Abstract
We perform a detailed numerical study of the transition to equipartition in the Fermi-Pasta-Ulam quartic model and in a class of potentials of given symmetry using the normalized spectral entropy as a probe. We show that the typical time scale for the equipartition of energy among Fourier modes grows linearly with system size: this is the time scale associated with the smallest frequency present in the system. We obtain two different scaling behaviors, either with energy or with energy density, depending on the scaling of the initial condition with system size. These different scaling behaviors can be understood by a simple argument, based on the Chirikov overlap criterion. Some aspects of the universality of this result are investigated: symmetric potentials show a similar transition, regulated by the same time scale.
Similar content being viewed by others
References
E. Fermi, J. Pasta, and S. Ulam, inCollected Papers of E. Fermi (University of Chicago Press, Chicago, 1965), Vol. 2, p. 78.
F. Izrailev and B. Chirikov,Sov. Phys. Doklady 11:30 (1966).
B. V. Chirikov, F. U. Izrailev, and V. A. Tayurskij,Comp. Phys. Comm. 5:11 (1973).
P. Bocchieri, A. Scotti, B. Bearzi, and A. Loinger,Phys. Rev. A 2:2013 (1970) M. Casartelli, G. Casati, E. Diana, L. Galgani, and A. Scotti,Theor. Math. Phys. 29:205 (1976); G. Benettin and A. Tenenbaum,Phys. Rev. A 28:3020 (1983).
R. Livi, A. Politi, and S. Ruffo,J. Phys. A 19:2033 (1986).
R. Livi, M. Pettini, S. Ruffo, M. Sparpaglione, and A. Vulpiani,Phys. Rev. A 31:1039 (1985); R. Livi, M. Pettini, S. Ruffo, and A. Vulpiani,Phys. Rev. A 31:2740 (1985).
H. Kantz,Physica D 39:322–335 (1989); M. Pettini and M. Landolfi,Phys. Rev. A 41:768 (1980); M. Pettini and M. Cerruti-Sola,Phys. Rev. A 44:975 (1991).
M. Toda,Phys. Rep. 18:1 (1975).
N. N. Nekhoroshev,Russ. Math. Surv. 18:85 (1977); G. Benettin, L. Galgani, and A. Giorgilli,Celestical Mech. 37:1 (1985); G. Benettin, L. Galgani, and A. Giorgilli,Commun. Math. Phys. 11W3:87 (1987),121:557 (1989); L. Galgani, A. Giorgilli, A. Martinoli, and S. Vanzini,Physica D 59:334 (1992).
C. G. Goedde, A. J. Lichtenberg, and M. A. Lieberman,Physica D 59:200 (1992).
B. Chirikov,Phys. Rep. 52:263 (1979).
G. A. Vinogradov, M. M. El'yashevich, and V. N. Likhachev,Phys. Lett. A 151:515 (1990).
J. De Luca, A. J. Lichtenberg, and M. Lieberman, Time scale to ergodicity in the FPU system, preprint, University of California, Bekeley (1994).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kantz, H., Livi, R. & Ruffo, S. Equipartition thresholds in chains of anharmonic oscillators. J Stat Phys 76, 627–643 (1994). https://doi.org/10.1007/BF02188678
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02188678