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Phase Space (phase + space)
Selected AbstractsGeneral Gyrokinetic Equations for Edge PlasmasCONTRIBUTIONS TO PLASMA PHYSICS, Issue 7-9 2006H. Qin Abstract During the pedestal cycle of H-mode edge plasmas in tokamak experiments, large-amplitude pedestal build-up and destruction coexist with small-amplitude drift wave turbulence. The pedestal dynamics simultaneously includes fast time-scale electromagnetic instabilities, long time-scale turbulence-induced transport processes, and more interestingly the interaction between them. To numerically simulate the pedestal dynamics from first principles, it is desirable to develop an effective algorithm based on the gyrokinetic theory. However, existing gyrokinetic theories cannot treat fully nonlinear electromagnetic perturbations with multi-scale-length structures in spacetime, and therefore do not apply to edge plasmas. A set of generalized gyrokinetic equations valid for the edge plasmas has been derived. This formalism allows large-amplitude, time-dependent background electromagnetic fields to be developed fully nonlinearly in addition to small-amplitude, short-wavelength electromagnetic perturbations. It turns out that the most general gyrokinetic theory can be geometrically formulated. The Poincaré-Cartan-Einstein 1-form on the 7D phase space determines particles' worldlines in the phase space, and realizes the momentum integrals in kinetic theory as fiber integrals. The infinitesimal generator of the gyro-symmetry is then asymptotically constructed as the base for the gyrophase coordinate of the gyrocenter coordinate system. This is accomplished by applying the Lie coordinate perturbation method to the Poincaré-Cartan-Einstein 1-form. General gyrokinetic Vlasov-Maxwell equations are then developed as the Vlasov-Maxwell equations in the gyrocenter coordinate system, rather than a set of new equations. Because the general gyrokinetic system developed is geometrically the same as the Vlasov-Maxwell equations, all the coordinate-independent properties of the Vlasov-Maxwell equations, such as energy conservation, momentum conservation, and phase space volume conservation, are automatically carried over to the general gyrokinetic system. The pullback transformation associated with the coordinate transformation is shown to be an indispensable part of the general gyrokinetic Vlasov-Maxwell equations. As an example, the pullback transformation in the gyrokinetic Poisson equation is explicitly expressed in terms of moments of the gyrocenter distribution function, with the important gyro-orbit squeezing effect due to the large electric field shearing in the edge and the full finite Larmour radius effect for short wavelength fluctuations. The familiar "polarization drift density" in the gyrocenter Poisson equation is replaced by a more general expression. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] On the Effective Liapunov Exponent with Increasing Stochasticity for TokamapCONTRIBUTIONS TO PLASMA PHYSICS, Issue 4 2004A. Maluckov Abstract The effective Liapunov exponent statistical description of tokamap with increasing stochasticity is numerically developed. As the 2D Hamiltonian mapping, tokamap is characterized by the two Liapunov exponents lx = ,l, , 0, and 2D Liapunov exponent L = 0. The time development of the effective Liapunov exponent Lxe, the number of trajectories with lx > 0, Np, and the statistics of the non-negative Liapunov exponents are numerically considered following the test ensemble of N = 10000 trajectories. The sticking to regular structures in the phase space of tokamap, which is associated with the non-positive Liapunov exponent lx, clearly effects its statistics. Thus, the behavior of the effective Liapunov exponent Lxe may be taken as one of the qualitative measures of the degree of stochasticity for the observed mapping. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Dynamical similarity of explosions at Stromboli volcanoGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 3 2004S. De Martino SUMMARY We analyse Strombolian explosions recorded with broadband seismometers to quantify the degree of similarity among explosions. First, we construct the trajectory space. This reconstruction is analogous to that of phase space but, unlike the phase space which relies on the analysis of a single trajectory of a dynamic system, the trajectory space takes into account many trajectories of the dynamic process to study the overlapping properties. Based on the scaling of distances between each pair of histories, it is possible to evaluate the dimension of the trajectory space. We consider the different explosions as different trajectories and, after normalizing all the records, we find that the scaling region is spread over less than one order of magnitude. This absence of scaling implies that all the trajectories cover the same