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At least 775 records · Page 43Linked to original sources

arXiv1998

Analytic estimation of Lyapunov exponent in a mean-field model undergoing a phase transition

The parametric instability contribution to the largest Lyapunov exponent (LLE) is derived for a mean-field Hamiltonian model, with attractive long-range interactions. This uses a recent Riemannian approach to describe Hamiltonian chaos with a large number N of degrees of freedom. Through microcanonical estimates of suitable geometrical observables, the mean-field behavior of the LLE is analytically computed and related to the second order phase transition undergone by the system. It predicts that chaoticity drops to zero at the critical temperature and remains vanishing above it, with the LLE scaling as N^-1/3 to the leading order in N.

chao-dyn↗

arXiv1998

Universality for the breakup of invariant tori in Hamiltonian flows

In this article, we describe a new renormalization-group scheme for analyzing the breakup of invariant tori for Hamiltonian systems with two degrees of freedom. The transformation, which acts on Hamiltonians that are quadratic in the action variables, combines a rescaling of phase space and a partial elimination of irrelevant (non-resonant) frequencies. It is implemented numerically for the case applying to golden invariant tori. We find a nontrivial fixed point and compute the corresponding scaling and critical indices. If one compares flows to maps in the canonical way, our results are consistent with existing data on the breakup of golden invariant circles for area-preserving maps.

chao-dyn↗

arXiv1997

Random Matrix Elements and Eigenfunctions in Chaotic Systems

The expected root-mean-square value of a matrix element $A_{αβ}$ in a classically chaotic system, where $A$ is a smooth, $\hbar$-independent function of the coordinates and momenta, and $α$ and $β$ label different energy eigenstates, has been evaluated in the literature in two different ways: by treating the energy eigenfunctions as gaussian random variables and averaging $|A_{αβ}|^2$ over them; and by relating $|A_{αβ}|^2$ to the classical time-correlation function of $A$. We show that these two methods give the same answer only if Berry's formula for the spatial correlations in the energy eigenfunctions (which is based on a microcanonical density in phase space) is modified at large separations in a manner which we previously proposed.

chao-dyn↗

arXiv1998

Uniform semiclassical approximations for umbilic bifurcation catastrophes

Gutzwiller's trace formula for the semiclassical density of states diverges at the bifurcation points of periodic orbits and has to be replaced with uniform semiclassical approximations. We present a method to derive these expressions from the standard representations of the elementary catastrophes and to directly relate the uniform solutions to classical periodic orbit parameters, thereby circumventing the numerical application of normal form theory. The technique allows an easy handling of ungeneric bifurcations with corank 2 such as the umbilic catastrophes and is demonstrated on a hyperbolic umbilic in the diamagnetic Kepler problem.

chao-dyn↗

arXiv1998

Chaos in effective classical and quantum dynamics

We investigate the dynamics of classical and quantum N-component phi^4 oscillators in the presence of an external field. In the large N limit the effective dynamics is described by two-degree-of-freedom classical Hamiltonian systems. In the classical model we observe chaotic orbits for any value of the external field, while in the quantum case chaos is strongly suppressed. A simple explanation of this behaviour is found in the change in the structure of the orbits induced by quantum corrections. Consistently with Heisenberg's principle, quantum fluctuations are forced away from zero, removing in the effective quantum dynamics a hyperbolic fixed point that is a major source of chaos in the classical model.

hep-th↗

arXiv1998

Targetting Chaos through Adaptive Control

We describe adaptive control algorithms whereby a chaotic dynamical system can be steered to a target state with desired characteristics. A specific implementation considered has the objective of directing the system to a state which is more chaotic or mixed than the uncontrolled one. This methodology is easy to implement in discrete or continuous dynamical systems. It is robust and efficient, and has the additional advantage that knowledge of the detailed behaviour of the system is not required.

