Phase space transport and the continuum limit in nonlinear Hamiltonian systems

Details Phase space transport and the continuum limit in nonlinear Hamiltonian systems Phase space transport and the continuum limit in nonlinear Hamiltonian systems

Oct 2001

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2001phdt.........9p&link_type=abstract

Thesis (PhD). UNIVERSITY OF FLORIDA, Source DAI-B 62/04, p. 1904, Oct 2001, 70 pages.

Computer Science

1

Scientific paper

First passage time experiments were used to explore the effects of low amplitude noise as a source of accelerated phase space diffusion in two-dimensional Hamiltonian systems, and these effects were then compared with the effects of periodic driving. For both noise and periodic driving, the typical escape time scales logarithmically with the amplitude of the perturbation. For white noise, the details seem unimportant: Additive and multiplicative noise typically have very similar effects, and the presence or absence of a friction related to the noise by a Fluctuation-Dissipation Theorem is also largely irrelevant. Allowing for colored noise can significantly decrease the efficacy of the perturbation, but only when the autocorrelation time becomes so large that there is little power at frequencies comparable to the natural frequencies of the unperturbed orbit. Similarly, periodic driving is relatively inefficient when the driving frequency is not comparable to these natural frequencies. This suggests that noise-induced extrinsic diffusion, like modulational diffusion associated with periodic driving, is a resonance phenomenon. In the second part of the dissertation, the continuum limit of the gravitational N-body problem was investigated both numerically and analytically. For model systems described in the continuum limit by an integrable potential, numerically generated orbits become more nearly regular in terms of visual appearance and Fourier spectra as N increases. However, the values of the largest Lyapunov exponents actually grow larger as N increases. For one model N-body system, it was possible to construct a statistical description that explains qualitatively this behavior. It was demonstrated that in both two and three dimensions, the values of the largest Lyapunov exponents are bounded from below by a positive constant. In two dimensions, the largest Lyapunov exponent grows no faster than N3/4, while in three dimensions the growth ``rate'' of λ does not exceed N1/2 . It is also shown how Monte-Carlo simulations can be used to improve these estimates. Both numerical and analytic results agree that λ cannot converge, as N increases, to the zero value found in continuous integrable potentials. This implies that the spectra of Lyapunov exponents are of limited utility for describing chaos in N-body systems.

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