attractor in the trajectory space, and that the trajectories are generated by the same dynamic system. Accordingly, we conclude that all the events are very similar to each other. This result is confirmed by the energy distribution of these events. Standard techniques of energy estimation are inadequate in this case, and we propose an evaluation method based on a quantity that is proportional to energy. The distribution function of this quantity, calculated on 580 events, displays a log-normal behaviour with very low variability (less than two orders of magnitude in energy). These results provide quantitative support for the observation that there is a great degree of similarity among Strombolian explosions and support the idea that a dynamic model underlying these events can be elaborated through the study of just one event. [source] Online trained support vector machines-based generalized predictive control of non-linear systemsINTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, Issue 10 2006S. Iplikci Abstract In this work, an online support vector machines (SVM) training method (Neural Comput. 2003; 15: 2683,2703), referred to as the accurate online support vector regression (AOSVR) algorithm, is embedded in the previously proposed support vector machines-based generalized predictive control (SVM-Based GPC) architecture (Support vector machines based generalized predictive control, under review), thereby obtaining a powerful scheme for controlling non-linear systems adaptively. Starting with an initially empty SVM model of the unknown plant, the proposed online SVM-based GPC method performs the modelling and control tasks simultaneously. At each iteration, if the SVM model is not accurate enough to represent the plant dynamics at the current operating point, it is updated with the training data formed by persistently exciting random input signal applied to the plant, otherwise, if the model is accepted as accurate, a generalized predictive control signal based on the obtained SVM model is applied to the plant. After a short transient time, the model can satisfactorily reflect the behaviour of the plant in the whole phase space or operation region. The incremental algorithm of AOSVR enables the SVM model to learn the new training data pair, while the decremental algorithm allows the SVM model to forget the oldest training point. Thus, the SVM model can adapt the changes in the plant and also in the operating conditions. The simulation results on non-linear systems have revealed that the proposed method provides an excellent control quality. Furthermore, it maintains its performance when a measurement noise is added to the output of the underlying system. Copyright © 2006 John Wiley & Sons, Ltd. [source] Chemical reactions in the gas phase and in condensed matter: From wavefunctions to density operatorsINTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 13 2009David A. Micha Abstract This contribution generalizes the treatment of chemical reactions in the gas phase based on the reaction channel decomposition of the wavefunction, by introducing a similar channel decomposition of the statistical density operator valid also for condensed phases such as liquid solutions and solid surfaces. Coupled equations for the channel components of the density operator are derived and a brief presentation is given of their partial Wigner transform, which leads to a general treatment for coupling quantum and classical variables. This provides a general approach for reactions involving electronically excited states in many-atom systems. It is pointed out that reactions involving coupled quantal and classical variables can be correctly described provided (a) initial conditions for trajectories are generated from quantal distributions and (b) the bundle of trajectories for the whole initial classical phase space is propagated coupled to the quantal elements of the density matrix and used in the calculation of reaction flux averages. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009 [source] Symbolic methods for invariant manifolds in chemical kineticsINTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 1 2006Simon J. Fraser Abstract Chemical reactions show a separation of time scales in transient decay due to the stiffness of the ordinary differential equations (ODEs) that describe their evolution. This evolution can be represented as motion in the phase space spanned by the concentration variables of the chemical reaction. Transient decay corresponds to a collapse of the "compressible fluid" representing the continuum of possible dynamical states of the system. Collapse occurs sequentially through a hierarchy of nested, attracting, slow invariant manifolds (SIMs), i.e., sets that map into themselves under the action of the phase flow, eventually reaching the asymptotic attractor of the system. Using a symbolic manipulative language, explicit formulas for the SIMs can be found by iterating functional equations obtained from the system's ODEs. Iteration converges geometrically fast to a SIM at large concentrations and, if necessary, can be stabilized at small concentrations. Three different chemical models are examined in order to show how finding the SIM for a model depends on its underlying dynamics. For every model the iterative method provides a global SIM formula; however, formal series expansions for the SIM diverge in some models. Repelling SIMs can be also found by iterative methods because of the invariance of trajectory geometry under time reversal. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006 [source] Wigner function of the rotating Morse oscillatorINTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 1 2005Jerzy Stanek Abstract We present an analytical expression of the Wigner distribution function (WDF) for the bound eigenstates of the rotating Morse oscillator (RMO). The effect of rotational excitation on the WDF on the quantum phase space has been demonstrated. This effect has been visualized by a series of contour diagrams for given rovibrational quantum states. Rotations of the molecule have been proved to qualitatively and quantitatively change the Wigner function. As a result, the most probable distance between atoms in a rotating molecule changes, and depends on the parity of the vibrational quantum number. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005 [source] Coherent states for anharmonic diatomic moleculesINTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 6 2002J. Récamier Abstract In this work, we construct approximate coherent states for a Morse-like potential using the displacement operator method and a method recently proposed by Gazeau and Klauder. To test if these states are minimum uncertainty states, we evaluate the temporal evolution of the dispersions in position and momentum. We also construct the trajectories in the phase space and compare with the classic solution. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002 [source] The application of structure envelopes in structure determination from powder diffraction dataJOURNAL OF APPLIED CRYSTALLOGRAPHY, Issue 2 2002Simon Brenner A structure envelope is a special type of periodic nodal surface that separates regions of high electron density from those of low electron density. Once such a surface has been generated, it can be used in combination with direct-space methods to facilitate structure solution from powder data. To generate an informative structure envelope, the phases of the structure factors of a few strong low-order reflections must be determined; an algorithm has been developed for this purpose. The program SayPerm combines (a) the use of error-correcting codes (e.c.c.'s) to sample phase space efficiently, (b) a pseudo-atom approximation of structure fragments to simulate atomic resolution at ca 2.5,Å, and (c) phase extension and phase set ranking using the Sayre equation. The effect of using a structure envelope in structure solution was first tested in combination with a subroutine for finding zeolite topologies in the program FOCUS. Then extension to molecular structures in combination with a simulated-annealing program was explored. This resulted in the development of the program Safe and the subsequent determination of the structure of a tri-,-peptide (C32N3O6H53) with 17 variable torsion angles. [source] Subsolidus Phase Relationships in the ZnO,In2O3,SnO2 SystemJOURNAL OF THE AMERICAN CERAMIC SOCIETY, Issue 11 2008Steven P. Harvey The subsolidus phase relationships in the ZnO,InO1.5,SnO2 system were investigated at 1275°C using X-ray diffraction. Each of the end members of the ternary diagram is a transparent conducting oxide. There are two substantial solid solutions in the ternary phase space, the bixbyite solid solution In2,2xZnxSnxO3 (x=0,0.40), and the indium substituted zinc stannate spinel, Zn(2,x)Sn(1,x)In2xO4 (x=0,0.45). The bixbyite solid solution is an outstanding TCO, whereas the spinel is only moderately conducting. Along the ZnO,InO1.5 binary, there is a series of transparent conducting homologous compounds (ZnO)k·In2O3 (where k=3, 4, 5, 6, 7, 9, 11). Within ternary phase space, these homologous compounds were found to exhibit negligible Sn solubility, and were always found to be compatible with the spinel. Equilibrium was difficult to achieve in the phase space between the homologous series compounds and the spinel, owing to sluggish kinetics. A procedure involving mixtures of prereacted spinel and the Zn11In2O14 (k=11) compound was developed, which allowed for more rapid approach to thermodynamic equilibrium, thereby allowing for the establishment of phase relationships near the ZnO corner of the phase diagram. [source] Formation of High-Quality, Epitaxial La2Zr2O7 Layers on Biaxially Textured Substrates by Slot-Die Coating of Chemical Solution PrecursorsJOURNAL OF THE AMERICAN CERAMIC SOCIETY, Issue 11 2007Sung-Hun Wee Crystallization studies were performed of epitaxial La2Zr2O7 (LZO) films on biaxially textured Ni,3at.