- 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

Identities for nonlinear memory kernels

Juliana Caspers* and Matthias Krüger†

  • *Contact author: j.caspers@theorie.physik.uni-goettingen.de
  • †Contact author: matthias.kruger@uni-goettingen.de
PDF

Phys. Rev. E 112, 024124 – Published 18 August, 2025

DOI: https://doi.org/10.1103/pzs6-l3ws

Abstract

Perturbing a system far away from equilibrium via a time-dependent protocol can formally be described by a nonlinear Volterra series expansion. Here we derive identities for the nonlinear memory kernels arising in such nonlinear expansion, including the possibility of a nonlinear coupling between perturbation and system. These identities rely on local detailed balance, and they include the fluctuation dissipation theorem as the lowest-order identity. We test them in simulations for driven over- and underdamped Brownian particles. These identities for memory kernels can be recast in a series relation for the nonequilibrium cumulants of the observable conjugate to the driving and the observable described by the Volterra series. We discuss the relation to previously found identities.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (78)

  1. L. Onsager, Phys. Rev. 37, 405 (1931).
  2. T. Kawasaki, Prog. Theor. Phys. 39, 331 (1968).
  3. D. J. Evans and G. Morriss, Statistical Mechanics of Nonequilibrium Liquids, 2nd ed. (Cambridge University Press, Cambridge, UK, 2008).
  4. M. Fuchs and M. E. Cates, J. Phys.: Condens. Matter 17, S1681 (2005).
  5. J. Dhont, An Introduction to Dynamics of Colloids (Elsevier Science, Amsterdam, 1996).
  6. R. Zwanzig, Phys. Rev. 124, 983 (1961).
  7. H. Mori, Prog. Theor. Phys. 33, 423 (1965).
  8. R. Zwanzig, Nonequilibrium Statistical Mechanics (Oxford University Press, Oxford, UK, 2001).
  9. H. Grabert, Projection Operator Techniques in Nonequilibrium Statistical Mechanics (Springer, Berlin, 1982).
  10. W. Götze, Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory (Oxford University Press, Oxford, UK, 2008).
  11. L. M. C. Janssen, Front. Phys. 6, 97 (2018).
  12. M. Fuchs, W. Götze, and M. R. Mayr, Phys. Rev. E 58, 3384 (1998).
  13. J. H. van Zanten and K. P. Rufener, Phys. Rev. E 62, 5389 (2000).
  14. Q. Lu and M. J. Solomon, Phys. Rev. E 66, 061504 (2002).
  15. J. van der Gucht, N. A. M. Besseling, W. Knoben, L. Bouteiller, and M. A. Cohen Stuart, Phys. Rev. E 67, 051106 (2003).
  16. J. Caspers, N. Ditz, K. Krishna Kumar, F. Ginot, C. Bechinger, M. Fuchs, and M. Krüger, J. Chem. Phys. 158, 024901 (2023).
  17. A. O. Caldeira and A. J. Leggett, Phys. Rev. Lett. 46, 211 (1981).
  18. T. J. Doerries, S. A. M. Loos, and S. H. L. Klapp, J. Stat. Mech. (2021) 033202.
  19. D. Venturelli and A. Gambassi, New J. Phys. 25, 093025 (2023).
  20. T. M. Squires and J. F. Brady, Phys. Fluids 17, 073101 (2005).
  21. I. Gazuz, A. M. Puertas, T. Voigtmann, and M. Fuchs, Phys. Rev. Lett. 102, 248302 (2009).
  22. C. J. Harrer, D. Winter, J. Horbach, M. Fuchs, and T. Voigtmann, J. Phys.: Condens. Matter 24, 464105 (2012).
  23. A. Jayaraman and A. Belmonte, Phys. Rev. E 67, 065301(R) (2003).
  24. N. Z. Handzy and A. Belmonte, Phys. Rev. Lett. 92, 124501 (2004).
  25. J. Berner, B. Müller, J. R. Gomez-Solano, M. Krüger, and C. Bechinger, Nat. Commun. 9, 999 (2018).
  26. R. Jain, F. Ginot, J. Berner, C. Bechinger, and M. Krüger, J. Chem. Phys. 154, 184904 (2021).
  27. D. Winter, J. Horbach, P. Virnau, and K. Binder, Phys. Rev. Lett. 108, 028303 (2012).
  28. O. Bénichou, A. Bodrova, D. Chakraborty, P. Illien, A. Law, C. Mejía-Monasterio, G. Oshanin, and R. Voituriez, Phys. Rev. Lett. 111, 260601 (2013).
  29. L. G. Wilson and W. C. K. Poon, Phys. Chem. Chem. Phys. 13, 10617 (2011).
  30. V. Démery and E. Fodor, J. Stat. Mech. (2019) 033202.
  31. L. F. Cugliandolo, J. Phys. A: Math. Theor. 44, 483001 (2011).
  32. A. Puglisi, A. Sarracino, and A. Vulpiani, Phys. Rep. 709-710, 1 (2017).
  33. J. M. Brader, M. E. Cates, and M. Fuchs, Phys. Rev. Lett. 101, 138301 (2008).
