Some Analytic Results on the FPU Paradox

  • Dario BambusiEmail author
  • Andrea Carati
  • Alberto Maiocchi
  • Alberto Maspero
Part of the Fields Institute Communications book series (FIC, volume 75)


We present some analytic results aiming at explaining the lack of thermalization observed by Fermi Pasta and Ulam in their celebrated numerical experiment. In particular we focus on results which persist as the number N of particles tends to infinity. After recalling the FPU experiment and some classical heuristic ideas that have been used for its explanation, we concentrate on more recent rigorous results which are based on the use of (i) canonical perturbation theory and KdV equation, (ii) Toda lattice, (iii) a new approach based on the construction of functions which are adiabatic invariants with large probability in the Gibbs measure.


Thermodynamic Limit Gibbs Measure Fourier Mode Toda Lattice Action Angle Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We thank Luigi Galgani for many interesting discussions and for his careful reading of the manuscript. This research was founded by the Prin project 2010–2011 “Teorie geometriche e analitiche dei sistemi Hamiltoniani in dimensioni finite e infinite”.


  1. 1.
    Bambusi, D., Giorgilli, A.: Exponential stability of states close to resonance in infinite-dimensional Hamiltonian systems. J. Statist. Phys. 71, 569–606 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bambusi, D., Kappeler, T., Paul, T.: De Toda à KdV. C. R. Math. Acad. Sci. Paris, 347, 1025–1030 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bambusi, D., Kappeler, T., Paul, T.: Dynamics of periodic Toda chains with a large number of particles (2013) (ArXiv e-prints, arXiv:1309.5441 [math.AP])Google Scholar
  4. 4.
    Bambusi, D., Kappeler, T., Paul, T.: From Toda to KdV. ArXiv e-prints, arXiv:1309.5324 [math.AP], Sept. (2013).Google Scholar
  5. 5.
    Bambusi D, Maspero A. Birkhoff coordinates for the Toda lattice in the limit of infinitely many particles with an application to FPU. Preprint (2014)Google Scholar
  6. 6.
    Bambusi, D., Ponno A.: On metastability in FPU. Comm. Math. Phys. 264, 539–561 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Benettin, G., Christodoulidi, H., Ponno, A.: The Fermi-Pasta-Ulam problem and its underlying integrable dynamics. J. Stat. Phys. 152, 195–212 (2013)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Benettin, G., Galgani, L., Giorgilli, A.: Classical perturbation theory for systems of weakly coupled rotators. Nuovo Cimento B (11) 89, 89–102 (1985)Google Scholar
  9. 9.
    Benettin, G., Galgani, L., Giorgilli, A.: Numerical investigations on a chain of weakly coupled rotators in the light of classical perturbation theory. Nuovo Cimento B (11), 89, 103–119 (1985)Google Scholar
  10. 10.
    Benettin, G., Galgani, L., Giorgilli, A.: Realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory. I. Comm. Math. Phys. 113, 87–103 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Benettin, G., Galgani, L., Giorgilli, A.: Realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory. II. Comm. Math. Phys. 121, 557–601 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Benettin, G., Ponno, A.: Time-scales to equipartition in the Fermi-Pasta-Ulam problem: finite-size effects and thermodynamic limit. J. Stat. Phys. 144, 793–812 (2011)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Berchialla, L., Galgani, L., Giorgilli, A.: Localization of energy in FPU chains. Discrete Contin. Dyn. Syst. 11, 855–866 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Berchialla, L., Giorgilli, A., Paleari, S.: Exponentially long times to equipartition in the thermodynamic limit. Phys. Lett. A 321, 167–172 (2004)zbMATHCrossRefGoogle Scholar
  15. 15.
    Bloch, A., Golse, F, Paul T., Uribe, A.: Dispersionless toda and toeplitz operators. Duke Math. J. 117, 157–196 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Bocchieri, P., Scotti, A., Bearzi, B., Loinger, A.: Anharmonic chains with Lenard–Jones interactions. Phys. Rev. A 2, 2013–2019 (1970)CrossRefGoogle Scholar
  17. 17.
    Carati, A.: An averaging theorem for Hamiltonian dynamical systems in the thermodynamic limit. J. Stat. Phys. 128, 1057–1077 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Carati, A., Maiocchi, A.M.: Exponentially long stability times for a nonlinear lattice in the thermodynamic limit. Comm. Math. Phys. 314, 129–161 (2012)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    De Roeck, W., Huveneers, F.: Asymptotic localization of energy in non-disordered oscillator chains (2013) [arXiv:1305.512]Google Scholar
  20. 20.
    Fermi, E., Pasta, J., Ulam, S.: Studies of nonlinear problems. In Collected works of E. Fermi, vol.2. Chicago University Press, Chicago (1965)Google Scholar
  21. 21.
    Fucito, F., Marchesoni, F., Marinari, E., Parisi, G., Peliti, L., Ruffo, S., Vulpiani, A.: Approach to equilibrium in a chain of nonlinear oscillators. J. de Physique 43, 707–713 (1982)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Galgani, L., Giorgilli, A., Martinoli, A., Vanzini, S.: On the problem of energy equipartition for large systems of the Fermi-Pasta-Ulam type: analytical and numerical estimates. Phys. D 59, 334–348 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Giorgilli, A., Galgani, L.: Formal integrals for an autonomous Hamiltonian system near an equilibrium point. Celestial Mech. 17, 267–280 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Hairer, E., Lubich, C.: On the energy distribution in Fermi-Pasta-Ulam lattices. Arch. Ration. Mech. Anal. 205, 993–1029 (2012)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Hénon, M.: Integrals of the Toda lattice. Phys. Rev. B (3), 9, 1921–1923 (1974)Google Scholar
  26. 26.
    Henrici, A., Kappeler, T.: Birkhoff normal form for the periodic Toda lattice. In: Integrable systems and random matrices, Contemporary Math, vol. 458, pp. 11–29. American Mathematical Society, Providence (2008)Google Scholar
  27. 27.
    Henrici, A., Kappeler, T.: Global action-angle variables for the periodic Toda lattice. Int. Math. Res. Not. IMRN 11, 52 (2008). (Art. ID rnn031)Google Scholar
  28. 28.
    Henrici, A., Kappeler, T.: Global Birkhoff coordinates for the periodic Toda lattice. Nonlinearity 21, 2731–2758 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Izrailev, F.M., Chirikov, B.V.: Statistical properties of a nonlinear string. Sov. Phys. Dokl. 11, 30–32 (1966)Google Scholar
  30. 30.
    Kappeler, T., Pöschel, J.: KdV & KAM. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 45 (A Series of Modern Surveys in Mathematics) [Results in Mathematics and Related Areas, 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Berlin (2003)Google Scholar
  31. 31.
    Maiocchi, A.M., Bambusi, D., Carati, A.: An averaging theorem for fpu in the thermodynamic limit. J. Stat. Phys. 155, 300–322 (2014)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Nishida T. A note on an existence of conditionally periodic oscillation in a one-dimensional anharmonic lattice. Mem. Fac. Eng. Kyoto Univ. 33, 27–34 (1971)MathSciNetGoogle Scholar
  33. 33.
    Ponno, A.: The Fermi-Pasta-Ulam problem in the thermodynamic limit. In: Chaotic dynamics and transport in classical and quantum systems, NATO Science Series II Mathematics Physics and Chemistry, vol. 182, pp. 431–440. Kluwer Acadamic, Dordrecht (2005)Google Scholar
  34. 34.
    Pöschel, J.: Hill’s potentials in weighted Sobolev spaces and their spectral gaps. Math. Ann. 349, 433–458 (2011)zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Rink, B.: Symmetry and resonance in periodic FPU chains. Comm. Math. Phys. 218, 665–685 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Schneider, G., Wayne, C.E.: Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model. In: International Conference on Differential Equations (Berlin, 1999), vols. 1, 2, pp. 390–404. World Science, River Edge (2000)Google Scholar
  37. 37.
    Shepelyansky, D.L.: Low-energy chaos in the Fermi–Pasta–Ulam problem. Nonlinearity 10, 1331–1338 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Toda, T.: Vibration of a chain with nonlinear interaction. J. Phys. Soc. Jpn. 22, 431 (1967)CrossRefGoogle Scholar
  39. 39.
    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)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Dario Bambusi
    • 1
    Email author
  • Andrea Carati
    • 1
  • Alberto Maiocchi
    • 2
  • Alberto Maspero
    • 1
  1. 1.Dipartimento di MatematicaUniversità degli Studi di MilanoMilanoItaly
  2. 2.Laboratoire de MathématiquesUniversité de Cergy-PontoiseCergy-PontoiseFrance

Personalised recommendations