Advertisement

Annales Henri Poincaré

, Volume 16, Issue 4, pp 897–959 | Cite as

An Extensive Adiabatic Invariant for the Klein–Gordon Model in the Thermodynamic Limit

  • Antonio Giorgilli
  • Simone Paleari
  • Tiziano PenatiEmail author
Article

Abstract

We construct an extensive adiabatic invariant for a Klein–Gordon chain in the thermodynamic limit. In particular, given a fixed and sufficiently small value of the coupling constant a, the evolution of the adiabatic invariant is controlled up to time scaling as β 1/a for any large enough value of the inverse temperature β. The time scale becomes a stretched exponential if the coupling constant is allowed to vanish jointly with the specific energy. The adiabatic invariance is exhibited by showing that the variance along the dynamics, i.e. calculated with respect to time averages, is much smaller than the corresponding variance over the whole phase space, i.e. calculated with the Gibbs measure, for a set of initial data of large measure. All the perturbative constructions and the subsequent estimates are consistent with the extensive nature of the system.

Keywords

Partition Function Poisson Bracket Thermodynamic Limit Homogeneous Polynomial Gibbs Measure 
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.

References

  1. 1.
    Arnol’d, V.: Chapitres supplémentaires de la théorie des équations différentielles ordinaires. “Mir”, Moscow (1984) (Translated from the Russian by Djilali Embarek, Reprint of the 1980 edition)Google Scholar
  2. 2.
    Arnol’d V.I.: Proof of a theorem of A N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian. Uspehi Mat. Nauk 18(5 (113)), 13–40 (1963)MathSciNetGoogle Scholar
  3. 3.
    Bambusi D., Giorgilli A.: Exponential stability of states close to resonance in infinite-dimensional Hamiltonian systems. J. Stat. Phys. 71(3–4), 569–606 (1993)CrossRefADSzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bambusi D, Nekhoroshev N.N.: A property of exponential stability in nonlinear wave equations near the fundamental linear mode. Phys. D 122(1–4), 73–104 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bambusi D., Ponno A.: On metastability in FPU. Commun. Math. Phys. 264(2), 539–561 (2006)CrossRefADSzbMATHMathSciNetGoogle Scholar
  6. 6.
    Benettin G., Christodoulidi H., Ponno A.: The Fermi–Pasta–Ulam problem and its underlying integrable dynamics. J. Stat. Phys. 152, 195–212 (2013)CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Benettin G., Livi R., Ponno A.: The Fermi–Pasta–Ulam problem: scaling laws vs. initial conditions. J. Stat. Phys. 135(5–6), 873–893 (2009)CrossRefADSzbMATHMathSciNetGoogle Scholar
  8. 8.
    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). doi: 10.1007/s10955-011-0277-9 CrossRefADSzbMATHMathSciNetGoogle Scholar
  9. 9.
    Benettin G., Fröhlich J., Giorgilli A.: A Nekhoroshev-type theorem for Hamiltonian systems with infinitely many degrees of freedom. Commun. Math. Phys. 119(1), 95–108 (1988)CrossRefADSzbMATHGoogle 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. II. Commun. Math. Phys. 121(4), 557–601 (1989)CrossRefADSzbMATHMathSciNetGoogle Scholar
  11. 11.
    Berchialla L., Giorgilli A., Paleari S.: Exponentially long times to equipartition in the thermodynamic limit. Phys. Lett. A 321(3–4), 167–172 (2004)CrossRefADSzbMATHGoogle Scholar
  12. 12.
    Bourgain, J.: Hamiltonian methods in nonlinear evolution equations. In: Fields Medallists’ lectures, pp. 542–554. World Scientific Publishing, River Edge, NJ, (1997)Google Scholar
  13. 13.
    Bourgain J.: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations. Ann. Math. (2) 148(2), 363–439 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Carati A., Galgani L., Giorgilli A., Paleari S.: Fermi–Pasta–Ulam phenomenon for generic initial data. Phys. Rev. E 76(2), 022104 (2007)CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Carati A.: An averaging theorem for Hamiltonian dynamical systems in the thermodynamic limit. J. Stat. Phys. 128, 1057–1077 (2007)CrossRefADSzbMATHMathSciNetGoogle Scholar
  16. 16.
    Carati A., Maiocchi A.M: Exponentially long stability times for a nonlinear lattice in the thermodynamic limit. Commun. Math. Phys. 314, 129–161 (2012). doi: 10.1007/s00220-012-1522-z CrossRefADSzbMATHMathSciNetGoogle Scholar
  17. 17.
