Advertisement

Journal of Statistical Physics

, Volume 174, Issue 1, pp 219–257 | Cite as

Memory Effects in the Fermi–Pasta–Ulam Model

  • Graziano AmatiEmail author
  • Hugues Meyer
  • Tanja Schilling
Article
  • 83 Downloads

Abstract

We study the intermediate scattering function (ISF) of the strongly-nonlinear Fermi–Pasta–Ulam Model at thermal equilibrium, using both numerical and analytical methods. From the molecular dynamics simulations we distinguish two limit regimes, as the system behaves as an ideal gas at high temperature and as a harmonic chain for low excitations. At intermediate temperatures the ISF relaxes to equilibrium in a nontrivial fashion. We then calculate analytically the Taylor coefficients of the ISF to arbitrarily high orders (the specific, simple shape of the two-body interaction allows us to derive an iterative scheme for these). The results of the recursion are in good agreement with the numerical ones. Via an estimate of the complete series expansion of the scattering function, we can reconstruct within a certain temperature range its coarse-grained dynamics. This is governed by a memory-dependent Generalized Langevin Equation (GLE), which can be derived via projection operator techniques. Moreover, by analyzing the first series coefficients of the ISF, we can extract a parameter associated to the strength of the memory effects in the dynamics.

Keywords

FPU model FPU system Fermi–Pasta–Ulam model Fermi–Pasta–Ulam system Coarse-graining Generalized Langevin equation Intermediate scattering function Memory kernel Harmonic chain 

Notes

Acknowledgements

We thank T. Voigtmann, T. Franosch and A. Zippelius for useful discussions. Computer simulations presented in this paper were carried out using the bwForCluster NEMO high-performance computing facility.

References

  1. 1.
    Peyrard, M., Farago, J.: Nonlinear localization in thermalized lattices: application to DNA. Phys. A 288(1), 199–217 (2000)CrossRefGoogle Scholar
  2. 2.
    Henry, A., Chen, G.: Anomalous heat conduction in polyethylene chains: theory and molecular dynamics simulations. Phys. Rev. B 79, 144305 (2009)ADSCrossRefGoogle Scholar
  3. 3.
    Spohn, H.: Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals. Phys. A 369(1), 71–99 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Gallavotti, G.: The Fermi–Pasta–Ulam Problem: A Status Report. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bambusi, D., Carati, A., Maiocchi, A., Maspero, A.: Some Analytic Results on the FPU Paradox, pp. 235–254. Springer, New York, NY (2015)zbMATHGoogle Scholar
  6. 6.
    Fermi, E., Pasta, J., Ulam, S.: Studies of nonlinear problems I, Los Alamos Report LA 1940, 1955 (1974)Google Scholar
  7. 7.
    Dauxois, T.: Fermi, Pasta, Ulam, and a mysterious lady. Phys. Today 61(1), 55–57 (2008)ADSCrossRefGoogle Scholar
  8. 8.
    Fermi, E.: Beweiss das ein mechanisches normal system im allgemeinen quasi-ergodisch ist. Phys. Z. 24, 261 (1923)zbMATHGoogle Scholar
  9. 9.
    Rink, B.: Proof of Nishida’s conjecture on anharmonic lattices. Commun. Math. Phys. 261(3), 613–627 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Friesecke, G., Mikikits-Leitner, A.: Cnoidal waves on Fermi-Pasta-Ulam lattices. J. Dyn. Differ. Equ. 27, 627–652 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hajnal, D., Schilling, R.: Delocalization-localization transition due to anharmonicity. Phys. Rev. Lett. 101, 124101 (2008)ADSCrossRefGoogle Scholar
  12. 12.
    Carati, A., Galgani, L.: Metastability in specific-heat measurements: simulations with the FPU model. EPL 75(4), 528 (2006)ADSCrossRefGoogle Scholar
  13. 13.
    Maiocchi, A., Bambusi, D., Carati, A.: An Averaging Theorem for FPU in the Thermodynamic Limit. J. Stat. Phys. 155, 300–322 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Maiocchi, A.M., Carati, A., Giorgilli, A.: A series expansion for the time autocorrelation of dynamical variables. J. Stat. Phys. 148, 1054–1071 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Carati, A., Maiocchi, A., Galgani, L., Amati, G.: The Fermi–Pasta–Ulam system as a model for glasses. Math. Phys. Anal. Geom. 18, 31 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mori, H.: Transport, collective motion, and Brownian motion. Prog. Theor. Phys. 33(3), 423–455 (1965)ADSCrossRefzbMATHGoogle Scholar
  17. 17.
    Zwanzig, R.: Memory effects in irreversible thermodynamics. Phys. Rev. 124, 983–992 (1961)ADSCrossRefzbMATHGoogle Scholar
  18. 18.
    Grabert, H.: Projection Operator Techniques in Nonequilibrium Statistical Mechanics. Springer, Berlin (1982)CrossRefGoogle Scholar
  19. 19.
    Wierling, A.: Dynamic structure factor of linear harmonic chain—a recurrence relation approach. Eur. Phys. J. B 85, 20 (2012)ADSCrossRefGoogle Scholar
  20. 20.
    Meyer, H., Voigtmann, T., Schilling, T.: On the non-stationary generalized Langevin equation. J. Chem. Phys. 147, 214110 (2017)ADSCrossRefGoogle Scholar
  21. 21.
    Allen, M.P., Tildesley, D.J.: Computer Simulation of Liquids. Clarendon Press, New York, NY (1989)zbMATHGoogle Scholar
  22. 22.
    Reichman, D.R., Charbonneau, P.: Mode-coupling theory. J. Stat. Mech. 2005(05), P05013 (2005)CrossRefGoogle Scholar
  23. 23.
    Hansen, J.-P., McDonald, I.R. (eds.): Theory of Simple Liquids, 4th edn. Academic Press, Oxford (2013)Google Scholar
  24. 24.
    Jean-Philippe, B.: Anomalous Relaxation in Complex Systems: From Stretched to Compressed Exponentials, pp. 327–345. Wiley-Blackwell, Hoboken (2008)Google Scholar
  25. 25.
    Yoshida, T., Shobu, K., Mori, H.: Dynamic properties of one-dimensional harmonic liquids. idensity correlation and transport coefficients. Prog. Theor. Phys. 66(3), 759–771 (1981)ADSCrossRefGoogle Scholar
  26. 26.
    Shobu, K., Yoshida, T., Mori, H.: Dynamic properties of one-dimensional harmonic liquids. ii—Energy density correlation and heat transport. Prog. Theor. Phys. 66(4), 1160–1168 (1981)ADSCrossRefGoogle Scholar
  27. 27.
    Radons, G., Keller, J., Geisel, T.: Dynamical structure factor of a one-dimensional harmonic liquid: comparison of different approximation methods. Zeitschrift für Physik B Condensed Matter 50, 289–296 (1983)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Physikalisches InstitutAlbert-Ludwigs-UniversitätFreiburgGermany
  2. 2.Unit in Engineering ScienceUniversité du LuxembourgEsch-sur-AlzetteLuxembourg

Personalised recommendations