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.
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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)
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)
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)
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)
Bambusi D., Ponno A.: On metastability in FPU. Commun. Math. Phys. 264(2), 539–561 (2006)
Benettin G., Christodoulidi H., Ponno A.: The Fermi–Pasta–Ulam problem and its underlying integrable dynamics. J. Stat. Phys. 152, 195–212 (2013)
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)
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
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)
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)
Berchialla L., Giorgilli A., Paleari S.: Exponentially long times to equipartition in the thermodynamic limit. Phys. Lett. A 321(3–4), 167–172 (2004)
Bourgain, J.: Hamiltonian methods in nonlinear evolution equations. In: Fields Medallists’ lectures, pp. 542–554. World Scientific Publishing, River Edge, NJ, (1997)
Bourgain J.: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations. Ann. Math. (2) 148(2), 363–439 (1998)
Carati A., Galgani L., Giorgilli A., Paleari S.: Fermi–Pasta–Ulam phenomenon for generic initial data. Phys. Rev. E 76(2), 022104 (2007)
Carati A.: An averaging theorem for Hamiltonian dynamical systems in the thermodynamic limit. J. Stat. Phys. 128, 1057–1077 (2007)
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
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)
Davis, Philip J.: Circulant matrices. A Wiley-Interscience Publication, Pure and Applied Mathematics. Wiley, New York (1979)
De Roeck, W., Huveneers, F.: Asymptotic localization of energy in non-disordered oscillator chains. ArXiv e-prints, May 2013. 1305.5127
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)
Gallavotti, G. (ed): The Fermi–Pasta–Ulam problem. Volume 728 of Lecture Notes in Physics. Springer, Berlin (2008) (A status report)
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)
Giorgilli A., Galgani L.: Formal integrals for an autonomous Hamiltonian system near an equilibrium point. Celest. Mech. 17(3), 267–280 (1978)
Giorgilli A., Paleari S., Penati T.: Extensive adiabatic invariants for nonlinear chains. J. Stat. Phys. 148(6), 1106–1134 (2012)
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)
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)
Kuksin S.B.: Analysis of Hamiltonian PDEs. The Clarendon Press, Oxford University Press, New York (2000)
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)
Lorenzoni P., Paleari S.: Metastability and dispersive shock waves in the Fermi–Pasta–Ulam system. Phys. D 221(2), 110–117 (2006)
Moser J.K.: A new technique for the construction of solutions of nonlinear differential equations. Proc. Nat. Acad. Sci. USA 47, 1824–1831 (1961)
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)
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)]
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)]
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)
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]
Pöschel J.: On Nekhoroshev estimates for a nonlinear Schrödinger equation and a theorem by Bambusi. Nonlinearity 12(6), 1587–1600 (1999)
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Communicated by Dmitry Dolgopyat.
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Giorgilli, A., Paleari, S. & Penati, T. An Extensive Adiabatic Invariant for the Klein–Gordon Model in the Thermodynamic Limit. Ann. Henri Poincaré 16, 897–959 (2015). https://doi.org/10.1007/s00023-014-0335-3
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DOI: https://doi.org/10.1007/s00023-014-0335-3