Localized behavior in the Lyapunov vectors for quasi-one-dimensional many-hard-disk systems

Abstract

We introduce a definition of a "localization width" whose logarithm is given by the entropy of the distribution of particle component amplitudes in the Lyapunov vector. Different types of localization widths are observed, for example, a minimum localization width where the components of only two particles are dominant. We can distinguish a delocalization associated with a random distribution of particle contributions, a delocalization associated with a uniform distribution and a delocalization associated with a wave-like structure in the Lyapunov vector. Using the localization width we show that in quasi-one-dimensional systems of many hard disks there are two kinds of dependence of the localization width on the Lyapunov exponent index for the larger exponents: one is exponential, and the other is linear. Differences, due to these kinds of localizations also appear in the shapes of the localized peaks of the Lyapunov vectors, the Lyapunov spectra and the angle between the spatial and momentum parts of the Lyapunov vectors. We show that the Krylov relation for the largest Lyapunov exponent $\lambda \sim -\rho \ln \rho$ as a function of the density $\rho$ is satisfied (apart from a factor) in the same density region as the linear dependence of the localization widths is observed. It is also shown that there are asymmetries in the spatial and momentum parts of the Lyapunov vectors, as well as in their $x$ and $y$-components.

arXiv comment41 pages, 21 figures, Manuscript including the figures of better quality is available from http://www.phys.unsw.edu.au/~gary/Research.html

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Cited by in corpus (16)

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• Lyapunov instabilities in lattices of interacting classical spins at infinite temperature
• Time-dependent mode structure for Lyapunov vectors as a collective movement in quasi-one-dimensional systems
• Hopping dynamics for localized Lyapunov vectors in many-hard-disk systems
• What does dynamical systems theory teach us about fluids?
• Lyapunov Modes for a Nonequilibrium System with a Heat Flux
• Estimating ergodization time of a chaotic many-particle system from a time reversal of equilibrium noise
• From Lyapunov modes to the exponents for hard disk systems
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• Critical bending point in the Lyapunov localization spectra of many-particle systems
• Lyapunov modes in three-dimensional Lennard-Jones fluids
• Equivalence of kinetic-theory and random-matrix approaches to Lyapunov spectra of hard-sphere systems
• Local integrability breaking and exponential localization of leading Lyapunov vectors