Lyapunov spectra of billiards with cylindrical scatterers: comparison with many-particle systems

The dynamics of a system consisting of many spherical hard particles can be described as a single point particle moving in a high-dimensional space with fixed hypercylindrical scatterers with specific orientations and positions. In this paper, the similarities in the Lyapunov exponents are investigated between systems of many particles and high-dimensional billiards with cylindrical scatterers which have isotropically distributed orientations and homogeneously distributed positions. The dynamics of the isotropic billiard are calculated using a Monte-Carlo simulation, and a reorthogonalization process is used to find the Lyapunov exponents. The results are compared to numerical results for systems of many hard particles as well as the analytical results for the high-dimensional Lorentz gas. The smallest three-quarters of the positive exponents behave more like the exponents of hard-disk systems than the exponents of the Lorentz gas. This similarity shows that the hard-disk systems may be approximated by a spatially homogeneous and isotropic system of scatterers for a calculation of the smaller Lyapunov exponents, apart from the exponent associated with localization. The method of the partial stretching factor is used to calculate these exponents analytically, with results that compare well with simulation results of hard disks and hard spheres.Comment: Submitted to PR

Dynamic fibrils in H-alpha and C IV

Aim: To study the interaction of the solar chromosphere with the transition region, in particular active-region jets in the transition region and their relation to chromospheric fibrils. Methods: We carefully align image sequences taken simultaneously in C IV with the Transition Region and Coronal Explorer and in H-alpha with the Swedish 1-m Solar Telescope. We examine the temporal evolution of "dynamic fibrils", i.e., individual short-lived active-region chromospheric jet-like features in H-alpha. Results: All dynamic fibrils appear as absorption features in H-alpha that progress from the blue to the red wing through the line, and often show recurrent behavior. Some of them, but not all, appear also as bright features in C IV which develop at or just beyond the apex of the H-alpha darkening. They tend to best resemble the H-alpha fibril at +700 mA half a minute earlier. Conclusions: Dynamic chromospheric fibrils observed in H-alpha regularly correspond to transition-region jets observed in the ultraviolet. This correspondence suggests that some plasma associated with dynamic fibrils is heated to transition-region temperatures.Comment: 8 pages, 8 figure

The Lyapunov spectrum of the many-dimensional dilute random Lorentz gas

For a better understanding of the chaotic behavior of systems of many moving particles it is useful to look at other systems with many degrees of freedom. An interesting example is the high-dimensional Lorentz gas, which, just like a system of moving hard spheres, may be interpreted as a dynamical system consisting of a point particle in a high-dimensional phase space, moving among fixed scatterers. In this paper, we calculate the full spectrum of Lyapunov exponents for the dilute random Lorentz gas in an arbitrary number of dimensions. We find that the spectrum becomes flatter with increasing dimensionality. Furthermore, for fixed collision frequency the separation between the largest Lyapunov exponent and the second largest one increases logarithmically with dimensionality, whereas the separations between Lyapunov exponents of given indices not involving the largest one, go to fixed limits.Comment: 8 pages, revtex, 6 figures, submitted to Physical Review

Observations of solar scattering polarization at high spatial resolution

The weak, turbulent magnetic fields that supposedly permeate most of the solar photosphere are difficult to observe, because the Zeeman effect is virtually blind to them. The Hanle effect, acting on the scattering polarization in suitable lines, can in principle be used as a diagnostic for these fields. However, the prediction that the majority of the weak, turbulent field resides in intergranular lanes also poses significant challenges to scattering polarization observations because high spatial resolution is usually difficult to attain. We aim to measure the difference in scattering polarization between granules and intergranules. We present the respective center-to-limb variations, which may serve as input for future models. We perform full Stokes filter polarimetry at different solar limb positions with the CN band filter of the Hinode-SOT Broadband Filter Imager, which represents the first scattering polarization observations with sufficient spatial resolution to discern the granulation. Hinode-SOT offers unprecedented spatial resolution in combination with high polarimetric sensitivity. The CN band is known to have a significant scattering polarization signal, and is sensitive to the Hanle effect. We extend the instrumental polarization calibration routine to the observing wavelength, and correct for various systematic effects. The scattering polarization for granules (i.e., regions brighter than the median intensity of non-magnetic pixels) is significantly larger than for intergranules. We derive that the intergranules (i.e., the remaining non-magnetic pixels) exhibit (9.8 \pm 3.0)% less scattering polarization for 0.2<u<0.3, although systematic effects cannot be completely excluded. These observations constrain MHD models in combination with (polarized) radiative transfer in terms of CN band line formation, radiation anisotropy, and magnetic fields.Comment: Accepted for publication in A&