Abstract
In this paper, we construct an adiabatic invariant for a large 1–d lattice of particles, which is the so called Klein Gordon lattice. The time evolution of such a quantity is bounded by a stretched exponential as the perturbation parameters tend to zero. At variance with the results available in the literature, our result holds uniformly in the thermodynamic limit. The proof consists of two steps: first, one uses techniques of Hamiltonian perturbation theory to construct a formal adiabatic invariant; second, one uses probabilistic methods to show that, with large probability, the adiabatic invariant is approximately constant. As a corollary, we can give a bound from below to the relaxation time for the considered system, through estimates on the autocorrelation of the adiabatic invariant.
Similar content being viewed by others
References
Kuksin S.B.: Analysis of Hamiltonian PDEs. Oxford University Press, Oxford (2000)
Bambusi D., Grébert B.: Birkhoff normal form for PDES with tame modulus. Duke Math. J. 135, 507–567 (2006)
Fröhlich J., Spencer T., Wayne C.E.: Localization in disordered, non–linear dynamical systems. J. Stat. Phys. 42, 247–274 (1986)
Bambusi D., Giorgilli A.: Exponential stability of states close to resonance in infinite dimensional Hamiltonian Systems. J. Stat. Phys. 71, 569–606 (1993)
Giorgilli A., Galgani L.: Formal integrals for an autonomous Hamiltonian system near an equilibrium point. Cel. Mech. 17, 267–280 (1978)
Dobrushin R.L.: The Description of a Random Field by Means of Conditional Probabilities and Conditions of Its Regularity. Theory Probab. Appl. 13, 197–224 (1968)
Bogolyubov, N.N., Khatset, B.I., Petrina, D.Ya.: Mathematical Description of the Equilibrium State of Classical Systems on the Basis of the Canonical Ensemble Formalism. Ukr. J. Phys. 53, Special Issue, 168–184 (2008), available at http://www.ujp.bitp.kiev.ua/files/file/papers/53/special_issue/53SI34p.pdf; Russian original in Teor. Mat. Fiz., 1,2, 251–274 (1969)
Dobrushin, R.L., Pechersky, E.A.: A criterion of the uniqueness of gibbsian fields in the non-compact case. In: Prokhorov, J.V., Ito, K. (eds.), Probability Theory and Mathematical Statistics, Springer: Berlin, 1983, pp. 97–110
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.–Paris 43, 707–714 (1982)
Parisi G.: On the approach to equilibrium of a Hamiltonian chain of anharmonic oscillators. Europhys. Lett. 40, 357–362 (1997)
Flach S.: Breathers on lattices with long range interaction. Phys. Rev. E 58, R4116–R4119 (1998)
Bambusi D., Muraro D., Penati T.: Numerical studies on boundary effects on the FPU paradox. Phys. Lett. A 372, 2039–2042 (2008)
Bambusi D., Carati A., Penati T.: Boundary effects on the dynamics of chains of coupled oscillators. Nonlinearity 22, 923–946 (2009)
Carati A., Galgani L., Santolini F.: On the energy transfer to small scales in a discrete model of one-dimensional turbulence. Chaos 19, 023123 (2009)
Carati A.: An averaging theorem for Hamiltonian dynamical systems in the thermodynamic limit. J. Stat. Phys. 128, 1057–1077 (2007)
Cherry T.M.: On integrals developable about a singular point of a Hamiltonian system of differential equations. Proc. Camb. Phil. Soc. 22, 325–349 (1924)
Giorgilli A.: Rigorous results on the power expansions for the integrals of a Hamiltonian system near an elliptic equilibrium point. Ann. Inst. Henri Poincaré 48, 423–439 (1988)
Ruelle D.: Probability estimates for continuous spin systems. Commun. Math. Phys. 50, 189–194 (1976)
Green M.S.: Markoff Random Processes and the Statistical Mechanics of Time-Dependent Phenomena. II. Irreversible Processes in Fluids. J. Chem. Phys 22, 398–413 (1954)
Kubo R.: Statistical-mechanical theory of irreversible processes. I. General theory and simple application to magnetic and conduction problems. J. Phys. Soc. Jpn. 12, 570–586 (1957)
Liverani C.: On contact Anosov flows. Ann. Math. 159, 1275–1312 (2004)
Keller G., Liverani C.: Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension. Commun. Math. Phys. 262, 33–50 (2006)
Chernov, N., Markarian, R.: Chaotic billiards. Providence, RJ: Amer. Math. Soc., 2006
Crawford J.D., Cary J.R.: Decay of correlations in a chaotic measure-preserving transformation. Physica D 6, 223–232 (1983)
Nekhoroshev N.N.: An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems. Russ. Math. Surv. 32(6), 1–65 (1977)
Giorgilli, A.: Studio con metodi perturbativi, Preprint
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. Gallavotti
Rights and permissions
About this article
Cite this article
Carati, A., Maiocchi, A.M. Exponentially Long Stability Times for a Nonlinear Lattice in the Thermodynamic Limit. Commun. Math. Phys. 314, 129–161 (2012). https://doi.org/10.1007/s00220-012-1522-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-012-1522-z