Abstract
We consider multiscale Hamiltonian systems in which harmonic oscillators with several high frequencies are coupled to a slow system. It is shown that the oscillatory energy is nearly preserved over long times \({\varepsilon^{-N}}\) for arbitrary N > 1, where \({\varepsilon^{-1}}\) is the size of the smallest high frequency. The result is uniform in the frequencies and does not require non-resonance conditions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aoki K., Lukkarinen J., Spohn H.: Energy transport in weakly anharmonic chains. J. Stat. Phys. 124, 1105–1129 (2006)
Bambusi D., Giorgilli A.: Exponential stability of states close to resonance in infinite-dimensional Hamiltonian systems. J. Stat. Phys. 71, 569–606 (1993)
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. Part I. Comm. Math. Phys. 113, 87–103 (1987)
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. Part II. Commun. Math. Phys. 121, 557–601 (1989)
Carati A., Galgani L.: Theory of dynamical systems and the relations between classical and quantum mechanics. Found. Phys. 31, 69–87 (2001)
Cohen D., Hairer E., Lubich C.: Modulated Fourier expansions of highly oscillatory differential equations. Found. Comput. Math. 3, 327–345 (2003)
Cohen D., Hairer E., Lubich C.: Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions. Arch. Ration. Mech. Anal. 187, 341–368 (2008)
Dolgopyat D., Liverani C.: Energy transfer in a fast-slow Hamiltonian system. Commun. Math. Phys. 308, 201–225 (2011)
Fermi, E., Pasta, J., Ulam, S.: Studies of non linear problems. Technical Report LA-1940, Los Alamos, 1955
Gallavotti, G. (ed.): The Fermi-Pasta-Ulam problem. A status report. Lecture Notes in Physics, Volume 728. Berlin: Springer, 2008
Gauckler L., Hairer E., Lubich C., Weiss D.: Metastable energy strata in weakly nonlinear wave equations. Comm. Partial Differ. Equ. 37, 1391–1413 (2012)
Hairer E., Lubich C.: On the energy distribution in Fermi–Pasta–Ulam lattices. Arch. Ration. Mech. Anal. 205, 993–1029 (2012)
Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations. \({2^{\rm nd}}\) ed., Springer Series in Computational Mathematics 31, Berlin: Springer-Verlag, 2006
Huveneers, F.: Thermal conductivity of disordered lattices in the weak coupling regime, Preprint (2012), available at http://arxiv.org/abs/1203.3587v3 [cond-mat.stat-mech], 2012
Shudo A., Ichiki K., Saito S.: Origin of slow relaxation in liquid-water dynamics: A possible scenario for the presence of bottleneck in phase space. Europhys. Lett. 73, 826–832 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H.-T. Yau
Rights and permissions
About this article
Cite this article
Gauckler, L., Hairer, E. & Lubich, C. Energy Separation in Oscillatory Hamiltonian Systems without any Non-resonance Condition. Commun. Math. Phys. 321, 803–815 (2013). https://doi.org/10.1007/s00220-013-1728-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-013-1728-8