Dirac fermion in a time-dependent spherical box

Adopting a simple independent 'particle in a box' model with a finite square barrier, Dirac's relativistic wave equation has been used to obtain exact results for the electron density and kinetic energy density in the semi-infinite inhomogeneous electron-gas limit. By utilising a suitable power series development, one can pass to the non-relativistic limit and subsequently to the limit of an infinite barrier, in which case well established results are regained. Finally, the relation to relativistic Thomas-Fermi theory is discussed.

The problem of the isotopic harmonic oscillator of time-dependent frequency confined in a spherical box with time-dependent radius is studied. We show that the exact solution and the Lewis invariant operator can be obtained by performing two consecutive gauge transformations on the time-dependent Schrödinger equation. On the basis of the exact solution the non-adiabatic Berry phases for the system are calculated.

One-electron 3+1 and 2+1 Dirac equations are used to calculate the motion of a relativistic electron in a vacuum in the presence of an external magnetic field. First, calculations are carried on an operator level and exact analytical results are obtained for the electron trajectories which contain both intraband frequency components, identified as the cyclotron motion, as well as interband frequency components, identified as the trembling motion (Zitterbewegung, ZB). Next, time-dependent Heisenberg operators are used for the same problem to compute average values of electron position and velocity employing Gaussian wave packets. It is shown that the presence of a magnetic field and the resulting quantization of the energy spectrum has pronounced effects on the electron Zitterbewegung: it introduces intraband frequency components into the motion, influences all the frequencies and makes the motion stationary (not decaying in time) in case of the 2+1 Dirac equation. Finally, simulations of the 2+1 Dirac equation and the resulting electron ZB in the presence of a magnetic field are proposed and described employing trapped ions and laser excitations. Using simulation parameters achieved in recent experiments of Gerritsma and coworkers we show that the effects of the simulated magnetic field on ZB are considerable and can certainly be observed.

The relativistic Dirac equation in four-dimensional spacetime reveals a coherent relation between the dimensions of spacetime and the degrees of freedom of fermionic spinors. A massless Dirac fermion generates new symmetries corresponding to chirality spin and charge spin as well as conformal scaling transformations. With the introduction of intrinsic W-parity, a massless Dirac fermion can be treated as a Majorana-type or Weyl-type spinor in a six-dimensional spacetime that reflects the intrinsic quantum numbers of chirality spin. A generalized Dirac equation is obtained in the six-dimensional spacetime with a maximal symmetry. Based on the framework of gravitational quantum field theory proposed in Ref. [] with the postulate of gauge invariance and coordinate independence, we arrive at a maximally symmetric gravitational gauge field theory for the massless Dirac fermion in six-dimensional spacetime. Such a theory is governed by the local spin gauge symmetry SP(1,5) and the global Poincaré symmetry P(1,5)=SO(1,5)⋉P1,5 as well as the charge spin gauge symmetry SU(2). The theory leads to the prediction of doubly electrically charged bosons. A scalar field and conformal scaling gauge field are introduced to maintain both global and local conformal scaling symmetries. A generalized gravitational Dirac equation for the massless Dirac fermion is derived in the six-dimensional spacetime. The equations of motion for gauge fields are obtained with conserved currents in the presence of gravitational effects. The dynamics of the gauge-type gravifield as a Goldstone-like boson is shown to be governed by a conserved energy-momentum tensor, and its symmetric part provides a generalized Einstein equation of gravity. An alternative geometrical symmetry breaking mechanism for the mass generation of Dirac fermions is demonstrated.

Measurements of strange hadron (KS0, Λ+Λ‾, and Ξ−+Ξ‾+) transverse momentum spectra in pp, pPb, and PbPb collisions are presented over a wide range of rapidity and event charged-particle multiplicity. The data were collected with the CMS detector at the CERN LHC in pp collisions at s=7TeV, pPb collisions at sNN=5.02TeV, and PbPb collisions at sNN=2.76TeV. The average transverse kinetic energy is found to increase with multiplicity, at a faster rate for heavier strange particle species in all systems. At similar multiplicities, the difference in average transverse kinetic energy between different particle species is observed to be larger for pp and pPb events than for PbPb events. In pPb collisions, the average transverse kinetic energy is found to be slightly larger in the Pb-going direction than in the p-going direction for events with large multiplicity. The spectra are compared to models motivated by hydrodynamics.

