Journal of Statistical Physics

, Volume 135, Issue 5–6, pp 873–893 | Cite as

The Fermi-Pasta-Ulam Problem: Scaling Laws vs. Initial Conditions

  • G. Benettin
  • R. LiviEmail author
  • A. Ponno


Numerical evidence on the relevance of the initial conditions to the Fermi-Pasta-Ulam problem is reported, supported by analytic estimates. In particular, we analyze the special, crucial role played by the phases of the low frequency normal modes initially excited, their energy being the same. The results found are the following. When the phases of the initially excited modes are randomly chosen, the parameter ruling the first stage of the transfer of energy to higher frequency modes turns out to be the energy per degree of freedom (or specific energy) of the system, i.e. an intensive parameter. On the other hand, if the initial phases are “coherently” selected (e.g. they are all equal or equispaced on the unit circle), then the energy cascade is ruled by the total energy of the system, i.e. an extensive parameter. Finally, when a few modes are initially excited, in which case specifying the randomness or coherence of the phases becomes meaningless, the relevant parameter turns out to be again the specific energy (this is the case of the original Fermi-Pasta-Ulam experiment).


Fermi-Pasta-Ulam problem Energy transfer Scaling laws 


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  1. 1.
    Fermi, E., Pasta, J., Ulam, S.: Studies of non linear problems, Los-Alamos internal report, document LA-1940 (1955). In: Enrico Fermi Collected Papers, vol. II, pp. 977–988. University of Chicago Press/Accad. Naz. Lincei, Chicago/Roma (1965). (Also reproduced in  [3]) Google Scholar
  2. 2.
    Berman, G.P., Izrailev, F.M.: The “Fermi–Pasta–Ulam” problem—the first 50 years. Chaos 15, 015104 (2005) CrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Gallavotti, G. (ed.): The Fermi-Pasta-Ulam Problem: A Status Report. Lect. Notes Phys., vol. 728. Springer, Berlin (2008) zbMATHGoogle Scholar
  4. 4.
    Livi, R., Pettini, M., Ruffo, S., Sparpaglione, M., Vulpiani, A.: Equipartition threshold in nonlinear large Hamiltonian systems: the Fermi–Pasta–Ulam model. Phys. Rev. A 31, 1039–1045 (1985) CrossRefADSGoogle Scholar
  5. 5.
    Goedde, C.G., Lichtenberg, A.J., Lieberman, M.A.: Chaos and the approach to equilibrium in a discrete Sine-Gordon equation. Physica D 59, 200–225 (1992) zbMATHADSMathSciNetGoogle Scholar
  6. 6.
    De Luca, J., Lichtenberg, A.J., Ruffo, S.: Energy transition and time scales to equipartition in the Fermi–Pasta–Ulam oscillator chain. Phys. Rev. E 51, 2877–2885 (1995) ADSGoogle Scholar
  7. 7.
    Fucito, E., Marchesoni, F., Marinari, E., Parisi, G., Peliti, L., Ruffo, S., Vulpiani, A.: Approach to equilibrium in a chain of nonlinear oscillators. J. Phys. 43, 707–713 (1982) MathSciNetGoogle Scholar
  8. 8.
    Livi, R., Pettini, M., Ruffo, S., Sparpaglione, M., Vulpiani, A.: Relaxation to different stationary states in the Fermi-Pasta-Ulam model. Phys. Rev. A 28, 3544–3552 (1983) CrossRefADSGoogle Scholar
  9. 9.
    Pettini, M., Landolfi, M.: Relaxation properties and ergodicity breaking in nonlinear Hamiltonian dynamics. Phys. Rev. A 41, 768–783 (1990) CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Berchialla, L., Giorgilli, A., Paleari, S.: Esponentially long times to equipartition in the thermodynamic limit. Phys. Lett. A 321, 167–172 (2004) zbMATHCrossRefADSGoogle Scholar
  11. 11.
