Appropriate Equations Of Motion Research Articles

Shooting at the nanoscale: Collection and acceleration of nanowires with an external electric field

We report an approach for collecting, charging, and exceedingly fast motion of silver nanowires (Ag NWs) using an external static electric field. With a proper choice of suspension medium, dispersed Ag NWs can be efficiently driven to align and accumulate vertically on the edges of two parallel gold microelectrodes on a glass substrate surface by dielectrophoresis. Then, at sufficiently high electric fields (>2.0×105 V/m), these NWs break at the electrode contact point while carrying some net charge. Afterwards, they immediately accelerate in the field direction and, despite an extremely low Reynolds number for the motion of NWs in viscous liquids, move with high speeds (>25 mm/s) toward the counter electrode. By solving the appropriate equation of motion, the amount of the net charge on the NWs in the beginning of the motion is estimated as ∼1×10−14 C. The described NW-shooting mechanism can be employed to construct a NW “gun” for piercing soft thin membranes at nanoscale. Furthermore, we show that the interplay of the competing dielectrophoretic and electric field forces leads to interesting dynamics for the NWs.

The finite-element time-domain method for elastic band-structure calculations

The finite-element time-domain method for elastic band-structure calculations is presented in this paper. The method is based on discretizing the appropriate equations of motion by finite elements, applying Bloch boundary conditions to reduce the analysis to a single unit cell, and conducting a simulation using a standard time-integration scheme. The unit cell is excited by a wide-band frequency signal designed to enable a large number of modes to be identified from the time-history response. By spanning the desired wave-vector space within the Brillouin zone, the band structure is then robustly generated. Bloch mode shapes are computed using the well-known concept of modal analysis, especially as implemented in an experimental setting. The performance of the method is analyzed in terms of accuracy, convergence, and computation time, and is compared to the finite-difference time-domain method as well as to a direct finite-element (FE) solution of the corresponding eigenvalue problem. The proposed method is advantageous over FD-based methods for unit cells with complex geometries, and over direct FE in situations where the formulation of an eigenvalue problem is not straightforward. For example, the new method makes it possible to accurately solve a time-dependent Bloch problem, such as the case of a complex unit cell model of a topological insulator where an internal fluid flow or other externally controlled physical fields are present.

Current-Voltage Characteristics of the Plasma Focus: A Deeper Look

A capacitor bank discharges a current which is a sinusoidal function, lightly damped by unavoidable circuit resistance. When powering a plasma focus, the current waveform is further damped by the axial motion typically during the rising part of the current. The radial phase, with severe rate of change of inductance due to a rapidly collapsing current sheet to a small radius, is so severely damped over a short period near the current peak that the waveform goes into the sharp dip. This produces the well-known signature current dip of the properly-operated plasma focus. Corresponding to the inductively-caused current dip is a sharp voltage spike which typically rises to a peak value in excess of the voltage to which the capacitor is charged. These features are adequately described by circuit equations coupled to appropriate equations of motion. The loading effect of different gases due to differences in mass, differences in compressibility and differences in radiation also produces differences in the current waveforms particularly in the current dips and voltage spikes. These differences could be subtle or dramatic, as are demonstrated in this paper.

Modeling the bending vibration of cross-laminated timber beams

Cross-laminated timber (CLT) is an economical construction material combining good structural properties and environmental advantages. Due to the layered geometry and the associated internal stress boundary conditions, modelling its dynamic behavior is complex. However, accurate vibration level estimations are necessary to estimate the sound insulation performance of CLT constructions. This paper outlines a method to determine the relevant elastic constants and the appropriate equations of motion to predict the vibration of beams cut from a three-ply CLT plate. Modal analysis performed on strips cut along the two principal directions provides insight in the material’s dynamics. The bending mode frequencies are determined experimentally by analyzing the sweep response spectrum using the linear prediction method. In a second step, analytical beam vibration models of increasing complexity are fitted to the measurements using a genetic optimization algorithm. Thin beam theory—the simplest model—does not predict the bending modes with sufficient accuracy. Depending on the strip direction, the bending vibrations can be modeled using Timoshenko’s theory for thick beams, or by a model for three layer sandwich beams respectively. The results show that taking the beam geometry into account is as important as estimating the orthotropic material constants to model the bending modes properly.

Information criteria and selection of vibration models

This paper presents a method of determining an appropriate equation of motion of two-dimensional plane structures like membranes and plates from vibration response measurements. The local steady-state vibration field is used as input for the inverse problem that approximately determines the dispersion curve of the structure. This dispersion curve is then statistically treated with Akaike information criterion (AIC), which compares the experimentally measured curve to several candidate models (equations of motion). The model with the lowest AIC value is then chosen, and the utility of other models can also be assessed. This method is applied to three experimental case studies: A red cedar wood plate for musical instruments, a thick paper subjected to unknown membrane tension, and a thick composite sandwich panel. These three cases give three different situations of a model selection.

