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arXiv1997

Invariant Correlation Entropy and Complexity of Quantum States

We define correlational (von Neumann) entropy for an individual quantum state of a system whose time-independent hamiltonian contains random parameters and is treated as a member of a statistical ensemble. This entropy is representation independent and can be calculated as a trace functional of the density matrix which describes the system in its interaction with the noise source. We analyze perturbation theory in order to show the evolution from the pure state to the mixed one. Exactly solvable examples illustrate the use of correlational entropy as a measure of the degree of complexity in comparison with other available suggestions such as basis-dependent information entropy. It is shown in particular that a harmonic oscillator in a uniform field of random strength comes to a quasithermal equilibrium; we discuss the relation between effective temperature and canonical equilibrium temperature. The notion of correlational entropy is applied to a realistic numerical caculation in the framework of the nuclear shell model. In this system, which reveals generic signatures of quantum chaos, correlational entropy and information entropy calculated in the mean field basis display similar qualitative behavior.

chao-dyn↗

arXiv1998

Dynamics of Dry Friction: A Numerical Investigation

We perform extended numerical simulation of the dynamics of dry friction, based on a model derived from the phenomenological description proposed by T. Baumberger et al.. In the case of small deviation from the steady sliding motion, the model is shown to be equivalent to the state- and rate-dependent friction law which was first introduced by Rice and Ruina on the basis of experiments on rocks. We obtain the dynamical phase diagram that agrees well with the experimental results on the paper-on-paper systems. In particular, the bifurcation between stick-slip and steady sliding are shown to change from a direct (supercritical) Hopf type to an inverted (subcritical) one as the driving velocity increases, in agreement with the experiments.

cond-mat↗

arXiv1998

Tunneling rate fluctuations induced by non-linear resonances: a quantitative treatment based on semiclassical arguments

We investigate the tunneling process between two symmetric stable islands of a forced pendulum Hamiltonian in the weak chaos regime. We show that when the tunneling doublet is quantized over a classical non-linear resonance the tunneling rate strongly deviates from the semiclassical prediction. This mechanism is responsible for the irregular dependence of the tunneling rate on the system parameters. The weak-chaos condition allows us to make a theoretical prediction which agrees very well with the numerical results. This opens up a possible avenue to a general theory on the dependence of quantum tunneling on classical chaos.

chao-dyn↗

arXiv1998

Integrability and level crossing manifolds in a quantum Hamiltonian system

We consider a two-spin model, represented classically by a nonlinear autonomous Hamiltonian system with two degrees of freedom and a nontrivial integrability condition, and quantum mechanically by a real symmetric Hamiltonian matrix with blocks of dimensionalities K=l(l+1)/2, l=1,2,... In the six-dimensional (6D) parameter space of this model, classical integrability is satisfied on a 5D hypersurface, and level crossings occur on 4D manifolds that are completely embedded in the integrability hypersurface except for some lower-D sub-manifolds. Under mild assumptions, the classical integrability condition can be reconstructed from a purely quantum mechanical study of level degeneracies in finite-dimensional invariant blocks of the Hamiltonian matrix. Our conclusions are based on rigorous results for K=3 and on numerical results for K=6,10.

chao-dyn↗

arXiv1998

Instanton for the Kraichnan Passive Scalar Problem

We consider high-order correlation functions of the passive scalar in the Kraichnan model. Using the instanton formalism we find the scaling exponents $ζ_n$ of the structure functions $S_n$ for $n\gg1$ under the additional condition $dζ_2\gg1$ (where $d$ is the dimensionality of space). At $n n_c$ they are $n$-independent: $ζ_n=ζ_2 n_c/4$. We also estimate $n$-dependent factors in $S_n$, particularly their behavior at $n$ close to $n_c$.

chao-dyn↗

arXiv1998

The Dynamics of Granular Segregation Patterns in a Long Drum Mixer

We have studied the early time evolution of granular segregation patterns in a horizontal rotating cylinder partially filled with a sand/salt mixture. The growth of concentration fluctuations starting from premixed initial conditions was analyzed using Fourier techniques. In one mixture, we observed generally merging dynamics in the segregated bands up to the onset of saturation. At the threshold of saturation, we found a spectrum of wavelengths with a broad peak. The peak position was nearly independent of the tube rotation rate. The overall growth rate of Fourier modes had a maximum at a particular value of the angular rotation frequency. In a slightly different mixture, we observed transient traveling waves when the larger grains were in the majority. We measured the wave speed as a function of several parameters using presegregated initial conditions to launch waves of various wavelengths.

