The effect of long-range interactions on the dynamics and statistics of 1D Hamiltonian lattices with on-site potential

Abstract

We examine the role of long-range interactions on the dynamical and statistical properties of two 1D lattices with on-site potentials that are known to support discrete breathers: the Klein–Gordon (KG) lattice which includes linear dispersion and the Gorbach–Flach (GF) lattice, which shares the same on-site potential but its dispersion is purely nonlinear. In both models under the implementation of long-range interactions (LRI), we find that single-site excitations lead to special low-dimensional solutions, which are well described by the undamped Duffing oscillator. For random initial conditions, we observe that the maximal Lyapunov exponent λ scales as N−0.12 in the KG model and as N−0.27 in the GF with LRI, suggesting in that case an approach to integrable behavior towards the thermodynamic limit. Furthermore, under LRI, their non-Gaussian momentum distributions are distinctly different from those of the FPU model.

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

References

  1. 1.

    C. Anteneodo, C. Tsallis, Phys. Rev. Lett. 80, 5313 (1998)

    ADS  Article  Google Scholar 

  2. 2.

    V. Latora, A. Rapisarda, S. Ruffo, Phys. Rev. Lett. 80, 692 (1998)

    ADS  Article  Google Scholar 

  3. 3.

    L.J.L. Cirto, V. Assis, C. Tsallis, Physica A 393, 286 (2013)

    ADS  Article  Google Scholar 

  4. 4.

    H. Christodoulidi, C. Tsallis, T. Bountis, Europhys. Lett. 108, 40006 (2014)

    ADS  Article  Google Scholar 

  5. 5.

    H. Christodoulidi, T. Bountis, C. Tsallis, L. Drossos, J. Stat. Mech. 2016, 123206 (2016)

    Article  Google Scholar 

  6. 6.

    G. Miloshevich, T. Dauxois, R. Khomeriki, S. Ruffo, Europhys. Lett. 104, 17011 (2013)

    ADS  Article  Google Scholar 

  7. 7.

    G. Miloshevich, J.P. Nguenang, T. Dauxois, R. Khomeriki, S. Ruffo, Phys. Rev. E 91, 032927 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  8. 8.

    G. Miloshevich, J.P. Nguenang, T. Dauxois, R. Khomeriki, S. Ruffo, J. Phys. A: Math. Theor. 50, 12LT02 (2017)

    Article  Google Scholar 

  9. 9.

    S. Gupta, S. Ruffo, Int. J. Mod. Phys. A 32, 1741018 (2017)

    ADS  Article  Google Scholar 

  10. 10.

    S. Iubini, P. Di Cintio, S. Lepri, R. Livi, L. Casetti, Phys. Rev. E 97, 032102 (2018)

    ADS  Article  Google Scholar 

  11. 11.

    D. Bagchi, C. Tsallis, Phys. Rev. E 93, 062213 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  12. 12.

    D. Bagchi, Phys. Rev. E 95, 032102 (2017)

    ADS  Article  Google Scholar 

  13. 13.

    D. Bagchi, Phys. Rev. E 96, 042121 (2017)

    ADS  Article  Google Scholar 

  14. 14.

    D. Bagchi, C. Tsallis, Physica A 491, 869 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  15. 15.

    E. Fermi, J. Pasta, S. Ulam, Los Alamos, Report No. LA-1940, 1955

  16. 16.

    A.V. Gorbach, S. Flach, Phys. Rev. E 72, 056607 (2005)

    ADS  MathSciNet  Article  Google Scholar 

  17. 17.

    M.V. Ivanchenko, O.I. Kanakov, V.D. Shalfeev, S. Flach, Physica D 198, 120 (2004)

    ADS  MathSciNet  Article  Google Scholar 

  18. 18.

    P. Maniadis, T. Bountis, Phys. Rev. E 73, 046211 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  19. 19.

    R.S. MacKay, S. Aubry, Nonlinearity 7, 1623 (1994)

    ADS  MathSciNet  Article  Google Scholar 

  20. 20.

    S. Flach, Phys. Rev. E 51, 3579 (1995)

    ADS  Article  Google Scholar 

  21. 21.

    J.M. Bergamin, T. Bountis, C. Jung, J. Phys. A: Math. Gen. 33, 8059 (2000)

    ADS  Article  Google Scholar 

  22. 22.

    J.M. Bergamin, T. Bountis, M.N. Vrahatis, Nonlinearity 15, 1603 (2002)

    ADS  MathSciNet  Article  Google Scholar 

  23. 23.

    S. Flach, Phys. Rev. E 58, R4116 (1998)

    ADS  Article  Google Scholar 

  24. 24.

    L. Casetti, R. Livi, M. Pettini, Phys. Rev. Lett. 74, 375 (1995)

    ADS  Article  Google Scholar 

  25. 25.

    C. Tsallis, Stat. Phys. 52, 479 (1988) First appeared as preprint in 1987: CBPF-NF-062/87, ISSN 0029-3865, Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro

    ADS  Article  Google Scholar 

  26. 26.

    M. Gell-Mann, C. Tsallis (Eds.),Nonextensive Entropy – Interdisciplinary Applications (Oxford University Press, New York, 2004)

  27. 27.

    C. Tsallis,Introduction to Nonextensive Statistical Mechanics – Approaching a Complex World (Springer, New York, 2009)

  28. 28.

    C. Tsallis, Contemp. Phys. 55, 179 (2014)

    ADS  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to A. Bountis.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Christodoulidi, H., Bountis, A. & Drossos, L. The effect of long-range interactions on the dynamics and statistics of 1D Hamiltonian lattices with on-site potential. Eur. Phys. J. Spec. Top. 227, 563–573 (2018). https://doi.org/10.1140/epjst/e2018-00003-9

Download citation