The Grand Challenge of Quantum Computing: Bridging the Capacity Gap

The fabrication and control of macroscopic artificial quantum structures, such as qubits (Mooij et al., 1999; Nakamura et al., 1999; Friedman et al., 2000), qubit arrays (Johnson et al., 2011; Barends et al., 2014), quantum annealers (Boixo et al., 2013) and, recently, quantum metamaterials (Macha et al., 2014), have witnessed significant progress over the last 15 years. This was a surprisingly quick evolution from theoretical musings to what can now be called quantum engineering [the observation of such phenomena even in a single superconducting device was considered a truly challenging task in 1980 (Leggett, 1980)]. And today, we stand at the point where existing theoretical and computational tools become inadequate for predicting, analyzing, and simulating the behavior of such structures, in which quantum superposition and entanglement play the key role (Zagoskin et al., 2014). The long-known fundamental impossibility of simulating large enough quantum systems by classical means (Feynman, 1982), unfortunately, manifests itself already at the level of systems containing as few as several hundreds of qubits. Such a system is still too small to be used as an efficient quantum simulator of comparable systems, but already too large for us to tell with certainty, using the existing classical tools, whether it behaves as a quantum system should (Smolin and Smith, 2014). Furthermore, the complexity of already existing quantum processor prototypes confronts us with an engineering problem designing a reliable quantum device and testing its reliability. What is even worse, if there are fundamental corrections to the laws of quantum mechanics for large enough systems, we will be unable to discover them because of our inability to tell what exactly quantum mechanics would predict. Let us take the optimistic view that quantum computing is not fundamentally restricted by, for example, the size of a system capable of demonstrating quantum behavior (Penrose, 1999). In this scenario, it would be possible to create quantum computing devices that will allow us to design and fabricate ever bigger and better quantum computers, as well as other macroscopic quantum devices, of a character and use of which we cannot even imagine at the moment. Alternatively, we may find fundamental limits to the applicability of quantum mechanics. Nevertheless, this can happen only if the gap between our current ability to characterize large quantum systems and the capacities of the smallest workable quantum computers is bridged. Bridging this capacity gap is thus the immediate grand challenge for the field: a challenge that must be met if we hope to make further progress in quantum computing and quantum engineering or if we hope to discover fundamentally new physics, or both. While it is impossible to efficiently simulate a large quantum system by classical means by directly solving the appropriate equations of motion, it is feasible that essential quantum properties of an ensemble of such systems will be reflected in certain higher-level, global characteristics. These properties should be insensitive to details of a particular instance, computable by classical tools and accessible to experimental investigation. This view of a system of qubits as a quantum many-body system should be amenable to the approaches that have proven to work very well in numerous applications in condensed matter physics and quantum statistical mechanics. Therefore, with such earlier breakthroughs in mind, the task at hand will be difficult yet not impossible, and more than worth the effort.

Quantum-Optical Foundations of Massive and Massless Particles

A source of the divergences in QED is proposed, and a theory in which the Lamb shift and electron’s anomalous magnetic moment are calculated free of divergences is reviewed. It is shown that Dirac’s equation for a relativistic electron can be inferred from a Lorentz invariant having the form of the Lorentz gauge equation, , on identifying a carrier-wave energy with the electron’s rest mass energy mc2, the vector potential’s polarization vector with Pauli’s vector σ, and the envelops of the scalar and vector potentials with the four components of Dirac’s vector wave function. The same methodology is used to infer relativistic equations of motion having the Dirac form for a neutrino accompanied by ab initio neutrino-matter interaction terms. Then it is shown that the theory, which comprises Dirac’s equation plus the relativistic equations of motion for the neutrino, supports binding on a nuclear scale of energy and length. The experimentally observed weakness of the interaction energy of free neutrinos and matter is due to the smallness of the rate of tunnelling of free neutrinos through a potential barrier which exists in the interaction of free neutrinos and matter. Models are also proposed for the proton and neutron, and good agreement is obtained for the neutron-proton rest mass energy difference in view of the approximations made to solve the appropriate equations of motion.

Transverse flow in relativistic viscous hydrodynamics

Hydrodynamic model simulations of Au-Au collisions at RHIC have indicated recently, that with improved simulations in the coming years, it may be feasible to quantify the viscosity of the matter produced in heavy ion collisions. To this end, a consistent fluid dynamic description of viscous effects is clearly needed. Here, we report and extend on recent work in this direction [R. Baier, P. Romatschke and U.A. Wiedemann, arXiv:hep-ph/0602249 ]. In particular, we observe that a recently used, approximate form of the 2nd order Israel-Stewart viscous hydrodynamic equations of motion cannot account consistently for the transverse flow fields, which are expected to develop in heavy ion collisions. We identify the appropriate equations of motion.