  34. I. Gazuz and M. Fuchs, Phys. Rev. E 87, 032304 (2013).
  35. M. Gruber, G. C. Abade, A. M. Puertas, and M. Fuchs, Phys. Rev. E 94, 042602 (2016).
  36. H. Meyer, T. Voigtmann, and T. Schilling, J. Chem. Phys. 147, 214110 (2017).
  37. M. te Vrugt and R. Wittkowski, Phys. Rev. E 99, 062118 (2019).
  38. H. Meyer, T. Voigtmann, and T. Schilling, J. Chem. Phys. 150, 174118 (2019).
  39. F. Glatzel and T. Schilling, Europhys. Lett. 136, 36001 (2021).
  40. H. Vroylandt and P. Monmarché, J. Chem. Phys. 156, 244105 (2022).
  41. T. Schilling, Phys. Rep. 972, 1 (2022).
  42. R. R. Netz, Phys. Rev. E 110, 014123 (2024).
  43. W. Bernard and H. B. Callen, Rev. Mod. Phys. 31, 1017 (1959).
  44. G. F. Efremov, Zh. Eksp. Teor. Fiz. 55, 2322 (1968).
  45. G. N. Bochkov and Y. E. Kuzovlev, Zh. Eksp. Teor. Fiz. 72, 238 (1977).
  46. G. N. Bochkov and Y. E. Kuzovlev, Physica A 106, 443 (1981).
  47. R. L. Stratonovich, Nonlinear Nonequilibrium Thermodynamics I: Linear and Nonlinear Fluctuation-Dissipation Theorems (Springer, Berlin, 1992).
  48. C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997).
  49. G. E. Crooks, Phys. Rev. E 60, 2721 (1999).
  50. D. Andrieux and P. Gaspard, Phys. Rev. Lett. 100, 230404 (2008).
  51. U. Seifert, Rep. Prog. Phys. 75, 126001 (2012).
  52. F. Penna, J. Dzubiella, and P. Tarazona, Phys. Rev. E 68, 061407 (2003).
  53. M. Rauscher, A. Domínguez, M. Krüger, and F. Penna, J. Chem. Phys. 127, 244906 (2007).
  54. D. de las Heras and M. Schmidt, Phys. Rev. Lett. 120, 028001 (2018).
  55. M. Schmidt, Rev. Mod. Phys. 94, 015007 (2022).
  56. S. Leitmann and T. Franosch, Phys. Rev. Lett. 111, 190603 (2013).
  57. S. Leitmann, O. Bénichou, and T. Franosch, J. Phys. A: Math. Theor. 51, 375001 (2018).
  58. K. Asheichyk, M. Fuchs, and M. Krüger, J. Phys.: Condens. Matter 33, 405101 (2021).
  59. M. Krüger and C. Maes, J. Phys.: Condens. Matter 29, 064004 (2017).
  60. B. Müller, Brownian particles in nonequilibrium solvents, Ph.D. thesis, Georg-August-Universität Göttingen, 2020.
  61. M. Colangeli, C. Maes, and B. Wynants, J. Phys. A: Math. Theor. 44, 095001 (2011).
  62. U. Basu, M. Krüger, A. Lazarescu, and C. Maes, Phys. Chem. Chem. Phys. 17, 6653 (2015).
  63. C. Maes, Front. Phys. 8, 229 (2020).
  64. J. Caspers and M. Krüger, J. Chem. Phys. 161, 124109 (2024).
  65. B. Müller, J. Berner, C. Bechinger, and M. Krüger, New J. Phys. 22, 023014 (2020).
  66. It will be detailed below why the expansion is performed around the equilibrium state corresponding to xt.
  67. G. Peccati and M. S. Taqqu, Wiener Chaos: Moments, Cumulants and Diagrams (Springer, Milan, 2011).
  68. H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951).
  69. R. Kubo, Rep. Prog. Phys. 29, 255 (1966).
  70. C. Maes, SciPost Phys. Lect. Notes 32 (2021).
  71. M. Baiesi, C. Maes, and B. Wynants, Phys. Rev. Lett. 103, 010602 (2009).
  72. Local detailed balance amounts to assuming that all hidden degrees of freedom are equilibrated and in contact with heat baths. Consider, for example, a driven Brownian particle in a fluid. In a good approximation, the fluid molecules remain in thermal equilibrium [77]. For explanations and counter-examples of systems where local detailed balance is broken, the reader is referred to Ref. [70].
  73. We note that instead of local detailed balance some literature use, possibly interchangeably, the notions microscopically reversible condition [49] or microreversibility condition [50, 78].
  74. T. Holsten and M. Krüger, Phys. Rev. E 103, 032116 (2021).
  75. Appendix C in Ref. [64] explains in detail how the memory kernels can be calculated numerically.
  76. L. Onsager and S. Machlup, Phys. Rev. 91, 1505 (1953).
  77. G. Falasco and M. Esposito, Phys. Rev. E 103, 042114 (2021).
  78. D. Andrieux and P. Gaspard, J. Stat. Mech. (2007) P02006.

Outline

Information

Phys. Rev. E 112, 024124– Published 18 August, 2025
  • Received 14 February 2025
  • Accepted 28 July 2025
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