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arXiv1999

Distributions of Transition Matrix Elements in Classically Mixed Quantum Systems

The quantitative contributions of a mixed phase-space to the mean characterizing the distribution of diagonal transition matrix elements and to the variance characterizing the distributions of non-diagonal transition matrix elements are studied. It is shown that the mean can be expressed as the sum of suitably weighted classical averages along an ergodic trajectory and along the stable periodic orbits. Similarly, it is shown that the values of the variance are well reproduced by the sum of the suitably weighted Fourier transforms of classical autocorrelation functions along an ergodic trajectory and along the stable periodic orbits. The illustrative numerical computations are done in the framework of the Hydrogen atom in a strong magnetic field, for three different values of the scaled energy.

chao-dyn↗

arXiv1999

Anderson Localization in a String of Microwave Cavities

The field distributions and eigenfrequencies of a microwave resonator which is composed of 20 identical cells have been measured. With external screws the periodicity of the cavity can be perturbed arbitrarily. If the perturbation is increased a transition from extended to localized field distributions is observed. For very large perturbations the field distributions show signatures of Anderson localization, while for smaller perturbations the field distribution is extended or weakly localized. The localization length of a strongly localized field distribution can be varied by adjusting the penetration depth of the screws. Shifts in the frequency spectrum of the resonator provide further evidence for Anderson localization.

chao-dyn↗

arXiv1999

Semiclassical Treatment of Diffraction in Billiard Systems with a Flux Line

In billiard systems with a flux line semiclassical approximations for the density of states contain contributions from periodic orbits as well as from diffractive orbits that are scattered on the flux line. We derive a semiclassical approximation for diffractive orbits that are scattered once on a flux line. This approximation is uniformly valid for all scattering angles. The diffractive contributions are necessary in order that semiclassical approximations are continuous if the position of the flux line is changed.

chao-dyn↗

arXiv1999

Classical Diffusion and Quantum Level Velocities: Systematic Deviations from Random Matrix Theory

We study the response of the quasi-energy levels in the context of quantized chaotic systems through the level velocity variance and relate them to classical diffusion coefficients using detailed semiclassical analysis. The systematic deviations from random matrix theory, assuming independence of eigenvectors from eigenvalues, is shown to be connected to classical higher order time correlations of the chaotic system. We study the standard map as a specific example, and thus the well known oscillatory behavior of the diffusion coefficient with respect to the parameter is reflected exactly in the oscillations of the variance of the level velocities. We study the case of mixed phase-space dynamics as well and note a transition in the scaling properties of the variance that occurs along with the classical transition to chaos.

chao-dyn↗

arXiv1999

Distribution of Husimi Zeroes in Polygonal Billiards

The zeroes of the Husimi function provide a minimal description of individual quantum eigenstates and their distribution is of considerable interest. We provide here a numerical study for pseudo- integrable billiards which suggests that the zeroes tend to diffuse over phase space in a manner reminiscent of chaotic systems but nevertheless contain a subtle signature of pseudo-integrability. We also find that the zeroes depend sensitively on the position and momentum uncertainties with the classical correspondence best when the position and momentum uncertainties are equal. Finally, short range correlations seem to be well described by the Ginibre ensemble of complex matrices.

chao-dyn↗

arXiv1999

Kinetic approach to the Gaussian thermostat in a dilute sheared gas in the thermodynamic limit

A dilute gas of particles with short range interactions is considered in a shearing stationary state. A Gaussian thermostat keeps the total kinetic energy constant. For infinitely many particles it is shown that the thermostat becomes a friction force with constant friction coefficient. For finite number of particles N, the fluctuations around this constant are of order one over the square root of N, and distributed approximately Gaussian with deviations for large fluctuations. These deviations prohibit a derivation of fluctuation-dissipation relations far from equilibrium, based on the Fluctuation Theorem.

cond-mat.stat-mech↗

arXiv1999

Universal long-time properties of Lagrangian statistics in the Batchelor regime and their application to the passive scalar problem

We consider transport of dynamically passive quantities in the Batchelor regime of smooth in space velocity field. For the case of arbitrary temporal correlations of the velocity we formulate the statistics of relevant characteristics of Lagrangian motion. This allows to generalize many results obtained previously for the delta-correlated in time strain, thus answering the question of universality of these results.