    Craig, W.: KAM theory in infinite dimensions. In: Dynamical systems and probabilistic methods in partial differential equations (Berkeley, CA, 1994), pp. 31–46. American Mathematical Society, Providence, RI (1996)Google Scholar
  18. 18.
    Davis, Philip J.: Circulant matrices. A Wiley-Interscience Publication, Pure and Applied Mathematics. Wiley, New York (1979)Google Scholar
  19. 19.
    De Roeck, W., Huveneers, F.: Asymptotic localization of energy in non-disordered oscillator chains. ArXiv e-prints, May 2013. 1305.5127Google Scholar
  20. 20.
    Fröhlich, J., Spencer, T., Wayne, C.E.: An invariant torus for nearly integrable Hamiltonian systems with infinitely many degrees of freedom. In: Stochastic processes in classical and quantum systems (Ascona, 1985), pp. 256–268. Springer, Berlin (1986)Google Scholar
  21. 21.
    Gallavotti, G. (ed): The Fermi–Pasta–Ulam problem. Volume 728 of Lecture Notes in Physics. Springer, Berlin (2008) (A status report)Google Scholar
  22. 22.
    Genta T., Giorgilli A., Paleari S., Penati T.: Packets of resonant modes in the Fermi–Pasta–Ulam system. Phys. Lett. Sect. A Gen. At. Solid State Phys. 376(28–29), 2038–2044 (2012)zbMATHGoogle Scholar
  23. 23.
    Giorgilli A., Galgani L.: Formal integrals for an autonomous Hamiltonian system near an equilibrium point. Celest. Mech. 17(3), 267–280 (1978)CrossRefADSzbMATHMathSciNetGoogle Scholar
  24. 24.
    Giorgilli A., Paleari S., Penati T.: Extensive adiabatic invariants for nonlinear chains. J. Stat. Phys. 148(6), 1106–1134 (2012)CrossRefADSzbMATHMathSciNetGoogle Scholar
  25. 25.
    Kolmogorov A.N.: On conservation of conditionally periodic motions for a small change in Hamilton’s function. Dokl. Akad. Nauk SSSR (N.S.) 98, 527–530 (1954)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Kuksin, S.B.: KAM-theory for partial differential equations. In: First European Congress of Mathematics, vol. II (Paris, 1992), pp. 123–157. Birkhäuser, Basel, (1994)Google Scholar
  27. 27.
    Kuksin S.B.: Analysis of Hamiltonian PDEs. The Clarendon Press, Oxford University Press, New York (2000)Google Scholar
  28. 28.
    Lanford, O.E.: Entropy and equilibrium states in classical statistical mechanics. In: Statistical mechanics and mathematical problems (Battelle Seattle 1971 Rencontre), volume 20 of Lecture Notes in Physics, pp. 1–113. Springer, Berlin (1973)Google Scholar
  29. 29.
    Lorenzoni P., Paleari S.: Metastability and dispersive shock waves in the Fermi–Pasta–Ulam system. Phys. D 221(2), 110–117 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Moser J.K.: A new technique for the construction of solutions of nonlinear differential equations. Proc. Nat. Acad. Sci. USA 47, 1824–1831 (1961)CrossRefADSzbMATHGoogle Scholar
  31. 31.
    Moser J.K.: On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1962, 1–20 (1962)zbMATHGoogle Scholar
  32. 32.
    Nekhoroshev, N.N.: An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. Uspehi Mat. Nauk 32 (6(198)), 5–66, 287 (1977). [English translation: Russian Math. Surveys 32 no. 6, 1–65 (1977)]Google Scholar
  33. 33.
    Nekhoroshev, N.N.: An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. II. Trudy Sem. Im. G. Petrovskogo 5, 5–50 (1979). [English translation: Topics in modern Mathematics, Petrovskij Semin. 5, 1–58 (1985)]Google Scholar
  34. 34.
    Paleari S., Bambusi D., Cacciatori S.: Normal form and exponential stability for some nonlinear string equations. Z. Angew. Math. Phys. 52(6), 1033–1052 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Poincaré, H.: Les méthodes nouvelles de la mécanique céleste. Tome II. Librairie Scientifique et Technique Albert Blanchard, Paris. Méthodes de MM. Newcomb, Gyldén, Lindstedt et Bohlin (1987) [The methods of Newcomb, Gyldén, Lindstedt and Bohlin, Reprint of the 1893 original, Bibliothèque Scientifique Albert Blanchard. Albert Blanchard Scientific Library]Google Scholar
  36. 36.
    Pöschel J.: On Nekhoroshev estimates for a nonlinear Schrödinger equation and a theorem by Bambusi. Nonlinearity 12(6), 1587–1600 (1999)CrossRefADSzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Antonio Giorgilli
    • 1
  • Simone Paleari
    • 1
  • Tiziano Penati
    • 1
    Email author
  1. 1.Dipartimento di MatematicaUniversità degli Studi di MilanoMilanItaly

Personalised recommendations