Anomalous Diffusion in Infinite Horizon Billiards

We consider the long time dependence for the moments of displacement < |r|^q > of infinite horizon billiards, given a bounded initial distribution of particles. For a variety of billiard models we find ~ t^g(q) (up to factors of log t). The time exponent, g(q), is piecewise linear and equal to q/2 for q2. We discuss the lack of dependence of this result on the initial distribution of particles and resolve apparent discrepancies between this time dependence and a prior result. The lack of dependence on initial distribution follows from a remarkable scaling result that we obtain for the time evolution of the distribution function of the angle of a particle's velocity vector.Comment: 11 pages, 7 figures Submitted to Physical Review

Invasion Percolation Between two Sites

We investigate the process of invasion percolation between two sites (injection and extraction sites) separated by a distance r in two-dimensional lattices of size L. Our results for the non-trapping invasion percolation model indicate that the statistics of the mass of invaded clusters is significantly dependent on the local occupation probability (pressure) Pe at the extraction site. For Pe=0, we show that the mass distribution of invaded clusters P(M) follows a power-law P(M) ~ M^{-\alpha} for intermediate values of the mass M, with an exponent \alpha=1.39. When the local pressure is set to Pe=Pc, where Pc corresponds to the site percolation threshold of the lattice topology, the distribution P(M) still displays a scaling region, but with an exponent \alpha=1.02. This last behavior is consistent with previous results for the cluster statistics in standard percolation. In spite of these discrepancies, the results of our simulations indicate that the fractal dimension of the invaded cluster does not depends significantly on the local pressure Pe and it is consistent with the fractal dimension values reported for standard invasion percolation. Finally, we perform extensive numerical simulations to determine the effect of the lattice borders on the statistics of the invaded clusters and also to characterize the self-organized critical behavior of the invasion percolation process.Comment: 7 pages, 11 figures, submited for PR

Chaotic motions of Prometheus and Pandora

Recent HST images of the Saturnian satellites Prometheus and Pandora show that their longitudes deviate from predictions of ephemerides based on Voyager images. Currently Prometheus is lagging and Pandora leading these predictions by somewhat more than 20◦. We show that these discrepancies are fully accounted for by gravitational interactions between the two satellites. These peak every 24.8 d at conjunctions and excite chaotic perturbations. The Lyapunov exponent for the Prometheus-Pandora system is of order 0.35 yr^−1 for satellite masses based on a nominal density of 1.3 g cm^−3. Interactions are strongest when the orbits come closest together. This happens at intervals of 6.2 yr when their apses are anti-aligned. In this context we note the sudden changes of opposite signs in the mean motions of Prometheus and Pandora at the end of 2000 occured shortly after their apsidal lines were anti-aligned

Growth rate of Rayleigh-Taylor turbulent mixing layers with the foliation approach

For years, astrophysicists, plasma fusion and fluid physicists have puzzled over Rayleigh-Taylor turbulent mixing layers. In particular, strong discrepancies in the growth rates have been observed between experiments and numerical simulations. Although two phenomenological mechanisms (mode-coupling and mode-competition) have brought some insight on these differences, convincing theoretical arguments are missing to explain the observed values. In this paper, we provide an analytical expression of the growth rate compatible with both mechanisms and is valide for a self-similar, low Atwood Rayleigh-Taylor turbulent mixing subjected to a constant or time-varying acceleration. The key step in this work is the introduction of {\it foliated} averages and {\it foliated} turbulent spectra highlighted in our three dimensional numerical simulations. We show that the exact value of the Rayleigh-Taylor growth rate not only depends upon the acceleration history but is also bound to the power-law exponent of the {\it foliated} spectra at large scales

Finite Temperature and Dynamical Properties of the Random Transverse-Field Ising Spin Chain

We study numerically the paramagnetic phase of the spin-1/2 random transverse-field Ising chain, using a mapping to non-interacting fermions. We extend our earlier work, Phys. Rev. 53, 8486 (1996), to finite temperatures and to dynamical properties. Our results are consistent with the idea that there are ``Griffiths-McCoy'' singularities in the paramagnetic phase described by a continuously varying exponent $z(\delta)$, where $\delta$ measures the deviation from criticality. There are some discrepancies between the values of $z(\delta)$ obtained from different quantities, but this may be due to corrections to scaling. The average on-site time dependent correlation function decays with a power law in the paramagnetic phase, namely $\tau^{-1/z(\delta)}$, where $\tau$ is imaginary time. However, the typical value decays with a stretched exponential behavior, $\exp(-c\tau^{1/\mu})$, where $\mu$ may be related to $z(\delta)$. We also obtain results for the full probability distribution of time dependent correlation functions at different points in the paramagnetic phase.Comment: 10 pages, 14 postscript files included. The discussion of the typical time dependent correlation function has been greatly expanded. Other papers of APY are available on-line at http://schubert.ucsc.edu/pete