Pair of null gravitating shells: II. Canonical theory and embedding variables

The study of the two shell system started in our first paper, ‘Pair of null gravitating shells I’, is continued. An action functional for a single shell given by Louko, Whiting and Friedman is generalized to give appropriate equations of motion for two and, in fact, any number of spherically symmetric null shells, including the cases when the shells intersect. In order to find the symplectic structure for the space of solutions described in paper I, the pull back to the constraint surface of the Liouville form determined by the action is transformed into new variables. They consist of Dirac observables, embeddings and embedding momenta (the so-called Kuchar decomposition). The calculation includes the integration of a set of coupled partial differential equations. A general method of solving the equations is worked out.

Friction coefficient, mean velocity, and conductivity tensor of large, heavy ions in a gas in any regime in external fields

The general expression for the time-dependent mean velocity of large, heavy (l.h.) ions in a gas in any regime in time-varying (or static) electric fields and/or in static magnetic fields is obtained as solution of the appropriate equation of motion for the average l.h. ion. It is shown that such an equation follows directly from the Newton's law for a single ion once appropriate averages are carried out. The procedure puts in evidence that an effective friction coefficient can be defined whose expression tends, in the hydrodynamic regime, to the usual expression following from the Stokes' law, and, in the molecular regime, to the expression relevant to the heavy particles of a Rayleigh gas. In the case of l.h. ions in static or alternate electric fields and static magnetic fields, the conductivity tensor is also obtained. Moreover, the limits of validity of the theory are briefly discussed. Finally it is shown that, when the l.h. ions are supposed to move in a hard-sphere gas, all the obtained results can explicitly be expressed in terms of the Knudsen number.

Boundary Integral Methods for Multicomponent Fluids and Multiphase Materials

We present a brief review of the application of boundary integral methods in two dimensions to multicomponent fluid flows and multiphase problems in materials science. We focus on the recent development and outcomes of methods which accurately and efficiently include surface tension. In fluid flows, we examine the effects of surface tension on the Kelvin–Helmholtz and Rayleigh–Taylor instabilities in inviscid fluids, the generation of capillary waves on the free surface, and problems in Hele-Shaw flows involving pattern formation through the Saffman–Taylor instability, pattern selection, and singularity formation. In materials science, we discuss microstructure evolution in diffusional phase transformations, and the effects of the competition between surface and elastic energies on microstructure morphology. A common link between these different physical phenomena is the utility of an analysis of the appropriate equations of motion at small spatial scales to develop accurate and efficient time-stepping methods.

Two-dimensional response of crushable polyurethane foam to low velocity impact

An experimental and numerical study of the two-dimensional response of crushable foam to low velocity impact is undertaken. Rigid polyurethane foam blocks are subjected to normal impact by gravity-driven impactors of different geometries, at velocities ranging from 2 to 4 m/s. The impactors comprise a rectangular block, a wedge-tipped block and a cylinder. Quantities measured during impact are the impactor deceleration, velocity and displacement, and the energy dissipated. The effects of impact velocity and geometry on the deformation and energy absorbed are studied. A two-dimensional numerical model is proposed to simulate the gross deformation induced in the impact process. It employs a lumped mass approach and is formulated in terms of finite deformation. Appropriate equations of motion, stress–strain relations, failure criteria and failure patterns are developed. Results generated by this model exhibit good correlation with experiments, thus substantiating its validity. The proposed model demonstrates advantages over traditional finite element approaches, in that it accommodates severe deformation and extensive structural failure without the problems of excessive mesh distortion and untenable time step reduction which accompany finite element simulations.

An effective Hamiltonian-based method for mixed quantum-classical dynamics on coupled electronic surfaces

We describe an approximate method for treating the mixed quantum-classical (QC) dynamics of many-body systems on N coupled electronic surfaces. The approach is based on calculating N×N reduced Hamiltonian matrices for the classical and quantal degrees of freedom by partial averaging, and then solving the appropriate equations of motion—Hamilton’s equations or the Schrödinger equation—self-consistently. The degrees of freedom requiring a quantum mechanical description are treated using a multistate Schrödinger equation with classically averaged effective time-dependent Hamiltonians and off-diagonal couplings. The classical degrees of freedom are treated by propagating N ensembles of trajectories, one on each electronic surface, using N reduced classical Hamiltonians defined in terms of the expectation value of the full Hamiltonian calculated using the evolving quantum wave functions. An ansatz is proposed to approximately estimate classical off-diagonal density matrix elements required for calculating the classically averaged interactions that couple quantum wave functions on different electronic states. We present the theory and then test it for a simple two-dimensional and two-state model system. Exact quantum and multiconfiguration time-dependent self-consistent-field (MCTDSCF) calculations are carried out to evaluate the QC performance. Good agreement between the MCTDSCF and QC results is obtained for the model considered.

