- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Fluctuation-response relation for a nonequilibrium system with resolved Markovian embedding

Rémi Goerlich1,*, Antoine Tartar2, Yael Roichman1,3, and Igor M. Sokolov4,†

  • *Contact author: remigoerlich@tauex.tau.ac.il
  • †Contact author: igor.sokolov@physik.hu-berlin.de
PDF

Phys. Rev. E 114, 014138 – Published 20 July, 2026

DOI: https://doi.org/10.1103/kldw-kftz

Abstract

Fluctuation-response relations must be modified to describe nonequilibrium systems with non-Markovian dynamics. Here, we experimentally demonstrate that such relation is quantitatively recovered when the appropriate Markovian embedding of the dynamics is explicitly resolved. Using a colloidal particle optically trapped in a harmonic potential and driven out of equilibrium by a controlled colored noise, we study the response to a perturbation of the stiffness of the confining potential. While the reduced dynamics violates equilibrium fluctuation-response relations, we show that the dynamical response to the stiffness perturbation is fully determined by steady-state correlations involving the exact conjugate observable in the Markovian embedding.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (46)

  1. H. Takahasi, Generalized theory of thermal fluctuations, J. Phys. Soc. Jpn. 7, 439 (1952).
  2. R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys. 29, 255 (1966).
  3. U. M. B. Marconi, A. Puglisi, L. Rondoni, and A. Vulpiani, Fluctuation–dissipation: Response theory in statistical physics, Phys. Rep. 461, 111 (2008).
  4. L. Caprini, A. Puglisi, and A. Sarracino, Fluctuation–dissipation relations in active matter systems, Symmetry 13, 81 (2021).
  5. R. Lenk, A simple proof of the classical fluctuation dissipation theorem, Phys. Lett. A 25, 198 (1967).
  6. M. Guvendiren and J. A. Burdick, Stiffening hydrogels to probe short-and long-term cellular responses to dynamic mechanics, Nat. Commun. 3, 792 (2012).
  7. S. Hurst, B. E. Vos, M. Brandt, and T. Betz, Intracellular softening and increased viscoelastic fluidity during division, Nat. Phys. 17, 1270 (2021).
  8. L. Agozzino, G. Balázsi, J. Wang, and K. A. Dill, How do cells adapt? Stories told in landscapes, Annu. Rev. Chem. Biomol. Eng. 11, 155 (2020).
  9. R. Goerlich, L. B. Pires, G. Manfredi, P.-A. Hervieux, and C. Genet, Harvesting information to control nonequilibrium states of active matter, Phys. Rev. E 106, 054617 (2022).
  10. R. Goerlich, B. Sorkin, D. Boriskovsky, L. B. Pires, B. Lindner, C. Genet, and Y. Roichman, Consistent thermodynamics reconstructed from transitions between nonequilibrium steady-states, arXiv:2601.03245.
  11. 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).
  12. C. Maggi, M. Paoluzzi, L. Angelani, and R. Di Leonardo, Memory-less response and violation of the fluctuation-dissipation theorem in colloids suspended in an active bath, Sci. Rep. 7, 17588 (2017).
  13. P. Mestres, I. A. Martinez, A. Ortiz-Ambriz, R. A. Rica, and E. Roldan, Realization of nonequilibrium thermodynamic processes using external colored noise, Phys. Rev. E 90, 032116 (2014).
  14. J. A. C. Albay, Z.-Y. Zhou, C.-H. Chang, H.-Y. Chen, J. S. Lee, C.-H. Chang, and Y. Jun, Colloidal heat engine driven by engineered active noise, Phys. Rev. Res. 5, 033208 (2023).
  15. C. Sandford, A. Y. Grosberg, and J.-F. Joanny, Pressure and flow of exponentially self-correlated active particles, Phys. Rev. E 96, 052605 (2017).
  16. F. Ginot, J. Caspers, M. Krüger, and C. Bechinger, Barrier crossing in a viscoelastic bath, Phys. Rev. Lett. 128, 028001 (2022).
  17. F. Ginot and C. Bechinger, Energy recuperation of driven colloids in non-Markovian baths, Nat. Commun. 16, 10114 (2025).
  18. R. E. Kalman and R. S. Bucy, New results in linear filtering and prediction theory, J. Basic Eng. 83, 95 (1961).
  19. J. Bechhoefer, Hidden Markov models for stochastic thermodynamics, New J. Phys. 17, 075003 (2015).
  20. A. Puglisi, A. Sarracino, and A. Vulpiani, Temperature in and out of equilibrium: A review of concepts, tools and attempts, Phys. Rep. 709-710, 1 (2017).
  21. J. Prost, J.-F. Joanny, and J. M. R. Parrondo, Generalized fluctuation-dissipation theorem for steady-state systems, Phys. Rev. Lett. 103, 090601 (2009).
