Generalized quantum measurement

We overcome one of Bell's objections to ‘quantum measurement’ by generalizing the definition to include systems outside the laboratory. According to this definition, a generalized quantum measurement takes place when the value of a classical variable is influenced significantly by an earlier state of a quantum system. A generalized quantum measurement can then take place in equilibrium systems, provided the classical motion is chaotic. This Letter deals with this classical aspect of quantum measurement, assuming that the Heisenberg cut between the quantum dynamics and the classical dynamics is made at a very small scale. For simplicity, a gas with collisions is modelled by an “Arnold gas”.     

Deterministic chaos in entangled eigenstates

We investigate the problem of deterministic chaos in connection with entangled states using the Bohmian formulation of quantum mechanics. We show for a two particle system in a harmonic oscillator potential, that in a case of entanglement and three energy eigen-values the maximum Lyapunov-parameters of a representative ensemble of trajectories for large times develops to a narrow positive distribution, which indicates nearly complete chaotic dynamics. We also present in short results from two time-dependent systems, the anisotropic and the Rabi oscillator.     

Quantization of a free particle interacting linearly with a harmonic oscillator

We investigate the quantization of a free particle coupled linearly to a harmonic oscillator. This system, whose classical counterpart has clearly separated regular and chaotic regions, provides an ideal framework for studying the quantization of mixed systems. We identify key signatures of the classically chaotic and regular portions in the quantum system by constructing Husimi distributions and investigating avoided level crossings of eigenvalues as functions of the strength and range of the interaction between the system's two components. We show, in particular, that the Husimi structure becomes mixed and delocalized as the classical dynamics becomes more chaotic.     

Wigner's pseudo-particle relativistic dynamics in external potential field

The Wigner's pseudo-particle formalism has been generalized to describe quantum dynamics of relativistic particle in external potential field. As a simplest application of the developed formalism the time evolution of the 1D relativistic quantum harmonic oscillator been considered. Due to the complex structure of the evolution equation for Wigner function, the only numerical treatment is possible by combining Monte Carlo and molecular dynamics methods. Relativistic dynamics results in appearance of the new physical effects as opposed to non-relativistic case. Interesting is the complete changing of the shape of the momentum and coordinate distribution functions as well as formation of ‘unexpected’ protuberances. To analyze the influence of relativistic effects on average values of quantum operators, the dependencies on time of average momentum, position, their dispersions and energy have been compared for the non-relativistic and relativistic dynamics.     

Variants of the Nosé–Hoover oscillator

The Nosé–Hoover oscillator is a well-studied chaotic system originally proposed to model a harmonic oscillator in equilibrium with a heat bath at constant temperature. Although it is a simple three-dimensional system with five terms and two quadratic nonlinearities, it displays a rich variety of unusual dynamics, but it falls considerably short of its original purpose. This review describes two simple variants of the Nosé–Hoover oscillator, the first of which satisfies the original goal exactly, and the second of which exhibits a hidden global chaotic attractor that fills all of its three-dimensional state space.

     

New scheme of anticipating synchronization for arbitrary anticipation time and its application to long-term prediction of chaotic states

How to predict the dynamics of nonlinear chaotic systems is still a challenging subject with important real-life applications. The present paper deals with this important yet difficult problem via a new scheme of anticipating synchronization. A global, robust, analytical and delay-independent sufficient condition is obtained to guarantee the existence of anticipating synchronization manifold theoretically in the framework of the Krasovskii-Lyapunov theory. Different from ＇traditional techniques （or regimes）＇ proposed in the previous literature, the present scheme guarantees that the receiver system can synchronize with the future state of a transmitter system for an arbitrarily long anticipation time, which allows one to predict the dynamics of chaotic transmitter at any point of time if necessary. Also it is simple to implement in practice. A classical chaotic system is employed to demonstrate the application of the proposed scheme to the long-term prediction of chaotic states.[第一段]     

On the parameters identification of the Duffing's system by means of a reduced order observer

An on-line procedure for recovering the unknown parameters set of the Duffing's oscillator by means of a reduced order proportional integral observer is presented in this Letter. First, it is shown that the oscillator has the properties of being algebraically observable and algebraically identifiable with respect to a well-chosen output (which turns out to be the oscillator's position). Therefore, an extended differential parametrization of the output and its time derivatives can be obtained. This extended differential parametrization has the necessary information to estimate the output time derivatives and to recover the unknown parameters. The numerical implementation of this method is easily accomplished in a digital computer.     

