The Fermi-Pasta-Ulam problem revisited: Stochasticity thresholds in nonlinear Hamiltonian systems

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 55, Issue 6 SUPPL. A, 1997, Pages 6566-6574

  Casetti, Lapo ^a,d   Cerruti Sola, Monica ^b   Pettini, Marco ^b,d   Cohen, E G D ^c  

^a SCUOLA NORMALE SUPERIORE   (Italy)

^b INAF OSSERVATORIO ASTROFISICO DI ARCETRI   (Italy)

^c ROCKEFELLER UNIVERSITY   (United States)

^d INFN SEZIONE DI FIRENZE   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000313054     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/physreve.55.6566     Document Type: Article

References (42)

1. E. Fermi, J. Pasta, and S. Ulam, Los Alamos Report LA-1940 (1955) (unpublished); in Collected Papers of Enrico Fermi, edited by E. Segré (University of Chicago, Chicago, 1965), Vol. 2, p. 978.
2. E. Fermi, J. Pasta, and S. Ulam, Los Alamos Report LA-1940 (1955) (unpublished); in Collected Papers of Enrico Fermi, edited by E. Segré (University of Chicago, Chicago, 1965), Vol. 2, p. 978.
3. A.N. Kolmogorov, Dokl. Akad. Nauk. SSSR 98, 527 (1954); V.I. Arnold, Russ. Math. Surv. 18, 9 (1963); J. Moser, Nachr. Akad. Wiss. Goettingen Math. Phys, Kl. 2 1, 1 (1962).
4. A.N. Kolmogorov, Dokl. Akad. Nauk. SSSR 98, 527 (1954); V.I. Arnold, Russ. Math. Surv. 18, 9 (1963); J. Moser, Nachr. Akad. Wiss. Goettingen Math. Phys, Kl. 2 1, 1 (1962).
5. A.N. Kolmogorov, Dokl. Akad. Nauk. SSSR 98, 527 (1954); V.I. Arnold, Russ. Math. Surv. 18, 9 (1963); J. Moser, Nachr. Akad. Wiss. Goettingen Math. Phys, Kl. 2 1, 1 (1962).
6. N.J. Zabusky and M.D. Kruskal, Phys. Rev. Lett. 15, 240 (1965).
7. N.J. Zabusky and G.S. Deem, J. Comp. Phys. 2, 126 (1967); N.J. Zabusky, J. Phys. Soc. Jpn. 26, 196 (1969).
8. N.J. Zabusky and G.S. Deem, J. Comp. Phys. 2, 126 (1967); N.J. Zabusky, J. Phys. Soc. Jpn. 26, 196 (1969).
9. P.M. Izrailev and B.V. Chirikov, Dokl. Akad. Nauk SSSR 166, 57 (1966) [Sov. Phys. Dokl. 11, 30 (1966)].
10. P.M. Izrailev and B.V. Chirikov, Dokl. Akad. Nauk SSSR 166, 57 (1966) [Sov. Phys. Dokl. 11, 30 (1966)].
11. J. Ford and G.H. Lunsford, Phys. Rev. A 1, 59 (1970).
12. J. Ford, Phys. Rep. 213, 271 (1992).
13. H. Kantz, R. Livi, and S. Ruffo, J. Stat. Phys. 76, 627 (1994).
14. J. De Luca, A. Lichtenberg, and S. Ruffo, Phys. Rev. E 51, 2877 (1995).
15. P. Poggi, S. Ruffo, and H. Kantz. Phys. Rev. E 52, 307 (1995).
16. When the nonlinearity of the system is large, equipartition of energy is intended in its generalized form: see, e.g., A. Münster, Statistical Thermodynamics (Springer-Verlag, Berlin, 1969), Vol. 1.
17. L. Casetti, Phys. Scr. 51, 29 (1995).
18. R. Livi, M. Pettini, S. Ruffo, M. Sparpaglione, and A. Vulpiani, Phys. Rev. A 31, 1039 (1985).
19. When the anharmonicity in the system is large, one should define the weights on the basis of the generalized equipartition theorem; for a discussion of this point see Ref. [16].
20. M. Pettini and M. Landolfi, Phys, Rev. A 41, 768 (1990).
21. M. Pettini and M. Cerruti-Sola, Phys. Rev. A 44, 975 (1991).
22. G. Benettin, L. Galgani, and J.-M. Strelcyn, Phys. Rev. A 14, 2338 (1976).
23. 1(t) is a meaningful quantity also at finite time t; see M. Pettini, Phys. Rev, E 47, 828 (1993), and L. Casetti, C. Clementi, and M. Pettini, ibid. 54, 5969 (1996).
24. 1(t) is a meaningful quantity also at finite time t; see M. Pettini, Phys. Rev, E 47, 828 (1993), and L. Casetti, C. Clementi, and M. Pettini, ibid. 54, 5969 (1996).
25. The Poincaré-Fermi nonexistence theorem states that for a number N of degrees of freedom ≥ 3, no analytic or smooth integral of motion besides the energy exists for nonintegrable systems. Thus, the nonintegrable motions (those that do not live on KAM tori) belong to a connected component of phase space and the whole of any assigned constant energy surface of phase space is topologically accessible.
26. We assume that even for the small amplitude considered by FPU a finite threshold exists, so that we can extrapolate our results to the FPU case.
27. M. Hénon and C. Heiles, Astron. J. 69, 73 (1964).
28. C.P. Enz, M.O. Hongler, and C.V. Quach Thi, Helv, Phys. Acta 48, 787 (1975).
29. H. Rüssmann, in Dynamical Systems, Theory and Applications, edited by J. Moser, Lectures Notes in Physics Vol. 38 (Springer-Verlag, Berlin, 1975);
30. L. Chierchia and G. Gallavotti, Nuovo Cimento B 67, 277 (1982).
31. E. Wayne, Commun. Math. Phys. 96, 311 (1984); 96, 331 (1984).
32. E. Wayne, Commun. Math. Phys. 96, 311 (1984); 96, 331 (1984).
33. H. Kantz, Physica D 39, 322 (1989).
34. J. De Luca, A.J. Lichtenberg and M.A. Lieberman, Chaos 5, 283 (1995).
35. D.L. Shepelyansky (unpublished).
36. N.N. Nekhoroshev, Funct. Anal. Appl. 5, 338 (1971); Russ. Math. Surv. 32, 1 (1977).
37. N.N. Nekhoroshev, Funct. Anal. Appl. 5, 338 (1971); Russ. Math. Surv. 32, 1 (1977).
38. G. Benettin, L. Galgani, and A, Giorgilli, Celest. Mech. 37, 1 (1985); P. Lochak, Phys. Lett. A 143, 39 (1990); G. Benettin and G. Gallavotti, J. Stat. Phys. 44, 293 (1986).
39. G. Benettin, L. Galgani, and A, Giorgilli, Celest. Mech. 37, 1 (1985); P. Lochak, Phys. Lett. A 143, 39 (1990); G. Benettin and G. Gallavotti, J. Stat. Phys. 44, 293 (1986).
40. G. Benettin, L. Galgani, and A, Giorgilli, Celest. Mech. 37, 1 (1985); P. Lochak, Phys. Lett. A 143, 39 (1990); G. Benettin and G. Gallavotti, J. Stat. Phys. 44, 293 (1986).
41. L. Casetti, R. Livi, A. Macchi, and M. Pettini, Europhys. Lett. 32, 549 (1995).
42. J.L. Tuck and M.T. Menzel, Adv. Math. 9, 399 (1972).