  22. I. M. Sokolov, Linear response and fluctuation-dissipation relations for Brownian motion under resetting, Phys. Rev. Lett. 130, 067101 (2023).
  23. U. Seifert and T. Speck, Fluctuation-dissipation theorem in nonequilibrium steady states, Europhys. Lett. 89, 10007 (2010).
  24. J. R. Gomez-Solano, A. Petrosyan, S. Ciliberto, R. Chetrite, and K. Gawędzki, Experimental verification of a modified fluctuation-dissipation relation for a micron-sized particle in a nonequilibrium steady state, Phys. Rev. Lett. 103, 040601 (2009).
  25. 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).
  26. L. Dinis, P. Martin, J. Barral, J. Prost, and J. Joanny, Fluctuation-response theorem for the active noisy oscillator of the hair-cell bundle, Phys. Rev. Lett. 109, 160602 (2012).
  27. A. Dechant and S.-I. Sasa, Fluctuation–response inequality out of equilibrium, Proc. Natl. Acad. Sci. USA 117, 6430 (2020).
  28. A. Gnoli, A. Puglisi, A. Sarracino, and A. Vulpiani, Nonequilibrium Brownian motion beyond the effective temperature, PLoS ONE 9, e93720 (2014).
  29. V. Lucarini and M. D. Chekroun, Theoretical tools for understanding the climate crisis from Hasselmann's programme and beyond, Nat. Rev. Phys. 5, 744 (2023).
  30. I. A. Martínez, É. Roldán, L. Dinis, D. Petrov, and R. A. Rica, Adiabatic processes realized with a trapped Brownian particle, Phys. Rev. Lett. 114, 120601 (2015).
  31. L. B. Pires, R. Goerlich, A. L. da Fonseca, M. Debiossac, P.-A. Hervieux, G. Manfredi, and C. Genet, Optimal time-entropy bounds and speed limits for Brownian thermal shortcuts, Phys. Rev. Lett. 131, 097101 (2023).
  32. V. Blickle and C. Bechinger, Realization of a micrometre-sized stochastic heat engine, Nat. Phys. 8, 143 (2012).
  33. I. A. Martínez, É. Roldán, L. Dinis, D. Petrov, J. M. Parrondo, and R. A. Rica, Brownian Carnot engine, Nat. Phys. 12, 67 (2016).
  34. S. A. Loos and S. H. Klapp, Irreversibility, heat and information flows induced by non-reciprocal interactions, New J. Phys. 22, 123051 (2020).
  35. M. S. Gutiérrez and V. Lucarini, On some aspects of the response to stochastic and deterministic forcings, J. Phys. A: Math. Theor. 55, 425002 (2022).
  36. K. Mamis, G. Athanassoulis, and Z. Kapelonis, A systematic path to non-Markovian dynamics: New response probability density function evolution equations under Gaussian coloured noise excitation, Proc. R. Soc. A 475, 20180837 (2019).
  37. R. Goerlich, M. Li, S. Albert, G. Manfredi, P.-A. Hervieux, and C. Genet, Noise and ergodic properties of Brownian motion in an optical tweezer: Looking at regime crossovers in an Ornstein-Uhlenbeck process, Phys. Rev. E 103, 032132 (2021).
  38. H. Risken, The Fokker-Planck Equation (Springer, 1989), pp. 63–95.
  39. C. Maggi, M. Paoluzzi, N. Pellicciotta, A. Lepore, L. Angelani, and R. Di Leonardo, Generalized energy equipartition in harmonic oscillators driven by active baths, Phys. Rev. Lett. 113, 238303 (2014).
  40. E. Dieterich, J. Camunas-Soler, M. Ribezzi-Crivellari, U. Seifert, and F. Ritort, Single-molecule measurement of the effective temperature in non-equilibrium steady states, Nat. Phys. 11, 971 (2015).
  41. J. F. Boudet, T. Guérin, C. Tiffon, T. Barois, and H. Kellay, Non-Markovian dynamics of a torsion pendulum immersed in an active bath, Commun. Phys. 8, 460 (2025).
  42. 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).
  43. K. Engbring, D. Boriskovsky, Y. Roichman, and B. Lindner, A nonlinear fluctuation-dissipation test for Markovian systems, Phys. Rev. X 13, 021034 (2023).
  44. U. Seifert, Entropy production along a stochastic trajectory and an integral fluctuation theorem, Phys. Rev. Lett. 95, 040602 (2005).
  45. A. Crisanti, A. Puglisi, and D. Villamaina, Nonequilibrium and information: The role of cross correlations, Phys. Rev. E 85, 061127 (2012).
  46. R. Goerlich, A. Tartar, Y. Roichman, and I. M. Sokolov, Dataset to fluctuation-response relation for a nonequilibrium system with resolved Markovian embedding, Zenodo, 2026, https://doi.org/10.5281/zenodo.21011963.

Outline

Information

Phys. Rev. E 114, 014138– Published 20 July, 2026
  • Received 2 February 2026
  • Accepted 25 June 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation