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We present a canonically invariant form for the generalized Langevin and Fokker-Planck equations. We discuss the role of constants of motion, and the construction of conservative stochastic processes.
The master equation for a damped spin well known from the theory of superradiance, is written as a finite-difference equation and solved by a WKB-like method. The propagator thus obtained looks like the van Vleck propagator of a certain classical Hamiltonian system with one degree of freedom. A new interpretation is provided of the temporal broadening of initially sharp probability distributions as the analogue of the spreading of the quantum mechanical wave packet.
The microwave ionization of internally chaotic Rydberg atoms is studied analytically and numerically. The internal chaos is induced by magnetic or static electric fields. This leads to a chaotic enhancement of microwave excitation. The dynamical localization theory gives a detailed description of the excitation process even in a regime where up to few thousands photons are required to ionize one atom. Possible laboratory experiments are also discussed.
I investigate the propagator of the Wigner function for a dissipative chaotic quantum map. I show that a small amount of dissipation reduces the propagator of sufficiently smooth Wigner functions to its classical counterpart, the Frobenius-Perron operator, if $\hbar\to 0$. Several consequences arise: The Wigner transform of the invariant density matrix is a smeared out version of the classical strange attractor; time dependent expectation values and correlation functions of observables can be evaluated via hybrid quantum-classical formulae in which the quantum character enters only via the initial Wigner function. If a classical phase-space distribution is chosen for the latter or if the map is iterated sufficiently many times the formulae become entirely classical, and powerful classical trace formulae apply.
This paper demonstrates that rapid fracture of ideal brittle lattices naturally involves phenomena long seen in experiment, but which have been hard to understand from a continuum point of view. These idealized models do not mimic realistic microstructure, but can be solved exactly and understood completely. First it is shown that constant velocity crack solutions do not exist at all for a range of velocities starting at zero and ranging up to about one quarter of the shear wave speed. Next it is shown that above this speed cracks are by and large linearly stable, but that at sufficiently high velocity they become unstable with respect to a nonlinear micro-cracking instability. The way this instability works itself out is related to the scenario known as intermittency, and the basic time scale which governs it is the inverse of the amount of dissipation in the model. Finally, we compare the theoretical framework with some new experiments in Plexiglas, and show that all qualitative features of the theory are mirrored in our experimental results.
The squeezing in a nonlinear system with chaotic dynamics is considered. The model describing interaction of collection of two-level atoms with a single-mode of self-consistent field and an external field is analyzed. It is shown that in the semiclassical limit, in contrast to the regular behaviour, the chaotic dynamics result in: (i) an increase in squeezing, (ii) unstable squeezing and contraction of time intervals of squeezing on large enough times. The possibility of the experimental observation of the described effects is discussed.
The long-time behaviour of many dynamical systems may be effectively predicted by a low-dimensional model that describes the evolution of a reduced set of variables. We consider the question of how to equip such a low-dimensional model with appropriate initial conditions, so that it faithfully reproduces the long-term behaviour of the original high-dimensional dynamical system. Our method involves putting the dynamical system into normal form, which not only generates the low-dimensional model, but also provides the correct initial conditions for the model. We illustrate the method with several examples. Keywords: normal form, isochrons, initialisation, centre manifold
A coupled map lattice for convection is proposed, which consists of Eulerian and Lagrangian procedures. Simulations of the model not only reproduce a wide-range of phenomena in Rayleigh-Bénard convection experiments but also lead to several prediction of novel phenomena there: For small aspect ratios, the formation of convective rolls, their oscillation, many routes to chaos, and chaotic itinerancy are found, with the increase of the Rayleigh number. For large aspect ratios, the collective oscillation of convective rolls, travelling waves, coherent chaos, and spatiotemporal intermittency are observed. At high Rayleigh numbers, the transition from soft to hard turbulence is confirmed, as is characterized by the change of the temperature distribution from Gaussian to exponential. Roll formation in three-dimensional convection is also simulated, and found to reproduce experiments well.
We study the classical and quantum mechanics of a free particle that collides elastically with the walls of a circular disk with the radius varying periodically in time. The quasi-energy spectral properties of the model are obtained from evaluation of finite-dimensional approximations to the time evolution operator. As the scaled hbar is changed from large to small, the statistics of the Quasienergy Eigenvalues (QEE) change from Poisson to circular orthogonal ensemble (COE). Different statistical tests are used to characterize this transition. The transition of the Quasienergy Eigenfunctions (QEF) is also studied using the chi-squared test with nu degrees of freedom, which goes over to the Porter-Thomas distribution for nu=1. We find that the integrable regime is associated with exponentially localized QEF whereas the eigenfunctions are extended in the chaotic, COE regime. We change the representation of the model to one in which the boundary is fixed and the Hamiltonian acquires a quadratic forcing term. We then carry out a successful comparison between specific classical phase space solutions and their corresponding QEFs in the Husimi representation.
