Semiclassical theory of molecular collisions: Real and complex-valued classical trajectories for collinear atom Morse oscillator collisions

Real and complex-valued classical trajectories have been calculated for the collinear collision of an atom with a Morse oscillator. They are used in three semiclassical approximations for the transition probability: a Bessel uniform approximation, an Airy uniform approximation and a primitive semiclassical approximation. Comparison with exact quantum results shows that the Bessel uniform approximation is accurate even for near elastic collisions where the Airy and primitive approximations break down. The Airy and Bessel approximations agree quite closely for inelastic collisions however. The primitive semiclassical approximation is less accurate than either the Airy or Bessel approximation.     

Fluctuations of expectation values in chaotic systems and broken time-reversal symmetry

Fluctuations of expectation values of observables are calculated in complex quantum systems, such as disordered metallic grains or quantum systems with classical chaotic motion. We derive an exact expression for these fluctuations valid for systems with and without time-reversal symmetry, as well as in the transition region between these two cases. We compare our results with those of a semiclassical theory and with simulations of random matrices.     

Comparison between phase space structures in coupled Morse systems and in various su(2) approximations

While Hamiltonians written in terms of position and momentum provide a transparent picture of the motion of a system, Hamiltonians written in terms of Lie algebras are easier to handle quantum mechanically. Therefore we are interested to know how to transform one into the other. Since the exact transformation often leads to complicated expressions, we look for approximations which preserve the essential features. As basic criterion we look for the degree of equality of the classical phase space structures. We illustrate our ideas for the case of two coupled Morse systems and its approximation in terms of the Lie algebra su(2), which is relevant to anharmonic models of molecular spectroscopy. (c) 2001 American Institute of Physics.     

Photo control of transport properties in a disordered wire: Average conductance,conductance statistics,and time-reversal symmetry

In this paper, we study the full conductance statistics of a disordered 1D wire under the application of light. We develop the transfer matrix method for periodically driven systems to analyze the conductance of a large system with small frequency of light, where coherent photon absorptions play an important role to determine not only the average but also the shape of conductance distributions. The average conductance under the application of light results from the competition between dynamic localization and effective dimension increase, and shows non-monotonic behavior as a function of driving amplitude. On the other hand, the shape of conductance distribution displays a crossover phenomena in the intermediate disorder strength; the application of light dramatically changes the distribution from log-normal to normal distributions. Furthermore, we propose that conductance of disordered systems can be controlled by engineering the shape, frequency and amplitude of light. Change of the shape of driving field controls the time-reversals symmetry and the disordered system shows analogous behavior as negative magneto-resistance known in static weak localization. A small change of frequency and amplitude of light leads to a large change of conductance, displaying giant opto-response. Our work advances the perspective to control the mean as well as the full conductance statistics by coherently driving disordered systems.     

Entropy,information theory,and the approach to equilibrium of coupled harmonic oscillator systems

Finite segments of infinite chains of classical coupled harmonic oscillators are treated as models of thermodynamic systems in contact with a heat bath, i.e., canonical ensembles. The Liouville function for the infinite chain is reduced by integrating over the outside variables to a function _N of the variables of theN-particle segment that is the thermodynamic system. The reduced Liouville function _N which is calculated from the dynamics of the infinite chain and the statistical knowledge of the coordinates and momenta att = 0, is a time-dependent probability density in the 2N-dimensional phase space of the system. A Gibbs entropy defined in terms of _N measures the evolution of knowledge of the system (more accurately, the growth of missing pertinent information) in the sense of information theory. As ¦t ¦ , energy is equipartitioned, the entropy evolves to the value expected from equilibrium statistical mechanics, and _N evolves to an equilibrium distribution function. The simple chain exhibits diffusion in coordinate space, i.e., Brownian motion, and the diffusivity is shown to depend only on the initial distribution of momenta (not of coordinates) in the heat bath. The harmonically bound chain, in the limit of weak coupling, serves as an excellent model for the approach to equilibrium of a canonical ensemble of weakly interacting particles.     

