Abstract
The time scales for equipartition to be reached are studied using a generalization of the recently introduced measure of ergodicity in liquids. Theβ-Fermi-Pasta-Ulam model is chosen as an illustration. The measures are constructed by following the evolution of the systems using two independent initial conditions. The time-averaged property of an observable is calculated using the two dynamical trajectories. The measure is essentially the norm in the space of the observable obtained from the two trajectories. We show that the time-dependent behavior of the measure is a good indicator of the equipartition in large nonlinear systems. The numerical results show that equipartitioning critically depends on the initial conditions, and even when adequate mode mixing occurs the time scales appear to be extremely long.
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References
E. Diana, L. Galgani, M. Casartelli, G. Casati, and A. Scotti,Theor. Math. Phys. 29:1022 (1976).
P. Bocchieri, A. Scotti, B. Bearzi, and A. Loinger,Phys. Rev. A 2:2013 (1970).
G. Bennetin, G. Lo Vecchio, and A. Tenenbaum,Phys. Rev. A 22:1709 (1980).
R. Livi, M. Petini, S. Ruffo, M. Sparpaglione, and A. Vulpiani,Phys. Rev. A 31:1039, 2740 (1985).
A. J. Lichtenberg and M. A. Lieberman,Regular and Stochastic Motion (Springer-Verlag, New York, 1983).
M. Pettini and A. Vulpiani,Phys. Lett. 106A:207 (1980).
P. Ehrenfest and T. Ehrenfest,The Conceptual Foundations in Mechanics (Cornell University Press, Ithaca, New York, 1959).
A. Houghton, S. Jain, and A. P. Young,Phys. Rev. B 28:2630 (1983); D. Thirumalai and T. R. Kirkpatrick,Phys. Rev. B 38:4881 (1988).
F. M. Izrailev and B. V. Chirikov,Sov. Phys. Dokl. 11:30 (1966).
B. Callegari, M. Carotta, C. Ferrario, G. Lo Vecchio, and L. Galgani,Nuovo Cimento B 54:463 (1979); L. Galgani and G. Lo Vecchio,Nuovo Cimento B 52:1 (1979).
G. Bennetin and A. Tenenbaum,Phys. Rev. A 28:3020 (1983).
D. Thirumalai, R. D. Mountain, and T. R. Kirkpatrick,Phys. Rev. A 39:3563 (1989).
R. D. Mountain and D. Thirumalai,J. Phys. Chem. 93:6975 (1989).
E. Fermi, J. Pasta, and S. Ulam, Los Alamos Science Laboratory Report No. LA-1940 (1955) (unpublished); also inCollected works of Enrico Fermi, Vol. 2 (University of Chicago Press, Chicago, 1965), p. 978.
B. I. Henry and J. Grindlay,Phys. Rev. A 38:2594 (1988).
J. Ford,J. Math. Phys. 2:387 (1961); J. Ford and J. Waters,J. Math. Phys. 4:1293 (1963).
T. R. Kirkpatrick and D. Thirumalai,J. Phys. A 22:L149 (1989).
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Thirumalai, D., Mountain, R.D. Probes of equipartition in nonlinear Hamiltonian systems. J Stat Phys 57, 789–801 (1989). https://doi.org/10.1007/BF01022832
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DOI: https://doi.org/10.1007/BF01022832