%W substrates having thin Y2O3 (10 nm) seed layers. LZO films were deposited under controlled humid atmosphere using reel-to-reel slot-die coating of chemical solution precursors. Controlled crystallization under various processing conditions has revealed a broad phase space for obtaining high-quality, epitaxial LZO films without microcracks, with no degradation of crystallographic texture and with high surface crystallinity. Crack-free and strong c -axis aligned LZO films with no random orientation were obtained even at relatively low annealing temperatures of 850°,950°C in flowing one atmosphere gas mixtures of Ar,4% H2 with an effective oxygen partial pressure of P(O2),10,22 atm. Texture and reflection high-energy electron diffraction analyses reveal that low-temperature-annealed samples have strong cube-on-cube epitaxy and high surface crystallinity, comparable to those of LZO film annealed at high temperature of 1100°C. In addition, these samples have a smoother surface morphology than films annealed at higher temperatures. Ni diffusion rate into the LZO buffer film is also expected to be significantly reduced at the lower annealing temperatures. [source] Finite-dimensional attractors and exponential attractors for degenerate doubly nonlinear equationsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2009M. Efendiev Abstract We consider the following doubly nonlinear parabolic equation in a bounded domain ,,,3: where the nonlinearity f is allowed to have a degeneracy with respect to ,tu of the form ,tu|,tu|p at some points x,,. Under some natural assumptions on the nonlinearities f and g, we prove the existence and uniqueness of a solution of that problem and establish the finite-dimensionality of global and exponential attractors of the semigroup associated with this equation in the appropriate phase space. Copyright © 2009 John Wiley & Sons, Ltd. [source] Linear instability of ideal flows on a sphereMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2009Yuri N. Skiba Abstract A unified approach to the normal mode instability study of steady solutions to the vorticity equation governing the motion of an ideal incompressible fluid on a rotating sphere is considered. The four types of well-known solutions are considered, namely, the Legendre-polynomial (LP) flows, Rossby,Haurwitz (RH) waves, Wu,Verkley (WV) waves and modons. A conservation law for disturbances to each solution is derived and used to obtain a necessary condition for its exponential instability. By these conditions, Fjörtoft's (Tellus 1953; 5:225,230) average spectral number of the amplitude of an unstable mode must be equal to a special value. In the case of LP flows or RH waves, this value is related only with the basic flow degree. For the WV waves and modons, it depends both on the basic flow degree and on the spectral distribution of the mode energy in the inner and outer regions of the flow. Peculiarities of the instability conditions for different types of modons are discussed. The new instability conditions specify the spectral structure of growing disturbances localizing them in the phase space. For the LP flows, this condition complements the well-known Rayleigh,Kuo and Fjörtoft conditions related to the zonal flow profile. Some analytical and numerical examples are considered. The maximum growth rate of unstable modes is also estimated, and the orthogonality of any unstable, decaying and non-stationary mode to the basic flow is shown in the energy inner product. The analytical instability results obtained here can also be applied for testing the accuracy of computational programs and algorithms used for the numerical stability study. It should be stressed that Fjörtoft's spectral number appearing both in the instability conditions and in the maximum growth rate estimates is the parameter of paramount importance in the linear instability problem of ideal flows on a sphere. Copyright © 2008 John Wiley & Sons, Ltd. [source] Global and exponential attractors for 3-D wave equations with displacement dependent dampingMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2006Vittorino Pata Abstract A weakly damped wave equation in the three-dimensional (3-D) space with a damping coefficient depending on the displacement is studied. This equation is shown to generate a dissipative semigroup in the energy phase space, which possesses finite-dimensional global and exponential attractors in a slightly weaker topology. Copyright © 2006 John Wiley & Sons, Ltd. [source] Exchange of conserved quantities, shock loci and Riemann problemsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2001Michael Sever Systems of conservation laws admitting extensions, such as entropy density/flux