patt-sol↗

arXiv1998

Linear Stochastic Models of Nonlinear Dynamical Systems

We investigate in this work the validity of linear stochastic models for nonlinear dynamical systems. We exploit as our basic tool a previously proposed Rayleigh-Ritz approximation for the effective action of nonlinear dynamical systems started from random initial conditions. The present paper discusses only the case where the PDF-Ansatz employed in the variational calculation is ``Markovian'', i.e. is determined completely by the present values of the moment-averages. In this case we show that the Rayleigh-Ritz effective action of the complete set of moment-functions that are employed in the closure has a quadratic part which is always formally an Onsager-Machlup action. Thus, subject to satisfaction of the requisite realizability conditions on the noise covariance, a linear Langevin model will exist which reproduces exactly the joint 2-time correlations of the moment-functions. We compare our method with the closely related formalism of principal oscillation patterns (POP), which, in the approach of C. Penland, is a method to derive such a linear Langevin model empirically from time-series data for the moment-functions. The predictive capability of the POP analysis, compared with the Rayleigh-Ritz result, is limited to the regime of small fluctuations around the most probable future pattern. Finally, we shall discuss a thermodynamics of statistical moments which should hold for all dynamical systems with stable invariant probability measures and which follows within the Rayleigh-Ritz formalism.

chao-dyn↗

arXiv1998

Relaxation of classical many-body hamiltonians in one dimension

The relaxation of Fourier modes of hamiltonian chains close to equilibrium is studied in the framework of a simple mode-coupling theory. Explicit estimates of the dependence of relevant time scales on the energy density (or temperature) and on the wavenumber of the initial excitation are given. They are in agreement with previous numerical findings on the approach to equilibrium and turn out to be also useful in the qualitative interpretation of them. The theory is compared with molecular dynamics results in the case of the quartic Fermi-Pasta-Ulam potential.

cond-mat.stat-mech↗

arXiv1998

Coherent and incoherent chaotic tunneling near singlet-doublet crossings

In the spectrum of systems showing chaos-assisted tunneling, three-state crossings are formed when a chaotic singlet intersects a tunnel doublet. We study the dissipative quantum dynamics in the vicinity of such crossings. A harmonically driven double well coupled to a bath serves as a model. Markov and rotating-wave approximations are introduced with respect to the Floquet spectrum of the time-dependent central system. The resulting master equation is integrated numerically. We find various types of transient tunneling, determined by the relation of the level width to the inherent energy scales of the crossing. The decay of coherent tunneling can be significantly retarded or accelerated. Modifications of the quantum asymptotic state by the crossing are also studied. The comparison with a simple three-state model shows that in contrast to the undamped case, the participation of states outside the crossing cannot be neglected in the presence of dissipation.

quant-ph↗

arXiv1998

Critical Behavior of Period Doublings in Coupled Inverted Pendulums

We study the critical behaviors of period doublings in N (N=2,3,4,...) coupled inverted pendulums by varying the driving amplitude $A$ and the coupling strength $c$. It is found that the critical behaviors depend on the range of coupling interaction. In the extreme long-range case of global coupling, in which each inverted pendulum is coupled to all the other ones with equal strength, the zero-coupling critical point and an infinity of critical line segments constitute the same critical set in the $A-c$ plane, independently of $N$. However, for any other nonglobal-coupling cases of shorter-range couplings, the structure of the critical set becomes different from that for the global-coupling case, because of a significant change in the stability diagram of periodic orbits born via period doublings. The critical scaling behaviors on the critical set are also found to be the same as those for the abstract system of the coupled one-dimensional maps.

chao-dyn↗

arXiv1998

Noise rectification in quasigeostrophic forced turbulence

We study the appearance of large scale mean motion sustained by stochastic forcing on a rotating fluid (in the quasigeostrophic approximation) flowing over topography. As in other noise rectification phenomena, the effect requires nonlinearity and absence of detailed balance to occur. By application of an analytical coarse graining procedure we identify the physical mechanism producing such effect: It is a forcing coming from the small scales that manifests in a change in the effective viscosity operator and in the effective noise statistical properties.