chao-dyn↗

arXiv1999

Localization and Fluctuations in Quantum Kicked Rotors

We address the issue of fluctuations, about an exponential lineshape, in a pair of one-dimensional kicked quantum systems exhibiting dynamical localization. An exact renormalization scheme establishes the fractal character of the fluctuations and provides a new method to compute the localization length in terms of the fluctuations. In the case of a linear rotor, the fluctuations are independent of the kicking parameter $k$ and exhibit self-similarity for certain values of the quasienergy. For given $k$, the asymptotic localization length is a good characteristic of the localized lineshapes for all quasienergies. This is in stark contrast to the quadratic rotor, where the fluctuations depend upon the strength of the kicking and exhibit local "resonances". These resonances result in strong deviations of the localization length from the asymptotic value. The consequences are particularly pronounced when considering the time evolution of a packet made up of several quasienergy states.

chao-dyn↗

arXiv1999

Transition to turbulence in a shear flow

We analyze the properties of a 19 dimensional Galerkin approximation to a parallel shear flow. The laminar flow with a sinusoidal shape is stable for all Reynolds numbers Re. For sufficiently large Re additional stationary flows occur; they are all unstable. The lifetimes of finite amplitude perturbations show a fractal dependence on amplitude and Reynolds number. These findings are in accord with observations on plane Couette flow and suggest a universality of this transition szenario in shear flows.

chao-dyn↗

arXiv1999

An approximate renormalization-group transformation for Hamiltonian systems with three degrees of freedom

We construct an approximate renormalization transformation that combines Kolmogorov-Arnold-Moser (KAM)and renormalization-group techniques, to analyze instabilities in Hamiltonian systems with three degrees of freedom. This scheme is implemented both for isoenergetically nondegenerate and for degenerate Hamiltonians. For the spiral mean frequency vector, we find numerically that the iterations of the transformation on nondegenerate Hamiltonians tend to degenerate ones on the critical surface. As a consequence, isoenergetically degenerate and nondegenerate Hamiltonians belong to the same universality class, and thus the corresponding critical invariant tori have the same type of scaling properties. We numerically investigate the structure of the attracting set on the critical surface and find that it is a strange nonchaotic attractor. We compute exponents that characterize its universality class.

chao-dyn↗

arXiv1999

Two-Dimensional turbulence in the inverse cascade range

Numerical and physical experiments on the forced two-dimensional Navier-Stokes equations show that transverse velocity differences are described by ``normal'' Kolmogorov scaling $<(Δv)^{2n}> \propto r^{2n/3}$ and obey a gaussian statistics. Since the non-trivial scaling is a sign of a strong non-linearity of the problem, these two results seem to contradict each other. A theory, explaining this result is presented in this paper. Strong time-dependence of the large-scale features of the flow ($\bar{u^2}\propto t$) results in decoupling of the large-scale dynamics from statistically steady-state small-scale random processes. This time-dependence is also a reason for the localness of the pressure-gradient terms in the equations governing the small-scale velocity difference PDF's. The derived self-consistent expression for the pressure gradient contributions lead to the conclusion that the small-scale transverse velocity differences are governed by a linear Langevin-like equation, strirred by a non-local, universal, solution-depending gaussian random force. This explains the experimentally observed gaussian statistics of transverse velocity differences and their Kolmogorov scaling. The solution for the PDF of longitudinal velocity differences is based on a smallness of the energy flux in two-dimensional turbulence. The theory makes a few quantitative predictions which can be tested experimentally.

chao-dyn↗

arXiv1999

On The Three Point Velocity Correlation Functions in 2d Forced Turbulence

We present a simple exact formula for the three point velocity correlation functions in two dimensional turbulence which is valid on all scales and which interpolates between the direct and inverse cascade regimes. As expected, these correlation functions are universal in these extreme regimes. We also discuss the effect of anisotropy and friction.

