Fractal Weyl law for three-dimensional chaotic hard-sphere scattering systems

The fractal Weyl law connects the asymptotic level number with the fractal dimension of the chaotic repeller. We provide the first test for the fractal Weyl law for a three-dimensional open scattering system. For the four-sphere billiard, we investigate the chaotic repeller and discuss the semiclassical quantization of the system by the method of cycle expansion with symmetry decomposition. We test the fractal Weyl law for various symmetry subspaces and sphere-to-sphere separations.Comment: 12 pages, 8 figures, submitted to Phys. Rev.

Viscosity in the escape-rate formalism

We apply the escape-rate formalism to compute the shear viscosity in terms of the chaotic properties of the underlying microscopic dynamics. A first passage problem is set up for the escape of the Helfand moment associated with viscosity out of an interval delimited by absorbing boundaries. At the microscopic level of description, the absorbing boundaries generate a fractal repeller. The fractal dimensions of this repeller are directly related to the shear viscosity and the Lyapunov exponent, which allows us to compute its values. We apply this method to the Bunimovich-Spohn minimal model of viscosity which is composed of two hard disks in elastic collision on a torus. These values are in excellent agreement with the values obtained by other methods such as the Green-Kubo and Einstein-Helfand formulas.Comment: 16 pages, 16 figures (accepted in Phys. Rev. E; October 2003

Fractal Weyl law for chaotic microcavities: Fresnel's laws imply multifractal scattering

We demonstrate that the harmonic inversion technique is a powerful tool to analyze the spectral properties of optical microcavities. As an interesting example we study the statistical properties of complex frequencies of the fully chaotic microstadium. We show that the conjectured fractal Weyl law for open chaotic systems [W. T. Lu, S. Sridhar, and M. Zworski, Phys. Rev. Lett. 91, 154101 (2003)] is valid for dielectric microcavities only if the concept of the chaotic repeller is extended to a multifractal by incorporating Fresnel's laws.Comment: 8 pages, 12 figure

Chaotic Properties of Dilute Two and Three Dimensional Random Lorentz Gases II: Open Systems

We calculate the spectrum of Lyapunov exponents for a point particle moving in a random array of fixed hard disk or hard sphere scatterers, i.e. the disordered Lorentz gas, in a generic nonequilibrium situation. In a large system which is finite in at least some directions, and with absorbing boundary conditions, the moving particle escapes the system with probability one. However, there is a set of zero Lebesgue measure of initial phase points for the moving particle, such that escape never occurs. Typically, this set of points forms a fractal repeller, and the Lyapunov spectrum is calculated here for trajectories on this repeller. For this calculation, we need the solution of the recently introduced extended Boltzmann equation for the nonequilibrium distribution of the radius of curvature matrix and the solution of the standard Boltzmann equation. The escape-rate formalism then gives an explicit result for the Kolmogorov Sinai entropy on the repeller.Comment: submitted to Phys Rev

Short periodic orbits theory for partially open quantum maps

We extend the semiclassical theory of short periodic orbits [Phys. Rev. E {\bf 80}, 035202(R) (2009)] to partially open quantum maps. They correspond to classical maps where the trajectories are partially bounced back due to a finite reflectivity $R$. These maps are representative of a class that has many experimental applications. The open scar functions are conveniently redefined, providing a suitable tool for the investigation of these kind of systems. Our theory is applied to the paradigmatic partially open tribaker map. We find that the set of periodic orbits that belong to the classical repeller of the open map ($R=0$) are able to support the set of long-lived resonances of the partially open quantum map in a perturbative regime. By including the most relevant trajectories outside of this set, the validity of the approximation is extended to a broad range of $R$ values. Finally, we identify the details of the transition from qualitatively open to qualitatively closed behaviour, providing an explanation in terms of short periodic orbits.Comment: 6 pages, 4 figure

Fractal asymptotics

Recent advances in the periodic orbit theory of stochastically perturbed systems have permitted a calculation of the escape rate of a noisy chaotic map to order 64 in the noise strength. Comparison with the usual asymptotic expansions obtained from integrals and with a previous calculation of the electrostatic potential of exactly selfsimilar fractal charge distributions, suggests a remarkably accurate form for the late terms in the expansion, with parameters determined independently from the fractal repeller and the critical point of the map. Two methods give a precise meaning to the asymptotic expansion, Borel summation and Shafer approximants. These can then be compared with the escape rate as computed by alternative methods.Comment: 15 pages, 5 postscript figures incorporated into the text; v2: Quadratic Pade (Shafer) method added, also a few reference