The Clifford bundle and the dynamics of the superparticle

Using the Clifford bundle formalism we show that Frenet equations of classical differential geometry or its spinor version are the appropriate equations of motion for a classical spinning particle. We show that particular values of the curvatures appearin

Current Advance Method and Cyclic Leapfrog for 2D Multispecies Hybrid Plasma Simulations

CAM-CL (current advance method and cyclic leapfrog) is a new algorithm for hybrid plasma simulations. In common with existing methods, its physical basis is a "hybrid" plasma model which treats the ions as particles and the electrons as a massless fluid. CAM-CL is distinguished from previous 2D hybrid algorithms by four main features: (1) Multiple ion species may be treated with only a single computational pass through the particle data: this is achieved without extrapolation of the electric field in time. The particles are advanced by a leapfrog procedure which requires the electric field to be a half time-step ahead of the particle velocities. The electric field depends on the ionic current density and hence the particle velocities. In order to avoid a time-consuming "pre-push" of the velocities, CAM advances the ionic current density a half-step with an appropriate equation of motion. This is similar in concept to the moment method (D. Winske and K. B. Quest, J. Geophys, Res.93 (A9), 9681 (1988), Appendix A), except for the next two features: (2) CAM advances the ionic current density, whereas the moment method advances the fluid velocity. Consequently, multiple ion species may be easily treated (3) A free-streaming ionic current density is collected (velocities are collected at positions a half time-step ahead). An equation of motion is then applied, in which the advective term and the ionic stress tensor in the moment method are not needed, since transport effects are included in the free-streaming current. (4) CL is a leapfrog scheme for advancing the magnetic field, an adaptation of the modified midpoint method described by W. H. Press et al. (Numerical Recipes (Cambridge Univ. Press, Cambridge, 1986)). It is stable and allows sub-stepping of the magnetic field (the magnetic field time-step may be different to the particle time-step). A two-dimensional version of the algorithm has been tested on a quiet plasma, MHD wave propagation, and ion beam instabilities, the results of which are discussed.

An inexpensive, portable rain simulator: construction and test data

Construction details of an inexpensive, portable rain simulator are given. The dismountable components are easily transported on a light truck. The flat spray nozzles used produced rain with a >90% coefficient of uniformity over a 1-m2 test plot. Drop sizes and drop size distributions were determined using a flour pellet method. The appropriate equation of motion for accelerating drops and drag coefficients were used to calculate the impact velocity of individual drops and the kinetic energy of rain produced by various nozzles. Rainfall intensity can easily be varied in the range 10–150 mm h−1 and water consumption is low.

Excess electron motion in a molecular chain and nonlinear rate equations

The nonlinear quantum dynamics of the simultaneous transfer of two electrons in a molecular chain are studied. The discussion concentrates on the interplay between the mutual Coulomb interaction and the dephasing of the electronic wave function. A nonexponential redistribution of the electronic occupation probability has been obtained by the numerical solution of appropriate equations of motion of the electronic density matrix. Such a behavior can be explained in deriving a set of nonlinear rate equations which take into account the shift of the electronic site energies due to the mutual Coulomb interaction. This approximate description agrees well with the exact one for dephasing times equal and smaller than the coherent transfer time.

Computer modelling of fragmentation processes in radio frequency multipole collision cells

AbstractTo date, methods for calculating the trajectories of ions in radio frequency (RF) multipole fields have involved solving, analytically or numerically, the appropriate equations of motion. Another method is presented here whereby a non‐ideal distribution of field may be modelled and the position of an ion can be tracked, as it traverses the multipole. The validity of the calculations is discussed and some of the properties of collision cells are explained. Fragmentation processes within the cell are modelled showing that the focusing properties within the cell have a striking effect on parent‐ and daughter‐ion transmission.

DOMAIN STRUCTURE IN HELICAL MAGNETS

For a helical magnet the axis of the helix is a hard axis in most materials but the moments are never constrained so completely. Without that restriction the moments have two degrees of freedom. The two appropriate equations of motion are derived and it is show in the continuum approximation that within a domain a modulated structure is predicted for the variation of the out-of-plane angle, as found experimentally. The domain walls are probably similar in width to those in which the moments are constrained.

Theory of periodic ferromagnetic multilayers

We investigate the statics and dynamics of periodic multilayers consisting of two ferromagnetic materials. Our theory is based on a general Ginzburg-Landau functional for inhomogeneous systems and the appropriate equations of motion. We compute the transition temperature of the composite material, the temperature dependence of the magnetization and the magnon dispersion relation. The statics apply also to other multilayer systems such as ferroelectrics.