functions, generate related systems obtained by exchanging the extension with one of the constituent equations. Often if not always, the smooth solutions of the two systems coincide, and weak solutions of one system containing only small discontinuities are approximate weak solutions of the other. The adiabatic approximation for the Euler system illustrates the utility of this procedure. Such an exchange of conserved quantities preserves hyperbolicity and genuine non-linearity in the sense of Lax. On the other hand, the topological structure of the shock locus of a point in phase space and the solvability of Riemann problems in the large can be strongly affected. A discussion of when and how this occurs is given here. In this paper the exchange of conserved quantities is conveniently described by a simple homotopy in an extended version of the usual ,symmetric variables'. A dynamical system in phase space is constructed, the trajectories of which describe the Hugoniot locus of a fixed point in phase space at each state of the homotopy. The appearance of critical points for this dynamical system is identified with the alteration of the topological structure of the Hugoniot locus by the exchange of conserved quantities. Copyright © 2001 John Wiley & Sons, Ltd. [source] Fresnel coefficients for a permittivity-permeability phase space encompassing vacuum, anti-vacuum, and nihilityMICROWAVE AND OPTICAL TECHNOLOGY LETTERS, Issue 2 2006Akhlesh Lakhtakia Abstract The concept of the electromagnetic trinity comprising vacuum, anti-vacuum, and nihility is extended to a zone in the 4-D ,r -,r phase space that was largely inaccessible until the recent emergence of negative-phase-velocity materials. Expressions for Fresnel coefficients are derived for the reflection of plane waves at the planar interface of vacuum and a homogeneous, isotropic, dielectric-magnetic medium with constitutive properties lying in that zone. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 48: 265,270, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21323 [source] The resonant structure of Jupiter's Trojan asteroids , I. Long-term stability and diffusionMONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Issue 4 2006P. Robutel ABSTRACT We study the global dynamics of the jovian Trojan asteroids by means of the frequency map analysis. We find and classify the main resonant structures that serve as skeleton of the phase space near the Lagrangian points. These resonances organize and control the long-term dynamics of the Trojans. Besides the secondary and secular resonances, that have already been found in other asteroid sets in mean motion resonance (e.g. main belt, Kuiper belt), we identify a new type of resonance that involves secular frequencies and the frequency of the great inequality, but not the libration frequency. Moreover, this new family of resonances plays an important role in the slow transport mechanism that drives Trojans from the inner stable region to eventual ejections. Finally, we relate this global view of the dynamics with the observed Trojans, identify the asteroids that are close to these resonances and study their long-term behaviour. [source] From simple rules to cycling in community assemblyOIKOS, Issue 2 2004Sebastian J. Schreiber Simulation studies of community assembly have frequently observed two related phenomena: (1) the humpty dumpty effect in which communities can not be reconstructed by "sequential" invasions (i.e. single species invasions separated by long intervals of time) and (2) cycling between sub-communities. To better understand the mechanisms underlying these phenomena, we analyze a system consisting of two predators and two prey competing for a shared resource. We show how simple dominance rules (i.e. R* and P* rules) lead to cycling between sub-communities consisting of predator,prey pairs; predator and prey invasions alternatively lead to prey displacement via apparent competition and predator displacement via exploitative competition. We also show that these cycles are often dynamically unstable in the population phase space. More specifically, while for too slow invasion rates (i.e. "sequential" invasions) the system cycles indefinitely, faster invasion rates lead to coexistence of all species. In the later case, the assembly dynamics exhibit transient cycling between predator-prey subcommunities and the length of these transients decreases with the invasion rate and increases with habitat productivity. [source] The role of breathers in the anomalous decay of luminescencePHYSICA STATUS SOLIDI (C) - CURRENT TOPICS IN SOLID STATE PHYSICS, Issue 10 2006Eva Mihóková Abstract Luminescence of alkali halides doped by heavy ns2 ions exhibits an anomaly in the slow component emission decay. The anomaly is explained by the formation of a discrete breather in the immediate neighborhood of the impurity. We study properties of these breathers, their phase space structure, robustness, and propensity for formation. Under a wide range of parameters and interionic potentials they form 2-dimensional Kolmogorov-Arnold-Moser tori (less than generic) in phase space. We show strobed views of these tori, useful in quantization. All features support the thesis of breather formation as the explanation for the luminescence decay anomaly that first motivated our breather proposal. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Configurational-bias sampling technique for predicting side-chain conformations in proteinsPROTEIN SCIENCE, Issue 9 2006Tushar Jain Abstract Prediction of side-chain conformations is an important component of several biological modeling applications. In this work, we have developed and tested an advanced Monte Carlo sampling strategy for predicting side-chain conformations. Our method is based on a cooperative rearrangement of atoms that belong to a group of neighboring side-chains. This rearrangement is accomplished by deleting groups of atoms from the side-chains in a particular region, and regrowing them with the generation of trial positions that depends on both a rotamer library and a molecular mechanics potential function. This method allows us to incorporate flexibility about the rotamers in the library and explore phase space in a continuous fashion about the primary rotamers. We have tested our algorithm on a set of 76 proteins using the all-atom AMBER99 force field and electrostatics that are governed by a distance-dependent dielectric function. When the tolerance for correct prediction of the dihedral angles is a <20° deviation from the native state, our prediction accuracies for ,1 are 83.3% and for ,1 and ,2 are 65.4%. The accuracies of our predictions are comparable to the best results in the literature that often used Hamiltonians that have been specifically optimized for side-chain packing. We believe that the continuous exploration of phase space enables our method to overcome limitations inherent with using discrete rotamers as trials. [source] Prediction of High-Pressure Phase Equilibria using Cubic EOS: What Can Be Learned?THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, Issue 5 2002Ilya Polishuk Abstract Modern semi-empirical cubic EOSs are usually attached by complex functionalities, such as Huron-Vidal (HV)-type mixing rules. Although this practice improves the flexibility of the models, it also complicates consideration of an overall picture of phase behavior. As a result, GE -based equations are usually adjusted to the experimental data by way of a local fit. The latter approach tends to ignore that different regions of the thermodynamic phase space are closely inter-related. The present study demonstrates that the contribution of (HV)-type mixing rules to predicting high-pressure phase equilibria can be quite modest and that in addition they may generate non-realistic phase diagrams. Les équations d'état (EOS) cubiques semi-empiriques modernes s'accompagnent habituellement de fonctionnalités complexes, telles que les règles de mélange de type Huron-Vidal (HV). Bien que cette pratique améliore la flexibilité des modèles, elle complique aussi la prise en compte globale du comportement des phases. En conséquence, les équations basées sur GE sont généralement ajustées aux données expérimentales au moyen d'un calage local. Cette dernière approche tend à ignorer que les différentes régions de l'espace de phase thermodynamique sont étroitement intereliées. La présente étude montre que la contribution des règles de mélange de type HV pour prédire les équilibres de phases à haute pression peut être assez modeste et qu'en outre elles peuvent produire des diagrammes de phases non réalistes. [source] Semiclassical expansion of quantum characteristics for many-body potential scattering problemANNALEN DER PHYSIK, Issue 9 2007M.I. Krivoruchenko Abstract In quantum mechanics, systems can be described in phase space in terms of the Wigner function and the star-product operation. Quantum characteristics, which appear in the Heisenberg picture as the Weyl's symbols of operators of canonical coordinates and momenta, can be used to solve the evolution equations for symbols of other operators acting in the Hilbert space. To any fixed order in the Planck's constant, many-body potential scattering problem simplifies to a statistical-mechanical problem of computing an ensemble of quantum characteristics and their derivatives with respect to the initial canonical coordinates and momenta. The reduction to a system of ordinary differential equations pertains rigorously at any fixed order in ,. We present semiclassical expansion of quantum characteristics for many-body scattering problem and provide tools for calculation of average values of time-dependent physical observables and cross sections. The method of quantum characteristics admits the consistent