Instability of the Gravitational N-Body Problem in the Large-N Limit

We use a systolic N-body algorithm to evaluate the linear stability of the gravitational N-body problem for N up to 1.3 x 10^5, two orders of magnitude greater than in previous experiments. For the first time, a clear ~ln N-dependence of the perturbation growth rate is seen. The e-folding time for N = 10^5 is roughly 1/20 of a crossing time.Comment: Accepted for publication in The Astrophysical Journa

Effects of High Fat Diet and Exercise on the Metabolism of Maternal Hearts during Pregnancy

Obesity has become a major concern for developed nations across the world, and the United States is the country which is most affected by this pandemic. Excess adiposity is known to cause chronic inflammation, diabetes, cancer, and cardiovascular disease: the leading cause of death for over a decade. With many women of reproductive age considered overweight or obese, the association between obesity and metabolic disorder is concerning. Positive metabolic health outcomes of offspring due to maternal exercise have been documented; however, little is known about how maternal exercise modifies high fat diet associated metabolic dysregulation upon mothers during gestation. The aim of our study was to determine whether maternal exercise before and during pregnancy would alleviate high fat diet-associated glucose and insulin resistance in high fat fed pregnant mice. Using C57BL/6 virgin female mice as a model, we fed the animals either a low fat diet (LFD; 10% kcal from fat) or a high fat diet (HFD; 45% kcal from fat) for twelve weeks, with an exercise intervention after four weeks (HFD+Ex), and pregnancy initiation after eight weeks of diet consumption. Glucose and insulin tolerance tests were performed at day 15 of gestation. Prescribed diet and exercise (or sedentary) behavior continued throughout pregnancy until animals were sacrificed at the 19th day of gestation. The HFD animals experienced a significant increase in body weight, along with increased numbers of calories consumed per day, and exercise further increased body weight and food intake. Both the HFD and the HFD+Ex animals displayed impaired glucose and insulin tolerance testing when compared with the LFD animals. Interestingly, exercise improved serum insulin levels at termination. mRNA expression of genes involved in fatty acid and glucose metabolism were upregulated in the HFD+Ex animals compared with the HFD mice. Our study exhibits that the development of adiposity from the consumption of a high fat diet prior to pregnancy leads to detrimental maternal effects during late gestation, including higher body weight, and glucose tolerance. Surprisingly, the addition of exercise did not alter dam morphology or gestational glucose tolerance; however, it did improve serum insulin levels and metabolite handling in the heart

Chaos and stability in a two-parameter family of convex billiard tables

We study, by numerical simulations and semi-rigorous arguments, a two-parameter family of convex, two-dimensional billiard tables, generalizing the one-parameter class of oval billiards of Benettin--Strelcyn [Phys. Rev. A 17, 773 (1978)]. We observe interesting dynamical phenomena when the billiard tables are continuously deformed from the integrable circular billiard to different versions of completely-chaotic stadia. In particular, we conjecture that a new class of ergodic billiard tables is obtained in certain regions of the two-dimensional parameter space, when the billiards are close to skewed stadia. We provide heuristic arguments supporting this conjecture, and give numerical confirmation using the powerful method of Lyapunov-weighted dynamics.Comment: 19 pages, 13 figures. Submitted for publication. Supplementary video available at http://sistemas.fciencias.unam.mx/~dsanders

About ergodicity in the family of limacon billiards

By continuation from the hyperbolic limit of the cardioid billiard we show that there is an abundance of bifurcations in the family of limacon billiards. The statistics of these bifurcation shows that the size of the stable intervals decreases with approximately the same rate as their number increases with the period. In particular, we give numerical evidence that arbitrarily close to the cardioid there are elliptic islands due to orbits created in saddle node bifurcations. This shows explicitly that if in this one parameter family of maps ergodicity occurs for more than one parameter the set of these parameter values has a complicated structure.Comment: 17 pages, 9 figure

Sensitivity of the eigenfunctions and the level curvature distribution in quantum billiards

In searching for the manifestations of sensitivity of the eigenfunctions in quantum billiards (with Dirichlet boundary conditions) with respect to the boundary data (the normal derivative) we have performed instead various numerical tests for the Robnik billiard (quadratic conformal map of the unit disk) for 600 shape parameter values, where we look at the sensitivity of the energy levels with respect to the shape parameter. We show the energy level flow diagrams for three stretches of fifty consecutive (odd) eigenstates each with index 1,000 to 2,000. In particular, we have calculated the (unfolded and normalized) level curvature distribution and found that it continuously changes from a delta distribution for the integrable case (circle) to a broad distribution in the classically ergodic regime. For some shape parameters the agreement with the GOE von Oppen formula is very good, whereas we have also cases where the deviation from GOE is significant and of physical origin. In the intermediate case of mixed classical dynamics we have a semiclassical formula in the spirit of the Berry-Robnik (1984) surmise. Here the agreement with theory is not good, partially due to the localization phenomena which are expected to disappear in the semiclassical limit. We stress that even for classically ergodic systems there is no global universality for the curvature distribution, not even in the semiclassical limit.Comment: 19 pages, file in plain LaTeX, 15 figures available upon request Submitted to J. Phys. A: Math. Ge

