and viscoelastic properties of artificial and living systems from passive measurement, Nat. Mater. 23, 1283 (2024).
  80. D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud, and A. Petrosyan, Entropy production and time asymmetry in nonequilibrium fluctuations, Phys. Rev. Lett. 98, 150601 (2007).
  81. E. Roldan and J. M. R. Parrondo, Estimating dissipation from single stationary trajectories, Phys. Rev. Lett. 105, 150607 (2010).
  82. I. A. Martinez, 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).
  83. Y.-J. Yang and H. Qian, Bivectorial nonequilibrium thermodynamics: Cycle affinity, vorticity potential, and Onsager's principle, J. Stat. Phys. 182, 46 (2021).
  84. D. J. Skinner and J. Dunkel, Estimating entropy production from waiting time distributions, Phys. Rev. Lett. 127, 198101 (2021).
  85. P. E. Harunari, A. Dutta, M. Polettini, and E. Roldan, What to learn from a few visible transitions' statistics? Phys. Rev. X 12, 041026 (2022).
  86. 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).
  87. J. O'Byrne, Nonequilibrium currents in stochastic field theories: A geometric insight, Phys. Rev. E 107, 054105 (2023).
  88. H. Vroylandt, On the derivation of the generalized Langevin equation and the fluctuation-dissipation theorem, Europhys. Lett. 140, 62003 (2022).
  89. H. Kiefer, D. Furtel, C. Ayaz, A. Klimek, J. O. Daldrop, and R. R. Netz, Prediction of weather and financial time-series data via a Hamiltonian-based filter-projection approach, Newton 1, 100138 (2025).
  90. See Supplemental Material at http://link.aps.org/supplemental/10.1103/4rd5-7123 for the derivation of the multipoint joint distribution function.
  91. B. G. Mitterwallner, L. Lavacchi, and R. R. Netz, Negative friction memory induces persistent motion, Eur. Phys. J. E 43, 67 (2020).
  92. R. R. Netz, Fluctuation-dissipation relation and stationary distribution of an exactly solvable many-particle model for active biomatter far from equilibrium, J. Chem. Phys. 148, 185101 (2018).
  93. R. R. Netz, Approach to equilibrium and nonequilibrium stationary distributions of interacting many-particle systems that are coupled to different heat baths, Phys. Rev. E 101, 022120 (2020).
  94. I. Di Terlizzi, M. Gironella, D. Herraez-Aguilar, T. Betz, F. Monroy, M. Baiesi, and F. Ritort, Variance sum rule for entropy production, Science 383, 971 (2024).
  95. B. A. Dalton, H. Kiefer, and R. R. Netz, The role of memory- dependent friction and solvent viscosity in isomerization kinetics in viscogenic media, Nat. Commun. 15, 3761 (2024).
  96. S. Tuvia, A. Almagor, A. Bitler, S. Levin, R. Korenstein, and S. Yedgar, Cell membrane fluctuations are regulated by medium macroviscosity: Evidence for a metabolic driving force, Proc. Natl. Acad. Sci. 94, 5045 (1997).
  97. B. R. Ferrer, J. R. Gomez-Solano, and A. V. Arzola, Fluid viscoelasticity triggers fast transitions of a Brownian particle in a double well optical potential, Phys. Rev. Lett. 126, 108001 (2021).
  98. F. Ginot, J. Caspers, M. Krüger, and C. Bechinger, Barrier crossing in a viscoelastic bath, Phys. Rev. Lett. 128, 028001 (2022).

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Phys. Rev. E 112, 014132– Published 29 July, 2025
  • Received 1 July 2024
  • Revised 12 January 2025
  • Accepted 2 July 2025
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