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arXiv1993

Hypersensitivity to Perturbations in the Quantum Baker's Map

We analyze a randomly perturbed quantum version of the baker's transformation, a prototype of an area-conserving chaotic map. By numerically simulating the perturbed evolution, we estimate the information needed to follow a perturbed Hilbert-space vector in time. We find that the Landauer erasure cost associated with this information grows very rapidly and becomes much larger than the maximum statistical entropy given by the logarithm of the dimension of Hilbert space. The quantum baker's map thus displays a hypersensitivity to perturbations that is analogous to behavior found earlier in the classical case. This hypersensitivity characterizes ``quantum chaos'' in a way that is directly relevant to statistical physics.

chao-dyn↗

arXiv1994

Tunneling and the Band Structure of Chaotic Systems

We compute the dispersion laws of chaotic periodic systems using the semiclassical periodic orbit theory to approximate the trace of the powers of the evolution operator. Aside from the usual real trajectories, we also include complex orbits. These turn out to be fundamental for a proper description of the band structure since they incorporate conduction processes through tunneling mechanisms. The results obtained, illustrated with the kicked-Harper model, are in excellent agreement with numerical simulations, even in the extreme quantum regime.

cond-mat↗

arXiv1994

Mesoscopic Transport Through Ballistic Cavities: A Random S-Matrix Theory Approach

We deduce the effects of quantum interference on the conductance of chaotic cavities by using a statistical ansatz for the S matrix. Assuming that the circular ensembles describe the S matrix of a chaotic cavity, we find that the conductance fluctuation and weak-localization magnitudes are universal: they are independent of the size and shape of the cavity if the number of incoming modes, N, is large. The limit of small N is more relevant experimentally; here we calculate the full distribution of the conductance and find striking differences as N changes or a magnetic field is applied.

cond-mat↗

arXiv1994

Periodic Orbit Theory of Diffraction

An extension of the Gutzwiller trace formula is given that includes diffraction effects due to hard wall scatterers or other singularities. The new trace formula involves periodic orbits which have arcs on the surface of singularity and which correspond to creping waves. A new family of resonances in the two disk scattering system can be well described which is completely missing if only the traditional periodic orbits are used.

chao-dyn↗

arXiv1994

Crossover from high to low Reynolds number turbulence

The Taylor-Reynolds and Reynolds number ($Re_λ$ and $Re$) dependence of the dimensionless energy dissipation rate $\ceps =\eps L / \u1rms^3$ is derived for statistically stationary isotropic turbulence, employing the results of a variable range mean field theory. Here, $\eps$ is the energy dissipation rate, $L$ is the (fixed) outer length scale, and $\u1rms$ a rms velocity component. Our fit-parameter free results for $\ceps (Re_λ)$ and also for $Re_λ(Re)$ are in good agreement with experimental data. Using the $Re$-dependence of $\ceps$ we account for the time dependence of the mean vorticity $ω(t)$ for decaying isotropic turbulence, which was recently experimentally examined [M.\ Smith, R.\ J.\ Donelly, N.\ Goldenfeld, and W.\ F.\ Vinen, Phys.\ Rev.\ Lett.\ 71, 2583 (1993)]. The lifetime of decaying turbulence, depending on the initial $Re_{λ,0}$, is predicted and found to saturate at $0.18 L^2/ν\propto Re_{λ,0}^2$ ($ν$ is the viscosity) for large $Re_{λ,0}$.

chao-dyn↗

arXiv1994

Finite size corrections to scaling in high Reynolds number turbulence

We study analytically and numerically the corrections to scaling in turbulence which arise due to the finite ratio of the outer scale $L$ of turbulence to the viscous scale $η$, i.e., they are due to finite size effects as anisotropic forcing or boundary conditions at large scales. We find that the deviations $\dzm$ from the classical Kolmogorov scaling $ζ_m = m/3$ of the velocity moments $\langle |\u(\k)|^m\rangle \propto k^{-ζ_m}$ decrease like $δζ_m (Re) =c_m Re^{-3/10}$. Our numerics employ a reduced wave vector set approximation for which the small scale structures are not fully resolved. Within this approximation we do not find $Re$ independent anomalous scaling within the inertial subrange. If anomalous scaling in the inertial subrange can be verified in the large $Re$ limit, this supports the suggestion that small scale structures should be responsible, originating from viscosity either in the bulk (vortex tubes or sheets) or from the boundary layers (plumes or swirls).

chao-dyn↗

arXiv1995

Geometric phases and anholonomy for a class of chaotic classical systems

Berry's phase may be viewed as arising from the parallel transport of a quantal state around a loop in parameter space. In this Letter, the classical limit of this transport is obtained for a particular class of chaotic systems. It is shown that this ``classical parallel transport'' is anholonomic --- transport around a closed curve in parameter space does not bring a point in phase space back to itself --- and is intimately related to the Robbins-Berry classical two-form.

chao-dyn↗

arXiv1994

Iterated Moire Maps and Braiding of Chiral Polymer Crystals

In the hexagonal columnar phase of chiral polymers a bias towards cholesteric twist competes with braiding along an average direction. When the chirality is strong, screw dislocations proliferate, leading to either a tilt grain boundary phase or a new "moire state" with twisted bond order. Polymer trajectories in the plane perpendicular to their average direction are described by iterated moire maps of remarkable complexity.