Classification and unfolding of degenerate Hopf bifurcations with O(2) symmetry: No distinguished parameter

The Hopf bifurcation in the presence of O(2) symmetry is considered. When the bifurcation breaks the symmetry, the critical imaginary eigenvalues have multiplicity two and generically there are two primary branches of periodic orbits which bifurcate simultaneously. In applications these correspond to rotating (traveling) waves and standing waves. Using equivariant singularity theory a classification of all such bifurcations up to and including codimension three is presented. No distinguished parameter is assumed. The universal unfoldings reveal the existence of both 2-tori and 3-tori; corresponding to quasiperiodic waves with two and three independent frequencies, respectively.     

Book review: Nonlinear oscillations,dynamical systems,and bifurcations of vector fields

Journal of Statistical Physics -     

High resolution polar Kerr effect measurements of Sr2RuO4: evidence for broken time-reversal symmetry in the superconducting state

The polar Kerr effect in the spin-triplet superconductor Sr2RuO4 was measured with high precision using a Sagnac interferometer with a zero-area Sagnac loop. We observed nonzero Kerr rotations as big as 65 nanorad appearing below Tc in large domains. Our results imply a broken time-reversal symmetry state in the superconducting state of Sr2RuO4, similar to 3He-A.     

Nonlinear and nonlocal dynamics of spatially extended systems: Stationary states, bifurcations and stability

Spatially extended systems with nonlocal dynamics (e.g. ferromagnetic resonance or current instability) of the type

with uε ⁿ will be studied near the soft-mode instability (wave number k_c ≠ 0) of a stationary and uniform state. An amplitude equation is derived within the framework of a multiple-scale perturbation theory. A particular example of this class of nonlocal dynamics is also treated numerically. As the main result we find that in contrast to the well-known supercritical bifurcation into a stable periodic state, the uniform state can bifurcate supercritically into a stationary state of an amplitude-modulated fast oscillation in space.     

Conformal invariance and Noether symmetry, Lie symmetry of holonomic mechanical systems in event space

This paper is devoted to studying the conformal invariance and Noether symmetry and Lie symmetry of a holonomic mechanical system in event space. The definition of the conformal invariance and the corresponding conformal factors of the holonomic system in event space are given. By investigating the relation between the conformal invariance and the Noether symmetry and the Lie symmetry, expressions of conformal factors of the system under these circumstances are obtained, and the Noether conserved quantity and the Hojman conserved quantity directly derived from the conformal invariance are given. Two examples are given to illustrate the application of the results.     

Theory of symmetry in the quantum mechanics of infinite systems: II. Isotropic representations of the Galilei group and its central extensions

The isotropic bundle representations of the Galilei group and its central extension are classified, and the natural cross-section, action of the group on the base manifold and the canonical cocyle are determined for all cases. Projective bundle representations of the Galilei group are defined and the extension of Bargmann's superselection rule is established. Coordinate transformations on the base space are discussed in all cases, and the notion of generalized coordinate transformations is introduced. It is then shown that the bundle representations being considered do not violate the principle of Galilean relativity as it is commonly understood. The physical interpretation of the irreducible and some reducible representations is discussed. It is found that some bundle representations might correspond to objects which can act as sources or sinks of linear and/or angular momentum.     

Definition of corresponding orbitals and the diradical character in broken symmetry DFT calculations on spin coupled systems

The broken symmetry (BS) concept is an extremely useful tool for the prediction of exchange coupling constants in molecules with interacting paramagnetic centers. An analysis of the BS wavefunctions is presented and the relationship between the overlap of magnetic orbitals and the exchange coupling is stressed. The corresponding orbital transformation is introduced as a useful tool in order to determine the non-orthogonal ‘valence bond’-like magnetic orbital pairs in many electron systems.     

Existence and analytical predictions of periodic motions in a periodically forced,nonlinear friction oscillator

The grazing bifurcation, stick phenomena and periodic motions in a periodically forced, nonlinear friction oscillator are investigated. The nonlinear friction force is approximated by a piecewise linear, kinetic friction model with the static force. The total forces for the input and output flows to the separation boundary are introduced, and the force criteria for the onset and vanishing of stick motions are developed through such input and output flow forces. The periodic motions of such an oscillator are predicted analytically through the corresponding mapping structure. Illustrations of the periodic motions in such a piecewise friction model are given for a better understanding of the stick motion with the static friction. The force responses are presented, which agreed very well with the force criteria. If the fully nonlinear friction force is modeled by several portions of piecewise linear functions, the periodically forced, nonlinear friction oscillator can be predicted more accurately. However, for the fully nonlinear friction force model, only the numerical investigation can be carried out.