Energy transfer to an anharmonic diatomic system

We model an anharmonic diatomic molecule using deformed creation and annihilation operators such that the energy spectrum generated by a Hamiltonian of the harmonic oscillator's form written in terms of deformed operators is similar to that of a Morse potential. We construct an approximate time evolution operator and evaluate transition probabilities which are compared with those obtained by an expansion in a basis of Morse eigenfunctions. The algebraic results compare favorably with the numerical results.     

Algebraic methods applied to the study of energy transfer in anharmonic systems

We make use of two different methodologies to study the transition probabilities in a molecular anharmonic system in the presence of an external perturbation. For the first method, we use a series expansion of the displacement coordinate keeping up to fourth order terms; for the second method we use a deformed algebra to approximate the anharmonic Hamiltonian via a harmonic oscillator's Hamiltonian written in terms of deformed operators. We evaluate vibrational transition probabilities as a function of the collision energy and compare the results obtained with the two approaches.     

Energy features of an adiabatically loaded anharmonic oscillator

An excited anharmonic oscillator is considered under conditions of adiabatic (i.e., slow, as compared to the oscillation period) loading with an external force tending to a constant value at long times. The energy characteristics of the adiabatically loaded anharmonic oscillator, such as the instantaneous energy of the oscillator, the maximum kinetic (oscillation) energy, and the kinetic and potential energies averaged over the period, are analytically calculated as a function of the steady-state force. The analytical results are confirmed by the data of numerical calculations. It is established that the external force gives rise to a redistribution of the average kinetic and potential components of the initial energy of the anharmonic oscillator and that the transferred energy portions at a small external force considerably exceed the average work done by the external force.     

A patch that imparts unconditional stability to explicit integrators for Langevin-like equations

This paper extends the results in [8] to stochastic differential equations (SDEs) arising in molecular dynamics. It implements a patch to explicit integrators that consists of a Metropolis–Hastings step. The ‘patched integrator’ preserves the SDE’s equilibrium distribution and is accurate on finite time intervals. As a corollary this paper proves the integrator’s accuracy in estimating finite-time dynamics along an infinitely long solution - a first in molecular dynamics. The paper also covers multiple time-steps, holonomic constraints and scalability. Finally, the paper provides numerical tests supporting the theory.     

一个一维周期驱动哈密顿系统的实例及混沌控制

提出一种新的周期驱动非线性不可积哈密顿系统模型,并对其特性进行了讨论.通过简单的非反馈控制装置对这一系统进行混沌控制,将其混沌轨道分别控制在周期,准周期及指定混沌轨道上.与以往的控制方法不同的是,控制项仅是一结构简单、可调节的限位装置.为保守系统混沌控制的实际应用提供可供选择的途径     

Quasi-stationary chaotic states in multi-dimensional Hamiltonian systems

We study numerically statistical distributions of sums of chaotic orbit coordinates, viewed as independent random variables, in weakly chaotic regimes of three multi-dimensional Hamiltonian systems: Two Fermi-Pasta-Ulam (FPU-β) oscillator chains with different boundary conditions and numbers of particles and a microplasma of identical ions confined in a Penning trap and repelled by mutual Coulomb interactions. For the FPU systems we show that, when chaos is limited within “small size” phase space regions, statistical distributions of sums of chaotic variables are well approximated for surprisingly long times (typically up to t≈10⁶) by a q-Gaussian (1<q<3) distribution and tend to a Gaussian (q=1) for longer times, as the orbits eventually enter into “large size” chaotic domains. However, in agreement with other studies, we find in certain cases that the q-Gaussian is not the only possible distribution that can fit the data, as our sums may be better approximated by a different so-called “crossover” function attributed to finite-size effects. In the case of the microplasma Hamiltonian, we make use of these q-Gaussian distributions to identify two energy regimes of “weak chaos”—one where the system melts and one where it transforms from liquid to a gas state-by observing where the q-index of the distribution increases significantly above the q=1 value of strong chaos.     

Harmonic oscillator Wigner function extension to exceptional polynomials

Continuous energy supply is critical and important to support oscillating behaviour; otherwise, the oscillator will die. For nonlinear and chaotic circuits, enough energy supply is also important to keep electric devices working. In this paper, Hamilton energy is calculated for dimensionless dynamical system (e.g., the chaotic Lorenz system) using Helmholtz’s theorem. The Hamilton energy is considered as a new variable and then the dynamical system is controlled by using the scheme of energy feedback. It is found that chaos can be suppressed even when intermittent feedback scheme is applied. This scheme is effective to control chaos and to stabilise other dynamical systems.     