chao-dyn↗

arXiv1998

Extended Self-similarity in Kinetic Surface Roughening

We show from numerical simulations that a limited mobility solid-on-solid model of kinetically rough surface growth exhibits extended self-similarity analogous to that found in fluid turbulence. The range over which scale-independent power-law behavior is observed is significantly enhanced if two correlation functions of different order, such as those representing two different moments of the difference in height between two points, are plotted against each other. This behavior, found in both one and two dimensions, suggests that the `relative' exponents may be more fundamental than the `absolute' ones.

cond-mat.stat-mech↗

arXiv1998

Anomalous diffusion as a signature of collapsing phase in two dimensional self-gravitating systems

A two dimensional self-gravitating Hamiltonian model made by $N$ fully-coupled classical particles exhibits a transition from a collapsing phase (CP) at low energy to a homogeneous phase (HP) at high energy. From a dynamical point of view, the two phases are characterized by two distinct single-particle motions : namely, superdiffusive in the CP and ballistic in the HP. Anomalous diffusion is observed up to a time $τ$ that increases linearly with $N$. Therefore, the finite particle number acts like a white noise source for the system, inhibiting anomalous transport at longer times.

cond-mat.stat-mech↗

arXiv1998

Intensity distribution for waves in disordered media: deviations from Rayleigh statistics

We study the intensity distribution function, P(I), for monochromatic waves propagating in quasi one-dimensional disordered medium, assuming that a point source and a point detector are embedded in the bulk of the medium. We find deviations from the Rayleigh statistics at moderately large I and a logarithmically-normal asymptotic behavior of P(I). When the radiation source and the detector are located close to the opposite edges of the sample (on a distance much less then the sample length), an intermediate regime with a stretched-exponential behavior of P(I) emerges.

cond-mat↗

arXiv1998

Driven Maps and the Emergence of Ordered Collective Behavior in Globally Coupled Maps

A method to predict the emergence of different kinds of ordered collective behaviors in systems of globally coupled chaotic maps is proposed. The method is based on the analogy between globally coupled maps and a map subjected to an external drive. A vector field which results from this analogy appears to govern the transient evolution of the globally coupled system. General forms of global couplings are considered. Some simple applications are given.

chao-dyn↗

arXiv1998

Mode Fluctuation Distribution for Spectra of Superconducting Microwave Billiards

High resolution eigenvalue spectra of several two- and three-dimensional superconducting microwave cavities have been measured in the frequency range below 20 GHz and analyzed using a statistical measure which is given by the distribution of the normalized mode fluctuations. For chaotic systems the limit distribution is conjectured to show a universal Gaussian, whereas integrable systems should exhibit a non-Gaussian limit distribution. For the investigated Bunimovich stadium and the 3D-Sinai billiard we find that the distribution is in good agreement with this prediction. We study members of the family of limacon billiards, having mixed dynamics. It turns out that in this case the number of approximately 1000 eigenvalues for each billiard does not allow to observe significant deviations from a Gaussian, whereas an also measured circular billiard with regular dynamics shows the expected difference from a Gaussian.

chao-dyn↗

arXiv1998

Squeezed state dynamics of kicked quantum systems

We study kicked quantum systems by using the squeezed state approach. Taking the kicked quantum harmonic oscillator as an example, we demonstrate that chaos in an underlying classical system can be enhanced as well as suppressed by quantum fluctuations. Three different energy diffusions are observed in the kicked quantum harmonic oscillator, namely, localization, linear diffusion, and quadratic diffusion.

chao-dyn↗

arXiv1998

Free particle scattering off two oscillating disks

We investigate the two-dimensional classical dynamics of the scattering of point particles by two periodically oscillating disks. The dynamics exhibits regular and chaotic scattering properties, as a function of the initial conditions and parameter values of the system. The energy is not conserved since the particles can gain and loose energy from the collisions with the disks. We find that for incident particles whose velocity is on the order of the oscillating disk velocity, the energy of the exiting particles displays non-monotonic gaps of allowed energies, and the distribution of exiting particle velocities shows significant fluctuations in the low energy regime. We also considered the case when the initial velocity distribution is Gaussian, and found that for high energies the exit velocity distribution is Gaussian with the same mean and variance. When the initial particle velocities are in the irregular regime the exit velocity distribution is Gaussian but with a smaller mean and variance. The latter result can be understood as an example of stochastic cooling. In the intermediate regime the exit velocity distribution differs significantly from Gaussian. A comparison of the results presented in this paper to previous chaotic static scattering problems is also discussed.