Quantum chaos in a deformable billiard: Applications to quantum dots

We perform a detailed numerical study of energy-level and wavefunction statistics of a deformable quantum billiard focusing on properties relevant to semiconductor quantum dots. We consider the family of Robnik billiards generated by simple conformal maps of the unit disk; the shape of this family of billiards may be varied continuously at fixed area by tuning the parameters of the map. The classical dynamics of these billiards is well-understood and this allows us to study the quantum properties of subfamilies which span the transition from integrability to chaos as well as families at approximately constant degree of chaoticity (Kolmogorov entropy). In the regime of hard chaos we find that the statistical properties of interest are well-described by random-matrix theory and completely insensitive to the particular shape of the dot. However in the nearly-integrable regime non-universal behavior is found. Specifically, the level-width distribution is well-described by the predicted $\chi^2$ distribution both in the presence and absence of magnetic flux when the system is fully chaotic; however it departs substantially from this behavior in the mixed regime. The chaotic behavior corroborates the previously predicted behavior of the peak-height distribution for deformed quantum dots. We also investigate the energy-level correlation functions which are found to agree well with the behavior calculated for quasi-zero-dimensional disordered systems.Comment: 25 pages (revtex 3.0). 16 figures are available by mail or fax upon request at [email protected]

Edge Diffraction, Trace Formulae and the Cardioid Billiard

We study the effect of edge diffraction on the semiclassical analysis of two dimensional quantum systems by deriving a trace formula which incorporates paths hitting any number of vertices embedded in an arbitrary potential. This formula is used to study the cardioid billiard, which has a single vertex. The formula works well for most of the short orbits we analyzed but fails for a few diffractive orbits due to a breakdown in the formalism for certain geometries. We extend the symbolic dynamics to account for diffractive orbits and use it to show that in the presence of parity symmetry the trace formula decomposes in an elegant manner such that for the cardioid billiard the diffractive orbits have no effect on the odd spectrum. Including diffractive orbits helps resolve peaks in the density of even states but does not appear to affect their positions. An analysis of the level statistics shows no significant difference between spectra with and without diffraction.Comment: 25 pages, 12 Postscript figures. Published versio

Linear stability in billiards with potential

A general formula for the linearized Poincar\'e map of a billiard with a potential is derived. The stability of periodic orbits is given by the trace of a product of matrices describing the piecewise free motion between reflections and the contributions from the reflections alone. For the case without potential this gives well known formulas. Four billiards with potentials for which the free motion is integrable are treated as examples: The linear gravitational potential, the constant magnetic field, the harmonic potential, and a billiard in a rotating frame of reference, imitating the restricted three body problem. The linear stability of periodic orbits with period one and two is analyzed with the help of stability diagrams, showing the essential parameter dependence of the residue of the periodic orbits for these examples.Comment: 22 pages, LaTex, 4 Figure

Relevance of Chaos in Numerical Solutions of Quantum Billiards

In this paper we have tested several general numerical methods in solving the quantum billiards, such as the boundary integral method (BIM) and the plane wave decomposition method (PWDM). We performed extensive numerical investigations of these two methods in a variety of quantum billiards: integrable systens (circles, rectangles, and segments of circular annulus), Kolmogorov-Armold-Moser (KAM) systems (Robnik billiards), and fully chaotic systems (ergodic, such as Bunimovich stadium, Sinai billiard and cardiod billiard). We have analyzed the scaling of the average absolute value of the systematic error $\Delta E$ of the eigenenergy in units of the mean level spacing with the density of discretization $b$ (which is number of numerical nodes on the boundary within one de Broglie wavelength) and its relationship with the geometry and the classical dynamics. In contradistinction to the BIM, we find that in the PWDM the classical chaos is definitely relevant for the numerical accuracy at a fixed density of discretization $b$. We present evidence that it is not only the ergodicity that matters, but also the Lyapunov exponents and Kolmogorov entropy. We believe that this phenomenon is one manifestation of quantum chaos.Comment: 20 Revtex pages, 10 Eps figure

Separating the regular and irregular energy levels and their statistics in Hamiltonian system with mixed classical dynamics

We look at the high-lying eigenstates (from the 10,001st to the 13,000th) in the Robnik billiard (defined as a quadratic conformal map of the unit disk) with the shape parameter $\lambda=0.15$. All the 3,000 eigenstates have been numerically calculated and examined in the configuration space and in the phase space which - in comparison with the classical phase space - enabled a clear cut classification of energy levels into regular and irregular. This is the first successful separation of energy levels based on purely dynamical rather than special geometrical symmetry properties. We calculate the fractional measure of regular levels as $\rho_1=0.365\pm 0.01$ which is in remarkable agreement with the classical estimate $\rho_1=0.360\pm 0.001$. This finding confirms the Percival's (1973) classification scheme, the assumption in Berry-Robnik (1984) theory and the rigorous result by Lazutkin (1981,1991). The regular levels obey the Poissonian statistics quite well whereas the irregular sequence exhibits the fractional power law level repulsion and globally Brody-like statistics with $\beta = 0.286\pm0.001$. This is due to the strong localization of irregular eigenstates in the classically chaotic regions. Therefore in the entire spectrum we see that the Berry-Robnik regime is not yet fully established so that the level spacing distribution is correctly captured by the Berry-Robnik-Brody distribution (Prosen and Robnik 1994).Comment: 20 pages, file in plain LaTeX, 7 figures upon request submitted to J. Phys. A. Math. Gen. in December 199