Abstract
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).
Similar content being viewed by others
References
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])
Berman, G.P., Izrailev, F.M.: The “Fermi–Pasta–Ulam” problem—the first 50 years. Chaos 15, 015104 (2005)
Gallavotti, G. (ed.): The Fermi-Pasta-Ulam Problem: A Status Report. Lect. Notes Phys., vol. 728. Springer, Berlin (2008)
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)
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)
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)
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)
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)
Pettini, M., Landolfi, M.: Relaxation properties and ergodicity breaking in nonlinear Hamiltonian dynamics. Phys. Rev. A 41, 768–783 (1990)
Berchialla, L., Giorgilli, A., Paleari, S.: Esponentially long times to equipartition in the thermodynamic limit. Phys. Lett. A 321, 167–172 (2004)
Berchialla, L., Galgani, L., Giorgilli, A.: Localization of energy in FPU chains. Discrete Contin. Dyn. Syst. Ser. A 11, 855–866 (2004)
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)
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)
Izrailev, F.M., Chirikov, B.V.: Statistical properties of a nonlinear string. Sov. Phys. Dokl. 11, 30–32 (1966)
Kantz, H., Livi, R., Ruffo, S.: Equipartition thresholds in chains of anharmonic oscillators. J. Stat. Phys. 76, 627–643 (1994)
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)
Shepelyansky, D.L.: Low-energy chaos in the Fermi-Pasta-Ulam problem. Nonlinearity 10, 1331–1338 (1997)
Ponno, A.: A theory of the energy cascade in FPU models. Preprint (2008)
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)
Bambusi, D., Ponno, A.: On metastability in FPU. Commun. Math. Phys. 264, 539–561 (2006)
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)
Ponno, A.: Soliton theory and the Fermi-Pasta-Ulam problem in the thermodynamic limit. Europhys. Lett. 64, 606–612 (2003)
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)
Flach, S., Ponno, A.: The Fermi-Pasta-Ulam problem: Periodic orbits, normal forms and resonance overlap criteria. Physica D 237, 908–917 (2008)
Ford, J.: Equipartition of energy for nonlinear systems. J. Math. Phys. 2, 387–393 (1961)
Venakides, S.: The zero dispersion limit of the Korteweg-de Vries equation with periodic initial data. Trans. Am. Math. Soc. 301, 189–226 (1987)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Benettin, G., Livi, R. & Ponno, A. The Fermi-Pasta-Ulam Problem: Scaling Laws vs. Initial Conditions. J Stat Phys 135, 873–893 (2009). https://doi.org/10.1007/s10955-008-9660-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-008-9660-6