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arXiv1995

Semiclassical form factor of matrix element fluctuations

We analyze within a semiclassical approximation the form factor for the fluctuations of quantum matrix elements around their classical average. We find two contributions: one is proportional to the form factor for the density of states, with an amplitude determined by the squared average of the matrix elements. The other is constant and related to the fluctuations of finite time classical trajectory segments around the phase space average. The results are illustrated for an observable in the quadratic Zeeman effect.

chao-dyn↗

arXiv1995

Semiclassical Analysis of the Conductance of Mesoscopic Systems

The Kubo formula for the conductance of classically chaotic systems is analyzed semiclassically, yielding simple expressions for the mean and the variance of the quantum interference terms. In contrast to earlier work, here times longer than $O( \log \hbar^{-1} )$ give the dominant contributions, i.e. the limit $\hbar \rightarrow 0$ is not implied. For example, the result for the weak localization correction to the dimensionless conductance of a chain of $k$ classically ergodic scatterers connected in series is $-{1 \over 3} [ 1 - (k+1)^{-2} ]$, interpolating between the ergodic ($k = 1$) and the diffusive ($k \rightarrow \infty$) limits.

cond-mat↗

arXiv1995

Dynamical behavior of a dissipative particle in a periodic potential subject to chaotic noise: Retrieval of chaotic determinism with broken parity

Dynamical behaviors of a dissipative particle in a periodic potential subject to chaotic noise are reported. We discovered a macroscopic symmetry breaking effect of chaotic noise on a dissipative particle in a multi-stable systems emerging, even when the noise has a uniform invariant density with parity symmetry and white Fourier spectrum. The broken parity symmetry of the multi-stable potential is not necessary for the dynamics with broken symmetry. We explain the mechanism of the symmetry breaking and estimate the average velocity of a particle under chaotic noise in terms of unstable fixed points.

chao-dyn↗

arXiv1995

Universal Behavior of Lyapunov Exponents in Unstable Systems

We calculate the Lyapunov exponents in a classical molecular dynamics framework. The system is composed of few hundreds particles interacting either through Yukawa (Nuclear) or Slater-Kirkwood (Atomic) forces. The forces are chosen to give an Equation of State that resembles the nuclear and the atomic $^4He$ Equation Of State respectively near the critical point for liquid-gas phase transition. We find the largest fluctuations for an initial "critical temperature". The largest Lyapunov exponents $λ$ are always positive and can be very well fitted near this "critical temperature" with a functional form $λ\propto |T-T_c|^{-ω}$, where the exponent $ω=0.15$ is independent of the system and mass number. At smaller temperatures we find that $λ\propto T~ ^{0.4498}$, a universal behavior characteristic of an order to chaos transition.

chao-dyn↗

arXiv1995

Global Diffusion in a Realistic Three-Dimensional Time-Dependent Nonturbulent Fluid Flow

We introduce and study the first model of an experimentally realizable three-dimensional time-dependent nonturbulent fluid flow to display the phenomenon of global diffusion of passive-scalar particles at arbitrarily small values of the nonintegrable perturbation. This type of chaotic advection, termed {\it resonance-induced diffusion\/}, is generic for a large class of flows.

chao-dyn↗

arXiv1995

ANOMALOUS SCALING OF THE PASSIVE SCALAR

We establish anomalous inertial range scaling of structure functions for a model of advection of a passive scalar by a random velocity field. The velocity statistics is taken gaussian with decorrelation in time and velocity differences scaling as $|x|^{κ/2}$ in space, with $0\leqκ< 2$. The scalar is driven by a gaussian forcing acting on spatial scale $L$ and decorrelated in time. The structure functions for the scalar are well defined as the diffusivity is taken to zero and acquire anomalous scaling behavior for large pumping scales $L$. The anomalous exponent is calculated explicitly for the $4^{\m\rm th}$ structure function and for small $κ$ and it differs from previous predictions. For all but the second structure functions the anomalous exponents are nonvanishing.

chao-dyn↗

arXiv1995

Spectral Statistics: From Disordered to Chaotic Systems

The relation between disordered and chaotic systems is investigated. It is obtained by identifying the diffusion operator of the disordered systems with the Perron-Frobenius operator in the general case. This association enables us to extend results obtained in the diffusive regime to general chaotic systems. In particular, the two--point level density correlator and the structure factor for general chaotic systems are calculated and characterized. The behavior of the structure factor around the Heisenberg time is quantitatively described in terms of short periodic orbits.

cond-mat↗

arXiv1995

Diffractive orbits in quantum billiards

We study diffractive effects in two dimensional polygonal billiards. We derive an analytical trace formula accounting for the role of non-classical diffractive orbits in the quantum spectrum. As an illustration the method is applied to a triangular billiard.

chao-dyn↗

arXiv1995

Experimental studies of Chaos and Localization in Quantum Wavefunctions

Wavefunctions in chaotic and disordered quantum billiards are studied experimentally using thin microwave cavities. The chaotic wavefunctions display universal density distributions and density auto-correlations in agreement with expressions derived fro m a 0-D nonlinear $σ$-model of supersymmetry, which coincides with Random Matrix Theory. In contrast, disordered wavefunctions show deviations from this universal behavior due to Anderson localization. A systematic behavior of the distribution function is studied as a function of the localization length, and can be understood in the framework of a 1-D version of the nonlinear $σ$-model.

