Abstract
In the present contribution we justify and discuss the scaling laws characterizing the first phase of the energy transfer from large to small spatial scales in a chain of nonlinear oscillators (the so-called Fermi-Pasta-Ulam α-model). By means of qualitative estimates, we show that large scale initial excitations (long wavelength Fourier modes) produce injection of energy into smaller scales on times t > τc ∼ ε−3/4 and up to a cutoff spatial scale \(\ell \)c ∼ ε−1/4, where ε is the energy per degree of freedom of the system.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Fermi E., Pasta J., Ulam S. (1954), reprinted in ”Non Linear Wave Motion”, ed. by A. Newell, Lectures in Applied Mathematics 15, 1974, pp. 143–156.
Benettin G. (1986), in ”Molecular dynamics simulation in classical statistical mechanical systems”, Proceedings of the E. Fermi school of Varenna, 1986, ed. by G. Ciccotti and W. G. Hoover.
Ford J. (1992), Phys. Rep. 213, 271–310.
Carati A., Galgani L., Ponno A., Giorgilli A. (2002), Il Nuovo Cimento B 117, 1017–1026.
Bocchieri P., Scotti A., Bearzi B., Loinger A. (1970), Phys. Rev. A 2, 2013–2019.
Biello J. A., Kramer P. R., Lvov Y. (2002), in ”Proceedings of the 4th International Conference on Dynamical Systems and Differential Equations”, Wilmington, NC, USA, 2002.
Berchialla L., Galgani L., Giorgilli A. (2003), DCDS-B, in print.
Zabusky N. J., Kruskal M. D. (1965), Phys. Rev. Lett. 15, 240–243.
Ponno A. (2003), Europhys. Lett. 64, in print.
Shepelyansky D. L. (1997), Nonlinearity 10, 1331–1338.
Izrailev F. M., Chirikov B. V. (1966), Sov. Phys. Dokl. 11, 30–32.
Chirikov B. V. (1969), reprinted and translated in CERN Trans. 71–40, Geneva, 1971, pp. 1–237.
Frisch U. (1995), Turbulence, Cambridge University Press, 1995.
Ford J. (1961), J. Math. Phys. 2, 387–393.
Pettini M., Landolfi M. (1990), Phys. Rev. A 41, 768–783.
Berchialla L., Giorgilli A., Paleari S. (2003), preprint.
Livi R., Pettini M., Ruffo S., Vulpiani A. (1985), Phys. Rev. A 31, 2740–2742.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Kluwer Academic Publishers
About this paper
Cite this paper
Ponno, A. (2005). The Fermi-Pasta-Ulam Problem in the Thermodynamic Limit. In: Collet, P., Courbage, M., Métens, S., Neishtadt, A., Zaslavsky, G. (eds) Chaotic Dynamics and Transport in Classical and Quantum Systems. NATO Science Series, vol 182. Springer, Dordrecht. https://doi.org/10.1007/1-4020-2947-0_20
Download citation
DOI: https://doi.org/10.1007/1-4020-2947-0_20
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-2945-5
Online ISBN: 978-1-4020-2947-9
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)