chao-dyn↗

arXiv1998

Chaotic Interaction of Langmuir Solitons and Long Wavelength Radiation

In this work we analyze the interaction of isolated solitary structures and ion-acoustic radiation. If the radiation amplitude is small solitary structures persists, but when the amplitude grows energy transfer towards small spatial scales occurs. We show that transfer is particularly fast when a fixed point of a low dimensional model is destroyed.

chao-dyn↗

arXiv1998

Separatrix Reconnections in Chaotic Regimes

In this paper we extend the concept of separatrix reconnection into chaotic regimes. We show that even under chaotic conditions one can still understand abrupt jumps of diffusive-like processes in the relevant phase-space in terms of relatively smooth realignments of stable and unstable manifolds of unstable fixed points.

chao-dyn↗

arXiv1999

Synchronisation of time--delay systems

We present the linear-stability analysis of synchronised states in coupled time-delay systems. There exists a synchronisation threshold, for which we derive upper bounds, which does not depend on the delay time. We prove that at least for scalar time-delay systems synchronisation is achieved by transmitting a single scalar signal, even if the synchronised solution is given by a high-dimensional chaotic state with a large number of positive Lyapunov-exponents. The analytical results are compared with numerical simulations of two coupled Mackey-Glass equations.

chao-dyn↗

arXiv1998

Extensive Scaling and Nonuniformity of the Karhunen-Loève Decomposition for the Spiral-Defect Chaos State

By analyzing large-aspect-ratio spiral-defect-chaos (SDC) convection images, we show that the Karhunen-Loève decomposition (KLD) scales extensively for subsystem-sizes larger than 4d (d is the fluid depth), which strongly suggests that SDC is extensively chaotic. From this extensive scaling, the intensive length ξ_KLD is computed and found to have a different dependence on the Rayleigh number than the two-point correlation length ξ_2. Local computations of ξ_KLD reveal a substantial spatial nonuniformity of SDC that extends over radii 18d< r < 45d in a Γ=109 aspect-ratio cell.

chao-dyn↗

arXiv1998

Equilibrium and dynamical properties of two dimensional self-gravitating systems

A system of N classical particles in a 2D periodic cell interacting via long-range attractive potential is studied. For low energy density $U$ a collapsed phase is identified, while in the high energy limit the particles are homogeneously distributed. A phase transition from the collapsed to the homogeneous state occurs at critical energy U_c. A theoretical analysis within the canonical ensemble identifies such a transition as first order. But microcanonical simulations reveal a negative specific heat regime near $U_c$. The dynamical behaviour of the system is affected by this transition : below U_c anomalous diffusion is observed, while for U > U_c the motion of the particles is almost ballistic. In the collapsed phase, finite $N$-effects act like a noise source of variance O(1/N), that restores normal diffusion on a time scale diverging with N. As a consequence, the asymptotic diffusion coefficient will also diverge algebraically with N and superdiffusion will be observable at any time in the limit N \to \infty. A Lyapunov analysis reveals that for U > U_c the maximal exponent λdecreases proportionally to N^{-1/3} and vanishes in the mean-field limit. For sufficiently small energy, in spite of a clear non ergodicity of the system, a common scaling law λ\propto U^{1/2} is observed for any initial conditions.

cond-mat.stat-mech↗

arXiv1998

Use of synchronization and adaptive control in parameter estimation from a time series

A technique is introduced for estimating unknown parameters when time series of only one variable from a multivariate nonlinear dynamical system is given. The technique employs a combination of two different control methods, a linear feedback for synchronizing system variables and an adaptive control. The technique is shown to work even when the unknown parameter appears in the evolution equations of the variables other than the one for which the time series is given. The technique not only esablishes that explicit detailed information about all system variables and parameters is contained in a scalar time series, but also gives a way to exract it out under suitable conditions. Illustrations are presented and effect of noise is studied.

chao-dyn↗

arXiv1999

Turbulence without pressure in d dimensions

The randomly driven Navier-Stokes equation without pressure in d-dimensional space is considered as a model of strong turbulence in a compressible fluid. We derive a closed equation for the velocity-gradient probability density function. We find the asymptotics of this function for the case of the gradient velocity field (Burgers turbulence), and provide a numerical solution for the two-dimensional case. Application of these results to the velocity-difference probability density function is discussed.

hep-th↗