    Berchialla, L., Galgani, L., Giorgilli, A.: Localization of energy in FPU chains. Discrete Contin. Dyn. Syst. Ser. A 11, 855–866 (2004) zbMATHMathSciNetGoogle Scholar
  12. 12.
    Biello, J.A., Kramer, P.R., L’vov, Y.V.: Stages of energy transfer in the FPU model. Discrete Contin. Dyn. Syst. 2003(Suppl.), 113–122 (2003). (Special number devoted to the Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, 24–27 May 2002, Wilmington, NC) zbMATHMathSciNetGoogle Scholar
  13. 13.
    Benettin, G., Carati, A., Galgani, L., Giorgilli, A.: The Fermi–Pasta–Ulam problem and the metastability perspective. In: Gallavotti, G. (ed.) The Fermi-Pasta-Ulam Problem: A Status Report. Lect. Notes Phys., vol. 728, pp. 151–189. Springer, Berlin (2008) CrossRefGoogle Scholar
  14. 14.
    Izrailev, F.M., Chirikov, B.V.: Statistical properties of a nonlinear string. Sov. Phys. Dokl. 11, 30–32 (1966) ADSGoogle Scholar
  15. 15.
    Kantz, H., Livi, R., Ruffo, S.: Equipartition thresholds in chains of anharmonic oscillators. J. Stat. Phys. 76, 627–643 (1994) CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Lichtenberg, A.J., Livi, R., Pettini, M., Ruffo, S.: Dynamics of oscillator chains. In: Gallavotti, G. (ed.) The Fermi-Pasta-Ulam Problem: A Status Report. Lect. Notes Phys., vol. 728, pp. 21–121. Springer, Berlin (2008) CrossRefGoogle Scholar
  17. 17.
    Shepelyansky, D.L.: Low-energy chaos in the Fermi-Pasta-Ulam problem. Nonlinearity 10, 1331–1338 (1997) zbMATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Ponno, A.: A theory of the energy cascade in FPU models. Preprint (2008) Google Scholar
  19. 19.
    Zabusky, N.J., Kruskal, M.D.: Interaction of “Solitons” in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243 (1965) CrossRefADSGoogle Scholar
  20. 20.
    Bambusi, D., Ponno, A.: On metastability in FPU. Commun. Math. Phys. 264, 539–561 (2006) CrossRefADSMathSciNetGoogle Scholar
  21. 21.
    Bambusi, D., Ponno, A.: Resonance, metastability and blow-up in FPU. In: Gallavotti, G. (ed.) The Fermi-Pasta-Ulam Problem: A Status Report. Lect. Notes Phys., vol. 728, pp. 191–205. Springer, Berlin (2008) CrossRefGoogle Scholar
  22. 22.
    Ponno, A.: Soliton theory and the Fermi-Pasta-Ulam problem in the thermodynamic limit. Europhys. Lett. 64, 606–612 (2003) CrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Ponno, A.: The Fermi-Pasta-Ulam problem in the thermodynamic limit: scaling laws of the energy cascade. In: Collet, P., et al. (eds.) Proceedings of the Cargèse Summer School 2003 on Chaotic Dynamics and Transport in Classical and Quantum Systems, pp. 431–440. Kluwer Academic, Dordrecht (2005) CrossRefGoogle Scholar
  24. 24.
    Flach, S., Ponno, A.: The Fermi-Pasta-Ulam problem: Periodic orbits, normal forms and resonance overlap criteria. Physica D 237, 908–917 (2008) zbMATHCrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Ford, J.: Equipartition of energy for nonlinear systems. J. Math. Phys. 2, 387–393 (1961) zbMATHCrossRefADSGoogle Scholar
  26. 26.
    Venakides, S.: The zero dispersion limit of the Korteweg-de Vries equation with periodic initial data. Trans. Am. Math. Soc. 301, 189–226 (1987) zbMATHCrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Dipartmento di Matematica Pura e ApplicataUniversità di PadovaPadovaItaly
  2. 2.Dipartimento di Fisica–CSDCUniversitá di Firenze and Sezione INFN di FirenzeSesto FiorentinoItaly

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