chao-dyn↗

arXiv1999

Frozen spatial chaos induced by boundaries

We show that rather simple but non-trivial boundary conditions could induce the appearance of spatial chaos (that is stationary, stable, but spatially disordered configurations) in extended dynamical systems with very simple dynamics. We exemplify the phenomenon with a nonlinear reaction-diffusion equation in a two-dimensional undulated domain. Concepts from the theory of dynamical systems, and a transverse-single-mode approximation are used to describe the spatially chaotic structures.

chao-dyn↗

arXiv1999

Escape from intermittent repellers- Periodic orbit theory for crossover from exponential to algebraic decay

We apply periodic orbit theory to study the asymptotic distribution of escape times from an intermittent map. The dynamical zeta function exhibits a branch point which is associated with an asymptotic power law escape. By an analytic continuation technique we compute a zero of the zeta function beyond its radius of convergence leading to a pre-asymptotic exponential decay. The time of crossover from an exponential to a power law is also predicted. The theoretical predictions are confirmed by numerical simulation. Applications to conductance fluctuations in quantum dots are discussed.

chao-dyn↗

arXiv1998

Continued Fraction Representation of Temporal Multi Scaling in Turbulence

It was shown recently that the anomalous scaling of simultaneous correlation functions in turbulence is intimately related to the breaking of temporal scale invariance, which is equivalent to the appearance of infinitely many times scales in the time dependence of time-correlation functions. In this paper we derive a continued fraction representation of turbulent time correlation functions which is exact and in which the multiplicity of time scales is explicit. We demonstrate that this form yields precisely the same scaling laws for time derivatives and time integrals as the "multi-fractal" representation that was used before. Truncating the continued fraction representation yields the "best" estimates of time correlation functions if the given information is limited to the scaling exponents of the simultaneous correlation functions up to a certain, finite order. It is worth noting that the derivation of a continued fraction representation obtained here for an operator which is not Hermitian or anti-Hermitian may be of independent interest.

chao-dyn↗

arXiv1999

Critical behavior of vorticity in two-dimensional turbulence

We point out some similitudes between the statistics of high Reynolds number turbulence and critical phenomena. An analogy is developed for two-dimensional decaying flows, in particular by studying the scaling properties of the two-point vorticity correlation function within a simple phenomenological framework. The inverse of the Reynolds number is the analogue of the small parameter that separates the system from criticality. It is possible to introduce a set of three critical exponents; for the correlation length, the autocorrelation function and a so-called susceptibility, respectively. The exponents corresponding to the well-known enstrophy cascade theory of Kraichnan and Batchelor are, remarkably, the same as the Gaussian approximation exponents for spin models. The limitations of the analogy, in particular the lack of universal scaling functions, are also discussed.

chao-dyn↗

arXiv1999

Quasilinear theory of collisionless Fermi acceleration in a multicusp magnetic confinement geometry

Particle motion in a cylindrical multiple-cusp magnetic field configuration is shown to be highly (though not completely) chaotic, as expected by analogy with the Sinai billiard. This provides a collisionless, linear mechanism for phase randomization during monochromatic wave heating. A general quasilinear theory of collisionless energy diffusion is developed for particles with a Hamiltonian of the form $H_0+H_1$, motion in the \emph{unperturbed} Hamiltonian $H_0$ being assumed chaotic, while the perturbation $H_1$ can be coherent (i.e. not stochastic). For the multicusp geometry, two heating mechanisms are identified --- cyclotron resonance heating of particles temporarily mirror-trapped in the cusps, and nonresonant heating of nonadiabatically reflected particles (the majority). An analytically solvable model leads to an expression for a transit-time correction factor, exponentially decreasing with increasing frequency. The theory is illustrated using the geometry of a typical laboratory experiment.

chao-dyn↗

arXiv1999

Characterization and control of small-world networks

Recently Watts and Strogatz have given an interesting model of small-world networks. Here we concretise the concept of a ``far away'' connection in a network by defining a {\it far edge}. Our definition is algorithmic and independent of underlying topology of the network. We show that it is possible to control spread of an epidemic by using the knowledge of far edges. We also suggest a model for better advertisement using the far edges. Our findings indicate that the number of far edges can be a good intrinsic parameter to characterize small-world phenomena.

chao-dyn↗