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 λ ∼ −ρ ln ρ as a function of the density ρ 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.

Lyapunov spectrum of forced homogeneous isotropic turbulent flows

“…Here, the statistical uncertainty of the quantity is not indicated for clarity. Similar to pre- vious results mentioned at the beginning of this section, the LVs are increasingly distributed in physical space as the LE index increases, although the range of variation is narrower than in other systems [63].…”

“…To do so, the entropy-like metric of localization introduced in Ref. [63] is used. This metric W N 3 , called the localization width, where N 3 is the number of entries in δξ 2 and is defined as W = exp(S), where S = − N 3 j=1 δξ 2 j log δξ 2 j , and δξ 2 j is the j-th entry of δξ 2 normalized by the L 2 -norm of δξ 2 .…”

Estimating ergodization time of a chaotic many-particle system from a time reversal of equilibrium noise

“…Why this approximation works so well for the DGPE on large lattices and whether it works for a more general class of systems needs further investigation. A possible explanation of equation (10) is that, in our simulations, the Lyapunov eigenvector corresponding to max l is usually localized at only a handful of sites, which is consistent with other observations of wandering localization of Lyapunov eigenvectors [36][37][38][39][40][41][42][43].…”

Lyapunov instabilities in lattices of interacting classical spins at infinite temperature

“…The classical spin lattices obviously belong to the latter group. We further remark on the existence of the delocalized Lyapunov-Goldstone modes, which were observed in dilute gases [4,30,31,14] and in some other extended systems [32]. If these modes exist in the spin systems, the projections on single spins of the Lyapunov vectors corresponding to the smallest nonzero Lyapunov exponents should exhibit a sinusoidal dependence on the positions of spins.…”