cond-mat↗

arXiv1995

Turbulent spectrum of the Earth's ozone field

The Total Ozone Mapping Spectrometer (TOMS) database is subjected to an analysis in terms of the Karhunen-Loeve (KL) empirical eigenfunctions. The concentration variance spectrum is transformed into a wavenumber spectrum, $% E_c(k)$. In terms of wavenumber $E_c(k)$ is shown to be $O(k^{-2/3})$ in the inverse cascade regime, $O(k^{-2})$ in the enstrophy cascade regime with the spectral {\it knee} at the wavenumber of barotropic instability.The spectrum is related to known geophysical phenomena and shown to be consistent with physical dimensional reasoning for the problem. The appropriate Reynolds number for the phenomena is $Re\approx 10^{10}$.

chao-dyn↗

arXiv1994

"Intermittency" in Hydrodynamic Turbulence as Intermediate Asymptotic to Kolmogorov41 Scaling

A synopsis of an analytical theory of scaling in developed turbulence is proposed on the basis of the Navier-Stokes equations. It is shown that corrections to the normal Kolmogorov 1941 scaling behavior of the $n$-th order velocity structure functions are finite Re effects which disappear when the inertial interval exceeds 5-6 decades. These corrections originate from the correlation between the velocity differences and energy dissipation which are characterized by an anomalous (subcritical) exponent. The values of the experimentally observed scaling indices for the $n$-th order structure functions for $n$ between 4 and 14 are in agreement with our findings.

chao-dyn↗

arXiv1995

Thermalization of a Brownian particle via coupling to low-dimensional chaos

It is shown that a paradigm of classical statistical mechanics --- the thermalization of a Brownian particle --- has a low-dimensional, deterministic analogue: when a heavy, slow system is coupled to fast deterministic chaos, the resultant forces drive the slow degrees of freedom toward a state of statistical equilibrium with the fast degrees. This illustrates how concepts useful in statistical mechanics may apply in situations where low-dimensional chaos exists.

chao-dyn↗

arXiv1995

Mean Field Theory for Lyapunov Exponents and KS Entropy in Lorentz Lattice Gases

Cellular automata lattice gases are useful systems for systematically exploring the connections between non-equilibrium statisitcal mechanics and dynamical systems theory. Here the chaotic properties of a Lorentz lattice gas are studied analytically and by computer simulations. The escape-rates, Lyapunov exponents, and KS entropies are estimated for a one-dimensional example using a mean-field theory, and the results are compared with simulations for a range of densities and scattering parameters of the lattice gas. The computer results show a distribution of values for the dynamical quantities with average values that are in good agreement with the mean field theory, and consistent with the escape-rate formalism for the coefficient of diffusion.

chao-dyn↗

arXiv1995

Universal Parametric Correlations of Eigenfunctions in Chaotic and Disordered Systems

This paper establishes the universality of parametric correlations of eigenfunctions in chaotic and weakly disordered systems. We demonstrate this universality in the framework of the gaussian random matrix process and obtain predictions for a number of parametric correlators, one of them analytically. We present numerical evidence from different models that verifies our predictions.

cond-mat↗

arXiv1995

Quantization of the three dimensional Sinai billiard

For the first time a three--dimensional (3D) chaotic billiard -- the 3D Sinai billiard -- was quantized, and high--precision spectra with thousands of eigenvalues were calculated. We present here a semiclassical and statistical analysis of the spectra, and point out some of the features which are genuine consequences of the three dimensionality of this chaotic billiard.

chao-dyn↗

arXiv1995

Statistical properties of energy levels of chaotic systems: Wigner or non-Wigner

For systems whose classical dynamics is chaotic, it is generally believed that the local statistical properties of the quantum energy levels are well described by Random Matrix Theory. We present here two counterexamples - the hydrogen atom in a magnetic field and the quartic oscillator - which display nearest neighbor statistics strongly different from the usual Wigner distribution. We interpret the results with a simple model using a set of regular states coupled to a set of chaotic states modeled by a random matrix.

chao-dyn↗

arXiv1993

Secondary Instabilities and Spatiotemporal Chaos in Parametric Surface Waves

A two dimensional model is introduced to study pattern formation, secondary instabilities and the transition to spatiotemporal chaos (weak turbulence) in parametric surface waves. The stability of a periodic standing wave state above onset is studied against Eckhaus, zig-zag and transverse amplitude modulations (TAM) as a function of the control parameter $\varepsilon$ and the detuning. A mechanism leading to a finite threshold for the TAM instability is identified. Numerical solutions of the model are in agreement with the stability diagram, and also reveal the existence of a transition to spatiotemporal chaotic states at a finite $\varepsilon$. Power spectra of temporal fluctuations in the chaotic state are broadband, decaying as a power law of the frequency $ω^{-z}$ with $z \approx 4.0$.

patt-sol↗

arXiv1995

Formation of Dynamic Domains in Strongly Driven Ferromagnets

Based on the dissipative Landau-Lifshitz equation, the spatiotemporal structure formation problem is investigated in the region far above the transverse ferromagnetic resonance instability. Apart from the external fields, the model contains an isotropic exchange field, a shape demagnetisation field and an anisotropy field. Numerical simulations exhibit in the rotating frame a stationary domain structure with a precessing motion in the walls regimes. Employing analytical methods, characteristic elements of this structure are explained. This driven dissipative system shows similarities to equilibrium systems of coexisting phases and organises itself in such a way that the local dynamics tends to become hamiltonian.

chao-dyn↗

arXiv1995

Chaos in Andreev Billiards

A new type of classical billiard - the Andreev billiard - is investigated using the tangent map technique. Andreev billiards consist of a normal region surrounded by a superconducting region. In contrast with previously studied billiards, Andreev billiards are integrable in zero magnetic field, {\it regardless of their shape}. A magnetic field renders chaotic motion in a generically shaped billiard, which is demonstrated for the Bunimovich stadium by examination of both Poincaré sections and Lyapunov exponents. The issue of the feasibility of certain experimental realizations is addressed.

cond-mat↗