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

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.Comment: 41 pages, 21 figures, Manuscript including the figures of better quality is available from http://www.phys.unsw.edu.au/~gary/Research.htm

Role of chaos for the validity of statistical mechanics laws: diffusion and conduction

Several years after the pioneering work by Fermi Pasta and Ulam, fundamental questions about the link between dynamical and statistical properties remain still open in modern statistical mechanics. Particularly controversial is the role of deterministic chaos for the validity and consistency of statistical approaches. This contribution reexamines such a debated issue taking inspiration from the problem of diffusion and heat conduction in deterministic systems. Is microscopic chaos a necessary ingredient to observe such macroscopic phenomena?Comment: Latex, 27 pages, 10 eps-figures. Proceedings of the Conference "FPU 50 years since" Rome 7-8 May 200

The role of skin ultrasound in systemic sclerosis: looking below the surface to understand disease evolution

Skin ultrasound has shown promising results in the evaluation of skin involvement in patients with systemic sclerosis, as substantiated by a recent systematic literature review from the World Scleroderma Foundation Skin Ultrasound Working Group. In this Viewpoint, we will discuss the role of ultrasound in evaluating skin involvement in patients with systemic sclerosis, particularly the possibility of using this technique to detect an early subclinical skin involvement from the very early phase, suggesting its possible use in both diagnosis and disease follow-up. To detect subclinical skin involvement, it is essential to understand the difference between the skin of patients with systemic sclerosis and that of healthy controls, including defining exactly which structures are affected by the disease and which are spared. The potential of this non-invasive technique might suggest its future role in both clinical practice and clinical trials, possibly replacing invasive and painful procedures such as skin biopsies and promoting patient retention in clinical trials

Dynamics of oscillator chains

The Fermi\u2013Pasta\u2013Ulam (FPU) nonlinear oscillator chain has proved to be a seminal system for investigating problems in nonlinear dynamics. First proposed as a nonlinear system to elucidate the foundations of statistical mechanics, the initial lack of confirmation of the researchers expectations eventually led to a number of profound insights into the behavior of high-dimensional nonlinear systems. The initial numerical studies, proposed to demonstrate that energy placed in a single mode of the linearized chain would approach equipartition through nonlinear interactions, surprisingly showed recurrences. Although subsequent work showed that the origin of the recurrences is nonlinear resonance, the question of lack of equipartition remained. The attempt to understand the regularity bore fruit in a profound development in nonlinear dynamics: the birth of soliton theory. A parallel development, related to numerical observations that, at higher energies, equipartition among modes could be approached, was the understanding that the transition with increasing energy is due to resonance overlap. Further numerical investigations showed that time-scales were also important, with a transition between faster and slower evolution. This was explained in terms of mode overlap at higher energy and resonance overlap at lower energy. Numerical limitations to observing a very slow approach to equipartition and the problem of connecting high-dimensional Hamiltonian systems to lower dimensional studies of Arnold diffusion, which indicate transitions from exponentially slow diffusion along resonances to power-law diffusion across resonances, have been considered. Most of the work, both numerical and theoretical, started from low frequency (long wavelength) initial conditions. Coincident with developments to understand equipartition was another program to connect a statistical phenomenon to nonlinear dynamics, that of understanding classical heat conduction. The numerical studies were quite different, involving the excitation of a boundary oscillator with chaotic motion, rather than the excitation of the entire chain with regular motion. Although energy transitions are still important, the inability to reproduce exactly the law of classical heat conduction led to concern for the generiticity of the FPU chain and exploration of other force laws. Important concepts of unequal masses, and \u201canti-integrability,\u201d i.e. isolation of some oscillators, were considered, as well as separated optical and acoustic modes that could only communicate through very weak interactions. The importance of chains that do not allow nonlinear wave propagation in producing the Fourier heat conduction law is now recognized. A more recent development has been the exploration of energy placed on the FPU or related oscillator chains in high-frequency (short wavelength) modes and the existence of isolated structures (breathers). Breathers are found as solutions to partial differential equations, analogous to solitons at lower frequency. On oscillator chains, such as the FPU, energy initially in a single high-frequency mode is found, at higher energies, to self-organize in oscillator space to form compact structures. These structures are \u201cchaotic breathers,\u201d i.e. not completely stable, and disintegrate on longer time-scales. With the significant progress in understanding this evolution, we now have a rather complete picture of the nonlinear dynamics of the FPU and related oscillator chains, and their relation to a wide range of concepts in nonlinear dynamics. This chapter\u2019s purpose is to explicate these many concepts. After a historical perspective the basic chaos theory background is reviewed. Types of oscillators, numerical methods, and some analytical results are considered. Numerical results of studies of equipartition, both from low-frequency and high-frequency modes, are presented, together with numerical studies of heat conduction. These numerical studies are related to analytical calculations and estimates of energy transitions and time-scales to equipartition