Behavior In Phase Space Research Articles

Explicit dissipative structures

Dissipative structures consisting of a few macrovariables arise out of a sea of reversible microvariables. Unexpected residual effects of the massive underlying reversibility, on the macrolevel, cannot therefore be excluded. In the age of molecular-dynamics simulations, explicit dissipative structures like excitable systems (“explicit observers”) can be generated in a computer from first reversible principles. A class of classical, 1-D Hamiltonian systems of chaotic type is considered which has the asset that the trajectorial behavior in phase space can be understood geometrically. If, as natural, the number of particle types is much smaller than that of particles, the Gibbs symmetry must be taken into account. The permutation invariance drastically changes the behavior in phase space (quasi-periodization). The explicit observer becomes effectively reversible on a short time scale. In consequence, his ability to measure microscopic motions is suspended in a characteristic fashion. Unlike quantum mechanics whose “holistic” nature cannot be transcended, the present holistic (internal-interface) effects—mimicking the former to some extent—can be understood fully in principle.

Estimating the Dimensions of Weather and Climate Attractors

The dimensions of attractors are estimated from phase space trajectories of observed weather and climate variables (local surface pressure and relative sunshine duration, zonal wave amplitude; a δ18O-record). They provide primary information for descriptions of properties of the attractors of dynamical systems and give a lower limit to the number of the essential variables necessary to model the dynamics. These estimates are based on distance distributions of pairs of points on the single variable trajectory evolving in phase spaces which embed the attractor. One observes a low fractal dimensionality between three and four for the weather attractor, if interannual variability and seasonal changes are eliminated. The physical interpretation is based on the three dominating scales of cyclones, cyclone families and index-cycle; the irregularity of the flow and strong dependence on initial conditions amount for the fractal value. The climate variable also reveals a low dimensionality (between four and five) of the climate attractor. This is supported by an independent estimate based on eigenfunction expansion of the embedded phase space trajectory. These types of analyses suggest how to extend the standard data evaluation and model verification techniques to an analysis of the phase space behavior of observed and simulated dynamical systems.

Geometrical structure of torsional and rotational degrees of freedom in defect systems

The geometrical and physical significance of the translational and rotational degrees of freedom of a dislocation in two-dimensional (2-D) systems is established. It is shown that the phase space accessible to the non-conservative translational motion of dislocations, in the presence of point defects, can also be realized by means of their conservative motion described by a set of torsion angles. This leads to the concept of torsional degrees of freedom providing wandering type slow motion in phase space and glass type behavior. The equation of motion of a N-particle torsional oscillator is derived and qualitatively discussed. Transitions between its rotational states being mediated by dislocation reaction processes, and its point defect generation regime leading to self-sustained non-conservative motion is studied. Physical consequences of the intricate geometry of the phase space for the melting transition are worked out. The three-dimensional (3-D) problem is studied in lesser detail.

Some results on the quantum dynamics of a particle in a Markovian potential

We consider the quantum dynamics of a particle in a time dependent potentialV(t), assuming it to be a Markovian random function of time. We derive a formula for the density matrix at timet averaged over the realisations of the potential. We then obtain a kind of RAGE theorem for the time evolution of compact observables, and some information on the phase space behaviour of the system.

Radial-Longitudinal Coupling in Proposed RCNP Injector Cyclotron

For an injector cyclotron to a ring cyclotron proposed at RCNP, the second, third and fourth harmonic modes are used for the light ion acceleration. To accelerate ions in these harmonic modes without phase space deformation, it is necessary to investigate the effect of radial-longitudinal coupling. For single, two and three dee configurations, the beam behavior in radial phase space was studied numerically for each harmonic mode and dee angle. In the case of the second and third harmonic accelerations of protons with two dees, dee angles between 60 and 75 degrees are acceptable. The results are compared with 180-degree single dee and 40-degree three dees for the third harmonic mode.

Stochastically-driven climatic fluctuations in the sea-ice, ocean temperature, CO<sub>2</sub> feedback system

A simple climatic feedback system described previously (Saltzman and Moritz, 1980), involving sea-ice extent, mean ocean temperature and atmospheric CO 2 , is modified in plausible ways to yield (1) a single stable equilibrium with slow periodic damping, and (2) auto-oscillatory (i.e., limit-cycle) behavior. These constitute two primary prototypical possibilities for the deterministic behavior of the real climatic system (others being the existence of multiple equilibria with assorted stabilities). The necessity for including stochastic forcing is discussed, and the role of this forcing in generating sustained irregular and quasi-periodic variability similar to that observed in the real system is illustrated. It is suggested that the spectral and “residence density” signatures of theoretical models such as these may be of value in identifying the deterministic phase space behaviour (e.g., number and nature of equilibria) underlying observed climatic variability, through an inspection of these signatures in real data. DOI: 10.1111/j.2153-3490.1982.tb01797.x

Finite-energy results of the parton model for the e+e− annihilation

A parton model for the e+e− annihilation which is valid at finite energies is constructed by taking into account the energy conservation and the effect of the finite transverse momentum relative to the jet axis. We study the production of the pseudoscalar nonet and of the baryon octet. By imposing constraints such as the Drell-Yan-West relation, the phase-space behaviour at smallx and the inclusive sum rules for energy and multiplicity, we are able to construct a simple model for the fragmentation function. Below the possible threshold of new particles, our model reproduces data on single-particle distributions and the charged-energy fraction.SU 3 symmetry relations for the fragmentation functions are satisfied after mass-breaking effect is taken into account. We throw light on various aspects of the e+e− annihilation in the course of approaching the scaling limit by studying the energy dependence of single-particle distributions, particle ratios, average charged and photon multiplicities and charged-energy fraction. Our model is supposed to describe the light-quark (u, d and s) component of the hadronic cross-section. We can learn about the nature of the new particles by comparing data above the new-particle threshold with our prediction (the difference between the two should be attributed to new degrees of freedom). In particular, we have estimated the average charged multiplicity and the charged-energy fraction of the charmed-quark component.