We consider the dynamical system consisting of a quantum degree of freedom $A$ interacting with $N$ quantum oscillators described by the Lagrangian \bq L = {1\over 2}\dot{A}^2 + \sum_{i=1}^{N} \left\{{1\over 2}\dot{x}_i^2 - {1\over 2}( m^2 + e^2 A^2)x_i^2 \right\}. \eq In the limit $N \rightarrow \infty$, with $e^2 N$ fixed, the quantum fluctuations in $A$ are of order $1/N$. In this limit, the $x$ oscillators behave as harmonic oscillators with a time dependent mass determined by the solution of a semiclassical equation for the expectation value $\VEV{A(t)}$. This system can be described, when $\VEV{x(t)}= 0$, by a classical Hamiltonian for the variables $G(t) = \VEV{x^2(t)}$, $\dot{G}(t)$, $A_c(t) = \VEV{A(t)}$, and $\dot{A_c}(t)$. The dynamics of this latter system turns out to be chaotic. We propose to study the nature of this large-$N$ limit by considering both the exact quantum system as well as by studying an expansion in powers of $1/N$ for the equations of motion using the closed time path formalism of quantum dynamics.
We address the problems in applying cycle expansions to bound chaotic systems, caused by e.g. intermittency and incompleteness of the symbolic dynamics. We discuss zeta functions associated with weighted evolution operators and in particular a one-parameter family of weights relevant for the calculation of classical resonance spectra, semiclassical spectra and topological entropy. For bound intermittent system we discuss an approximation of the zeta function in terms of probabilities rather than cycle instabilities. This approximation provides a generalization of the fundamental part of a cycle expansion for a finite subshift symbolic dynamics. This approach is particularly suitable for determining asymptotic properties of periodic orbits which are essential for scrutinizing the semiclassical limit of Gutzwiller's semiclassical trace formula. The Sinai billiard is used as model system. In particular we develope a crude approximation of the semiclassical zeta function which turns out to possess non analytical features. We also discuss the contribution to the semiclassical level density from the neutral orbits. Finally we discuss implications of our findings for the spectral form factor and compute the asymptotic behaviour of the spectral rigidity. The result is found to be consistent with exact quantum mechanical calculations.
Cycle expansions are an efficient scheme for computing the properties of chaotic systems. When enumerating the orbits for a cycle expansion not all orbits that one would expect at first are present --- some are pruned. This pruning leads to convergence difficulties when computing properties of chaotic systems. In numerical schemes, I show that pruning reduces the number of reliable eigenvalues when diagonalizing quantum mechanical operators, and that pruning slows down the convergence rate of cycle expansion calculations. I then exactly solve a diffusion model that displays chaos and show that its cycle expansion develops a branch point.
In relation to spatiotemporal intermittency, as it can be observed in coupled map lattices, we study the stability of different wavelengths in competition. Introducing a two dimensional map, we compare its dynamics with the one of the whole lattice. We conclude a good agreement between the two. The reduced model also allows to introduce an order parameter which combines the diffusion parameter and the spatial wavelength under consideration.
Orbital resonances are ubiquitous in the Solar system. They play a decisive role in the long term dynamics, and in some cases the physical evolution, of the planets and of their natural satellites, as well as the evolution of small bodies (including dust) in the planetary system. The few-body gravitational problem of hierarchical planetary-type systems allows for a complex range of dynamical timescales, from the fast orbital periods to the very slow orbit precession rates. The interaction of fast and slow degrees of freedom produces a rich diversity of resonance phenomena. Weak dissipative effects --- such as tides or radiation drag forces --- also produce unexpectedly rich dynamical behaviors. This paper provides a mostly qualitative discussion of simple dynamical models for the commonly encountered orbital resonance phenomena in the Solar system.
Global aspects of the motion of passive scalars in time-dependent incompressible fluid flows are well described by volume-preserving (Liouvillian) three-dimensional maps. In this paper the possible invariant structures in Liouvillian maps and the two most interesting nearly-integrable cases are investigated. In addition, the fundamental role of invariant lines in organizing the dynamics of this type of system is exposed. Bifurcations involving the destruction of some invariant lines and tubes and the creation of new ones are described in detail.
In this paper, we review the literature on statistical long-range correlation in DNA sequences. We examine the current evidence for these correlations, and conclude that a mixture of many length scales (including some relatively long ones) in DNA sequences is responsible for the observed 1/f-like spectral component. We note the complexity of the correlation structure in DNA sequences. The observed complexity often makes it hard, or impossible, to decompose the sequence into a few statistically stationary regions. We suggest that, based on the complexity of DNA sequences, a fruitful approach to understand long-range correlation is to model duplication, and other rearrangement processes, in DNA sequences. One model, called ``expansion-modification system", contains only point duplication and point mutation. Though simplistic, this model is able to generate sequences with 1/f spectra. We emphasize the importance of DNA duplication in its contribution to the observed long-range correlation in DNA sequences.
The pattern dynamics of the one-way coupled logistic lattice which can serve as a phenomenological model for open flow is investigated and shown to be extremely rich. For medium and large coupling strengths, we find spatially periodic, quasiperiodic and chaotic patterns with temporal periodicity and analyze their stability with the help of a newly introduced spatial map. Criteria are established for predicting the down-flow bifurcation behavior of the coupled map lattice, and for selecting attractors through modulation of the boundary. For smaller coupling strengths we find a novel regime in which chaotic defects form a periodic lattice that is shown to be the result of a boundary crisis.
Spatiotemporal chaos (STC) exhibited by the Kuramoto-Sivashinsky (KS) equation is investigated analytically and numerically. An effective stochastic equation belonging to the KPZ universality class is constructed by incorporating the chaotic dynamics of the small KS system in a coarse-graining procedure. The bare parameters of the effective theory are computed approximately. Stability of the system is shown to be mediated by space-time defects that are accompanied by stochasticity. The method of analysis and the mechanism of stability may be relevant to a class of STC problems.