chao-dyn↗

arXiv1995

Bubble Shape Oscillations and the Onset of Sonoluminescence

An air bubble trapped in water by an oscillating acoustic field undergoes either radial or nonspherical pulsations depending on the strength of the forcing pressure. Two different instability mechanisms (the Rayleigh--Taylor instability and parametric instability) cause deviations from sphericity. Distinguishing these mechanisms allows explanation of many features of recent experiments on sonoluminescence, and suggests methods for finding sonoluminescence in different parameter regimes.

chao-dyn↗

arXiv1996

Chaos around a Hénon-Heiles-inspired exact perturbation of a black hole

A solution of the Einstein's equations that represents the superposition of a Schwarszchild black hole with both quadrupolar and octopolar terms describing a halo is exhibited. We show that this solution, in the Newtonian limit, is an analog to the well known Hénon-Heiles potential. The integrability of orbits of test particles moving around a black hole representing the galactic center is studied and bounded zones of chaotic behavior are found.

gr-qc↗

arXiv1998

Overcoming the wall in the semiclassical baker's map

A major barrier in semiclassical calculations is the sheer number of terms that contribute as time increases; for classically chaotic dynamics, the proliferation is exponential. We have been able to overcome this ``exponential wall'' for the baker's map, using an analogy with spin-chain partition functions. The semiclassical sum is contracted so that only of order N T^(3/2) operations are needed for an N-state system evolved for T time steps. This is typically less than the computational load for quantum propagation, T N^2. This method enables us to obtain semiclassical results up to the Heisenberg time, which in our example would have required 10^90 terms if we were to evaluate the sum exactly. This calculation in turn provides new insight as to the accuracy of the semiclassical approximation at long times. The semiclassical result is often correct; its breakdown is nonuniform.

chao-dyn↗

arXiv1996

Wigner Random Banded Matrices with Sparse Structure: Local Spectral Density of States

Random banded matrices with linearly increasing diagonal elements are recently considered as an attractive model for complex nuclei and atoms. Apart from early papers by Wigner \cite{Wig} there were no analytical studies on the subject. In this letter we present analytical and numerical results for local spectral density of states (LDOS) for more general case of matrices with a sparsity inside the band. The crossover from the semicircle form of LDOS to that given by the Breit-Wigner formula is studied in detail.

chao-dyn↗

arXiv1995

The Role of Diffraction in the Quantization of Dispersing Billiards

We study diffraction corrections to the semiclassical spectral density of dispersing (Sinai) billiards. They modify the contributions of periodic orbits (PO's), with at least one segment which is almost tangent to the concave part of the boundary. Given a wavenumber $k$, all the PO's with length up to the Heisenberg length $O(k)$ are required for quantization. We show that most of the contributions of PO's which are longer than a limit $O(k^{2/3})$ must be corrected for diffraction effects. For orbits which just miss tangency, the corrections are of the same magnitude as the semiclassical contributions themselves. Orbits which bounce at extreme forward angles give very small terms in the standard semiclassical theory. The diffraction corrections increase their amplitude substantially.

chao-dyn↗

arXiv1995

Geometric Phase in Quantum Billiards with a Pointlike Scatterer

We examine the quantum energy levels of rectangular billiards with a pointlike scatterer in one and two dimensions. By varying the location and the strength of the scatterer, we systematically find diabolical degeneracies among various levels. The associated Berry phase is illustrated, and the existence of localized wave functions is pointed out. In one dimension, even the ground state is shown to display the sign reversal with a mechanism to circumvent the Sturm-Liouville theorem.

hep-th↗

arXiv1996

Extended Self-Similarity in Turbulent Systems: an Analytically Soluble Example

In turbulent flows the $n$'th order structure functions $S_n(R)$ scale like $R^{ζ_n}$ when $R$ is in the "inertial range". Extended Self-Similarity refers to the substantial increase in the range of power law behaviour of $S_n(R)$ when they are plotted as a function of $S_2(R)$ or $S_3(R)$. In this Letter we demonstrate this phenomenon analytically in the context of the ``multiscaling" turbulent advection of a passive scalar. This model gives rise to a series of differential equations for the structure functions $S_n(R)$ which can be solved and shown to exhibit extended self similarity. The phenomenon is understood by comparing the equations for $S_n(R)$ to those for $S_n(S_2)$.

chao-dyn↗

arXiv1995

Semiclassical Quantisation Using Diffractive Orbits

Diffraction, in the context of semiclassical mechanics, describes the manner in which quantum mechanics smooths over discontinuities in the classical mechanics. An important example is a billiard with sharp corners; its semiclassical quantisation requires the inclusion of diffractive periodic orbits in addition to classical periodic orbits. In this paper we construct the corresponding zeta function and apply it to a scattering problem which has only diffractive periodic orbits. We find that the resonances are accurately given by the zeros of the diffractive zeta function.

chao-dyn↗

arXiv1996

Anomalous Scaling Exponents of a White-Advected Passive Scalar

For Kraichnan's problem of passive scalar advection by a velocity field delta-correlated in time, the limit of large space dimensionality $d\gg1$ is considered. Scaling exponents of the scalar field are analytically found to be $ζ_{2n}=nζ_2-2(2-ζ_2)n(n-1)/d$, while those of the dissipation field are $μ_{n}=-2(2-ζ_2)n(n-1)/d$ for orders $n\ll d$. The refined similarity hypothesis $ζ_{2n}=nζ_2+μ_{n}$ is thus established by a straightforward calculation for the case considered.

chao-dyn↗