A method for prediction of heavy heavy-duty diesel vehicle emissions on a test cycle using data from dissimilar test cycles is examined and presented. Four dissimilar modes of a new California test schedule were used to generate emission predictions for the US heavy-duty urban dynamometer driving schedule (UDDS). Intensive properties of each mode and of the UDDS, including average speed, stops/mile, percentage idle and average kinetic energy, were chosen for further study. The four dissimilar modes were the idle, creep, transient and cruise modes, created by the California Air Resources Board (CARB) in a prior effort. The predictive weightings were applied to emissions from 11 heavy-duty vehicles measured in units of grams/second (g/s), grams litre of fuel consumed (g/l) and brake specific emissions (g/kW h). Predictions of emissions for the UDDS in units of grams second of NOx and CO2were acceptable, but particulate matter (PM) deviations were substantial. Errors for prediction of NOx did not exceed 7 per cent for any case, and errors for CO2 prediction did not exceed 15 per cent. PM errors in the g/s case varied substantially, depending on the weighting case, indicating instabilities in the predictions. Errors for g/l and g/kW h predictions were higher. A series of corrections was applied to each of the predictive cases for each emissions unit studied. These corrections are shown to improve the predictive ability of the weightings, but the fundamental nature of the prediction was eroded. A weighting using the intensive properties of average speed, percentage idle and average kinetic energy was found to yield the best uncorrected prediction for every case, regardless of the emissions unit considered. The best predictive method was also shown to work acceptably when applied to unseen data from another 12 heavy-duty vehicles.

Acoustical systems which are driven at a single terminal pair are considered. The acoustical driving-point impedance Z of such a system is the ratio of the pressure to the volume velocity at the driving terminal. If the system comprises a limited region R bounded by a surface S, Z can be expressed in terms of the volume integrals over R of the dissipation function, and the average lagrange density in addition to the surface integral of the complex power flowing across S. The acoustic impedance of a circular orifice, small compared with the wavelength, is evaluated by the methods discussed above. It is found that about 60 percent of the contribution to the mass reactance of a thin orifice arises from the average kinetic energy contained within the region bounded by two hemispherical caps which have radii equal to the radius of the orifice and which are placed concentrically on opposite sides of the orifice. At sound levels so great that the particle displacement amplitude through the orifice exceeds the radius of the orifice, the flow through the orifice may become jetlike in such a manner that vortex rings which form within the hemispherical caps detach themselves periodically. The coherence of the motion within the caps is destroyed by this jet-flow, so that the average kinetic energy in the caps no longer contributes to the reactance of the orifice. Measurements of the sound field in the neighborhood of the orifice indicate that the configuration of the field outside the jet region is essentially unaffected by the local jet action, thus the kinetic energy in this region will contribute to the mass reactance. The 60 percent reduction in mass reactance predicted by this theory will be compared with recent measurements performed by Ingard.

Nuclear charge distribution of mass-separated isobars from thermal-neutron-induced fission of 235U