incorporation of specific quantum effects, such as non-locality and coherence in propagation of particles, into the semiclassical transport models. We formulate the principle of stationary action for quantum Hamilton's equations and give quantum-mechanical extensions of the Liouville theorem on conservation of the phase-space volume and the Poincaré theorem on conservation of 2p -forms. The lowest order quantum corrections to the Kepler periodic orbits are constructed. These corrections show the resonance behavior. [source] A new look at the quantum mechanics of the harmonic oscillatorANNALEN DER PHYSIK, Issue 7-8 2007H.A. Kastrup Abstract In classical mechanics the harmonic oscillator (HO) provides the generic example for the use of angle and action variables and I > 0 which played a prominent role in the "old" Bohr-Sommerfeld quantum theory. However, already classically there is a problem which has essential implications for the quantum mechanics of the (,,I)-model for the HO: the transformation is only locally symplectic and singular for (q,p) = (0,0). Globally the phase space {(q,p)} has the topological structure of the plane ,2, whereas the phase space {(,,I)} corresponds globally to the punctured plane ,2 -(0,0) or to a simple cone with the tip deleted. From the properties of the symplectic transformations on that phase space one can derive the functions h0 = I, h1 = Icos , and h2 = - Isin , as the basic coordinates on {(,,I)}, where their Poisson brackets obey the Lie algebra of the symplectic group of the plane. This implies a qualitative difference as to the quantum theory of the phase space {(,,I)} compared to the usual one for {(q,p)}: In the quantum mechanics for the (,,I)-model of the HO the three hj correspond to the self-adjoint generators Kj, j = 0,1,2, of certain irreducible unitary representations of the symplectic group or one of its infinitely many covering groups, the representations being parametrized by a (Bargmann) index k > 0. This index k determines the ground state energy of the (,,I)-Hamiltonian . For an m -fold covering the lowest possible value for k is k = 1/m, which can be made arbitrarily small by choosing m accordingly! This is not in contradiction to the usual approach in terms of the operators Q and P which are now expressed as functions of the Kj, but keep their usual properties. The richer structure of the Kj quantum model of the HO is "erased" when passing to the simpler (Q,P)-model! This more refined approach to the quantum theory of the HO implies many experimental tests: Mulliken-type experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE-condensates, with charged HOs in external electric fields and the (Landau) levels of charged particles in external magnetic fields, with the propagation of light in vacuum, passing through strong external electric or magnetic fields. Finally it may lead to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant. [source] Analytic smoothing effect for the Schrödinger equation with long-range perturbationCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 9 2006Andre Martinez We study the microlocal analytic singularity of solutions to the Schrödinger equation with analytic coefficients. Using microlocal weight estimates developed for estimating phase space tunneling, we prove microlocal smoothing estimates that generalize results by Robbiano and Zuily. We show that the exponential decay of the initial state in a cone in the phase space implies microlocal analytic regularity of the solution at a positive time. We suppose the Schrödinger operator is a long-range-type perturbation of the Laplacian, and we employ positive commutator-type estimates to prove the smoothing property. © 2005 Wiley Periodicals, Inc. [source] Interfacial Thermodynamics of Polymeric MesophasesMACROMOLECULAR THEORY AND SIMULATIONS, Issue 8 2004Alejandro D. Rey Abstract Summary: A complete mechanical-thermodynamical formulation for multicomponent nematic polymer-isotropic fluid interfaces is derived, validated, and used to derive the structure and shape equations for these soft anisotropic polymer interfaces. The fundamental role of liquid crystalline order and long range effects in coupling bulk and interfacial effects, and in coupling thermodynamical/liquid crystalline order/geometrical variables is demonstrated, discussed, and validated. The Gibbs-Duhem nemato-thermodynamics equation emerges from an interfacial tension ,,=,,(,, ,, Q, ,sQ, k) that depends on temperature (,), chemical potential (,), nematic tensor order parameter Q, surface gradients of Q, and geometry k, and leads to new couplings in these enhanced phase spaces. The role of entropy and adsorption, and long range effects on interfacial shape and structure selection is revealed. For flat interfaces the preferred structure emerges from a competition between energy, entropy, and adsorption. [source] | |||||||||||||||||||||||||