Exact folded-band chaotic oscillator

An exactly solvable chaotic oscillator with folded-band dynamics is shown. The oscillator is a hybrid dynamical system containing a linear ordinary differential equation and a nonlinear switching condition. Bounded oscillations are provably chaotic, and successive waveform maxima yield a one-dimensional piecewise-linear return map with segments of both positive and negative slopes. Continuous-time dynamics exhibit a folded-band topology similar to Ro?ssler's oscillator. An exact solution is written as a linear convolution of a fixed basis pulse and a discrete binary sequence, from which an equivalent symbolic dynamics is obtained. The folded-band topology is shown to be dependent on the symbol grammar.     

Fluctuations, response and aging dynamics in a simple glass-forming liquid out of equilibrium

By means of molecular dynamics computer simulations we investigate the out of equilibrium relaxation dynamics of a simple glass former, a binary Lennard-Jones system, after a quench to low temperatures. We find that one-time quantities, such as the energy or the structure factor, show only a weak time dependence. By comparing the out of equilibrium structure factor with equilibrium data we find evidence that during the aging process the system remains in that part of phase space that mode-coupling theory classifies as liquid like. Two-times correlation functions show a strong time and waiting time dependence. For large and times corresponding to the early -relaxation regime the correlators approach the Edwards-Anderson value by means of a power-law in time. For large but fixed values of the relaxation dynamics in the -relaxation regime seems to be independent of the observable and temperature. The -relaxation shows a power-law dependence on time with an exponent which is independent of but depends on the observable. We find that at long times the correlation functions can be expressed as and compute the function h(t). This function is found to show a t-dependence which is a bit stronger than a logarithm and to depend on the observable considered. If the system is quenched to very low temperatures the relaxation dynamics at long times shows fast drops as a function of time. We relate these drops to relatively local rearrangements in which part of the sample relaxes its stress by a collective motion of 50-100 particles. Finally we discuss our measurements of the time dependent response function. We find that at long times the correlation functions and the response are not related by the usual fluctuation dissipation theorem but that this relation is similar to the one found for spin glasses with one step replica symmetry breaking. Received 17 May 1999     

Polar nano-rods under shear: From equilibrium to chaos

The orientational dynamics of rod-like particles with permanent (electric or magnetic) dipole moments in a plane Couette shear flow is investigated using mesoscopic relaxation equations combined with a generalized Landau free energy. The free energy contribution due to the coupling between average alignment and dipole orientation is derived on a microscopic basis. Numerical results of the resulting eight-dimensional dynamical system are presented for the case of longitudinal dipoles and thermodynamic conditions where the equilibrium state is a (polar or non-polar) nematic. Solution diagrams reveal presence of a large variety of periodic, transient chaotic, and chaotic dynamic states of the average alignment and dipole moment, respectively, appearing as a function of Deborah number and tumbling parameter. Compared to rods without dipoles we observe a significant preference of out-of-plane kayaking-tumbling states and, generally, a higher sensitivity to the initial conditions including bistability. We also demonstrate that the average (electric) dipole moment characterizing most of the observed states yields electrodynamic (magnetic) fields of measurable strength.     

Chaos in the quantum double well oscillator: the Ehrenfest view revisited

We treat the double well quantum oscillator from the standpoint of the Ehrenfest equation but in a manner different from Pattanayak and Schieve. We show that for short times there can be chaotic motion due to quantum fluctuations, but over sufficiently long time there will be quantum noise induced oscillations between the two wells, thus providing an alternative approach to the quantum noise induced chaotic oscillations found recently by Bag and Ray.     

Constant chemical potential,pressure and temperature profiles in liquid–vapour equilibrium obtained by spinodal decomposition

Constant chemical potential, pressure and temperature profiles across a slab of liquid in equilibrium with its vapour confirm that, the spinodal decomposition procedure carried on the NVT ensemble simulated via molecular dynamics produce an equilibrium system. An initial homogeneous crystalline configuration of fluid is kept in a cell with a parallelepiped shape at a density near the critical density and a temperature between the triple and critical temperatures, form a slab of liquid in equilibrium with its vapour by the spinodal decomposition phenomenon if the simulation is performed in the NVT ensemble. An elongated box favours the formation of two planar parallel surfaces along the largest side of the box. We show in this paper that the ‘three conditions’ for thermodynamic equilibrium: constant temperature, constant pressure and constant chemical potential are met for such a system.