cond-mat.mes-hall↗

arXiv1998

An optimal shell model of turbulence

We introduce a new shell model of turbulence which exhibits improved properties in comparison to the standard (and very popular) GOY model. The nonlinear coupling is chosen to minimize correlations between different shells. In particular the second order correlation function is diagonal in the shell index, the third order correlation exists only between three consecutive shells. Spurious oscillations in the scaling regime, which are an annoying feature of the GOY model, are eliminated by our choice of nonlinear coupling. We demonstrate that the model exhibits multi-scaling similarly to these GOY model. The scaling exponents are shown to be independent of the viscous mechanism as is expected for Navier-Stokes turbulence and other shell models. These properties of the new model make it optimal for further attempts to achieve understanding of multi-scaling in nonlinear dynamics.

chao-dyn↗

arXiv1997

Dynamics of a Simple Quantum System in a Complex Environment

We present a theory for the dynamical evolution of a quantum system coupled to a complex many-body intrinsic system/environment. By modelling the intrinsic many-body system with parametric random matrices, we study the types of effective stochastic models which emerge from random matrix theory. Using the Feynman-Vernon path integral formalism, we derive the influence functional and obtain either analytical or numerical solutions for the time evolution of the entire quantum system. We discuss thoroughly the structure of the solutions for some representative cases and make connections to well known limiting results, particularly to Brownian motion, Kramers classical limit and the Caldeira-Leggett approach.

nucl-th↗

arXiv1998

Wave-function entropy and dynamical-symmetry breaking in the interacting boson model

The degree of chaos in the Interacting Boson Model (IBM-1) is compared with what we call the "dynamical-symmetry content" of the system. The latter is represented by the information entropy of the eigenfunctions with respect to bases associated with dynamical symmetries of the IBM-1, and expresses thus the localization of actual eigenfunctions in these symmetry bases. The wave-function entropy is shown to be a sensitive tool for monitoring the processes of a single dynamical-symmetry breaking or transitions between two and more symmetries. For the IBM-1 hamiltonians studied here, the known features related to chaos, namely the dependence of chaotic measures on the hamiltonian parameters (position in the Casten triangle) and on the angular momentum, turn out to be correlated with the behaviour of the wave-function entropy. PACs 21.60.Fw; 05.45.+b

chao-dyn↗

arXiv1998

Fingerprints of Classical Instability in Open Quantum Dynamics

The dynamics near a hyperbolic point in phase space is modelled by an inverted harmonic oscillator. We investigate the effect of the classical instability on the open quantum dynamics of the oscillator, introduced through the interaction with a thermal bath, using both the survival probability function and the rate of von Neumann entropy increase, for large times. In this parameter range we prove, using influence functional techniques, that the survival probability function decreases exponentially at a rate, K', depending not only on the measure of instability in the model but also on the strength of interaction with the environment. We also show that K' determines the rate of von Neumann entropy increase and that this result is independent of the temperature of the environment. This generalises earlier results which are valid in the limit of vanishing dissipation. The validity of inferring similar rates of survival probability decrease and entropy increase for quantum chaotic systems is also discussed.

chao-dyn↗

arXiv1998

Spatial correlations of singularity strengths in multifractal branching processes

The n-point statistics of singularity strength variables for multiplicative branching processes is calculated from an analytic expression of the corresponding multivariate generating function. The key ingredient is a branching generating function which can be understood as a natural generalisation of the multifractal mass exponents. Various random multiplicative cascade processes pertaining to fully developed turbulence are discussed.

chao-dyn↗