Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Generic shape of multichromatic resonance peaks

  • 6 Accesses

Abstract

We investigate dissipative dynamical systems under the influence of an external driving with two or more frequencies. Our main quantities of interest are long-time averages of expectation values which turn out to exhibit universal features. In particular, resonance peaks in the vicinity of commensurable frequencies possess a generic enveloping function whose width is inversely proportional to the averaging time. While the universal features can be derived analytically, the transition from the specific short-time behavior to the long-time limit is illustrated for the examples of a classical random walk and a dissipative two-level system both with biharmonic driving. In these models, the dependence of the time-averaged response on the relative phase between the two driving frequencies changes with increasing integration time. For short times, it exhibits the 2π periodicity of the dynamic equations, while in the long-time limit, the period becomes a fraction of this value.

Graphical abstract

This is a preview of subscription content, log in to check access.

References

  1. 1.

    L. Gammaitoni, P. Hänggi, P. Jung, F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998)

  2. 2.

    J. Casado-Pascual, J. Gómez-Ordóñez, M. Morillo, P. Hänggi, Phys. Rev. Lett. 91, 210601 (2003)

  3. 3.

    V. Anishchenko, A. Neiman, A. Astakhov, T. Vadiavasova, L. Schimansky-Geier,Chaotic and Stochastic Processes in Dynamic Systems (Springer, Berlin, 2002)

  4. 4.

    J.A. Freund, L. Schimansky-Geier, P. Hänggi, Chaos 13, 225 (2003)

  5. 5.

    B. Lindner, J. Garcia-Ojalvo, A. Neiman, L. Schimansky-Geier, Phys. Rep. 392, 321 (2004)

  6. 6.

    J. Casado-Pascual, J. Gómez-Ordóñez, M. Morillo, J. Lehmann, I. Goychuk, P. Hänggi, Phys. Rev. E 71, 011101 (2005)

  7. 7.

    I. Goychuk, J. Casado-Pascual, M. Morillo, J. Lehmann, P. Hänggi, Phys. Rev. Lett. 97, 210601 (2006)

  8. 8.

    P. Reimann, Phys. Rep. 361, 57 (2002)

  9. 9.

    P. Hänggi, F. Marchesoni, Rev. Mod. Phys. 81, 387 (2009)

  10. 10.

    D. Cubero, F. Renzoni,Brownian Ratchets: From Statistical Physics to Bio and Nano-motors (Cambridge University Press, Cambridge, 2016)

  11. 11.

    S. Flach, O. Yevtushenko, Y. Zolotaryuk, Phys. Rev. Lett. 84, 2358 (2000)

  12. 12.

    D. Cubero, F. Renzoni, Phys. Rev. E 97, 062139 (2018)

  13. 13.

    D. Cubero, G.R. Robb, F. Renzoni, Phys. Rev. Lett. 121, 213904 (2018)

  14. 14.

    J. Casado-Pascual, D. Cubero, F. Renzoni, Phys. Rev. E 88, 062919 (2013)

  15. 15.

    J. Casado-Pascual, J.A. Cuesta, N.R. Quintero, R. Alvarez-Nodarse, Phys. Rev. E 91, 022905 (2015)

  16. 16.

    P. Reimann, M. Grifoni, P. Hänggi, Phys. Rev. Lett. 79, 10 (1997)

  17. 17.

    J. Lehmann, S. Kohler, P. Hänggi, A. Nitzan, J. Chem. Phys. 118, 3283 (2003)

  18. 18.

    S. Kohler, J. Lehmann, P. Hänggi, Phys. Rep. 406, 379 (2005)

  19. 19.

    F. Forster, M. Mühlbacher, R. Blattmann, D. Schuh, W. Wegscheider, S. Ludwig, S. Kohler, Phys. Rev. B 92, 245422 (2015)

  20. 20.

    S. Platonov, B. Kästner, H.W. Schumacher, S. Kohler, S. Ludwig, Phys. Rev. Lett. 115, 106801 (2015)

  21. 21.

    S. Morito, H.M. Salkin, Fibonacci Q. 17, 361 (1979)

  22. 22.

    S. Morito, H.M. Salkin, Acta Informatica 13, 379 (1980)

  23. 23.

    N.G. van Kampen,Stochastic processes in physics and chemistry (North-, Amsterdam, 1992)

  24. 24.

    P. Hänggi, P. Talkner, M. Borkovec, Rev. Mod. Phys. 62, 251 (1990)

  25. 25.

    S. Kohler, T. Dittrich, P. Hänggi, Phys. Rev. E 55, 300 (1997)

  26. 26.

    M. Grifoni, P. Hänggi, Phys. Rep. 304, 229 (1998)

  27. 27.

    H.P. Breuer, F. Petruccione,Theory of Open Quantum Systems (Oxford University Press, Oxford, 2003)

Download references

Author information

Correspondence to María Laura Olivera-Atencio or Jesús Casado-Pascual or Sigmund Kohler.

Additional information

Contribution to the Topical Issue “Advances in Quasi-Periodic and Non-Commensurate Systems”, edited by Tobias Stauber and Sigmund Kohler.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Olivera-Atencio, M.L., Casado-Pascual, J. & Kohler, S. Generic shape of multichromatic resonance peaks. Eur. Phys. J. B 93, 30 (2020). https://doi.org/10.1140/epjb/e2020-100595-0

Download citation