Double-porosity/naturally-fractured reservoir (NFRs) models have traditionally been used to represent the flow and pressure behavior for highly fractured carbonate reservoirs. Given that unconventional reservoirs such as shale oil/gas reservoirs may or may not be considered to be multi-porosity media, the use of the traditional/classical "double-porosity" models may not be adequate (or appropriate). The recent development of anomalous diffusion models has opened the possibility of adapting double-porosity models to estimate reservoir (and related) parameters for unconventional reservoirs. The primary objective of this work is to develop and demonstrate analytical reservoir models that provide (possible) physical explanations for the anomalous diffusion phenomenon. The models considering anomalous diffusion in reservoirs with Euclidean shape are developed using a convolved (i.e., time-dependent) version of Darcy's law. The use of these models can yield a power-law (straight-line) behavior for the pressure and/or rate performance — similar to the fractal reservoir models. The main advantage of using anomalous diffusion models compared to models considering fractal geometry is the reduction from two parameters (i.e., the Fractal Dimension and the Conductivity Index) to only one parameter (i.e., the anomalous diffusion exponent). However, the anomalous diffusion exponent does not provide information about the geometry or spatial distribution of the reservoir properties. To provide an alternative explanation for the anomalous diffusion phenomenon in petroleum reservoirs, we have developed double-porosity models considering matrix blocks with fractal geometry and fracture networks with either radial or fractal fracture networks. The flows inside the matrix blocks and the fractal fracture network assume that Darcy's Law is valid in its space-dependent (fractal) form, whereas the classical version of Darcy's Law is assumed for the radial fracture network case. The transient interporosity transfer is modeled using the classical convolution schemes given in the literature. For the system composed by a radial fracture network and fractal matrix blocks, we have investigated three cases by changing the producing conditions for the well and the flowing conditions for the matrix blocks: These three cases are: A well producing at a constant-rate and closed matrix blocks,A well producing at a constant-rate and "infinite-acting" matrix blocks, andA well producing at a variable-rate (time-dependent inner boundary condition) and "infinite-acting" matrix blocks. We have defined the matrix blocks to be "infinite-acting" in order to represent the nano/micro permeability of shale reservoirs. For the system defined by a fractal fracture network and infinite-acting fractal matrix blocks, we have investigated the influence of the fractal parameters (both matrix and fracture network) in the pressure- and rate-transient performance behaviors. We have defined the periods of flow that can be observed in these sorts of systems and we have developed analytical solutions for pressure-transient analysis. We demonstrate that the use of the convolved version of Darcy's Law results in a model very similar to the diffusivity equation for double-porosity systems (which incorporates transient interporosity flow). In performing this work, we establish the following obervations/conclusions using our new solutions: We have found that the assumption of a well producing at variable-rate (time-dependent inner boundary condition) has a more significant impact on the pressure (and derivative) functions and hinders the effects of the properties of the reservoir.We have demonstrated that the anomalous diffusion phenomena in unconventional reservoirs can be related to their multi-porosity nature.The pressure and pressure derivative responses may be used in the diagnosis of flow periods and the evaluation of reservoir parameters in unconventional reservoirs.

We consider the one-dimensional one phase Stefan problems with temperature or thermal flux boundary conditions. Exact and closed-form solutions for these two free boundary problems are very limited and highly restricted to particular initial and boundary conditions, and so are the similarity solutions. In general, only numerical methods allow one to completely solve such nonlinear problems. In this paper, the quasi-similarity solutions are defined and, for each problem, an analytical approximation of the interfacial position is given as function of the time-dependent boundary datum.

Analytical solutions of the one-dimensional advection–dispersion solute transport equation subject to time-dependent boundary conditions

An analytical method to solve heat conduction in layered spheres with time-dependent boundary conditions

A kinetic energy-based approach is developed to obtain the correct local temperature of nonequilibrium systems. We have demonstrated that as a temperature measure, the average disturbance kinetic energy of a sample is more applicable than its average total energy, and the latter has been widely used in most molecular dynamics software. However, it is proved and demonstrated by our simulation example that the average disturbance kinetic energy is sample-size dependent. By using a simple equilibrium system as a thermometer, we propose a calibration approach to eliminate this sample-size effect. A vibrating atomic bar is used as an example to test the validity and convergence of various temperature definitions.

Under classical particle dynamics, the interaction process between intense femtosecond laser pulses and icosahedral noble-gas atomic clusters was studied. Our calculated results show that ionization proceeds mainly through tunnel ionization in the combined field from ions, electrons and laser, rather than the electron-impact ionization. With increasing cluster size, the average and maximum kinetic energy of the product ion increases. According to our calculation, the expansion process of the clusters after laser irradiation is dominated by Coulomb explosion and the expansion scale increases with increasing cluster size. The dependence of average kinetic energy and average charge state of the product ions on laser wavelength is also presented and discussed. The dependence of average kinetic energy on the number of atoms inside the cluster was studied and compared with the experimental data. Our results agree with the experimental results reasonably well.

The mass-transfer in an electrorefiner for the pyroprocessing of spent fuel is investigated using multiphysics simulations, and a homogenous magnetic field is applied to accelerate the mass-transfer. The fields of current density, Lorentz force density, convective velocity, kinetic energy density, concentration and concentration gradient, and molar flux of uranium cations are calculated by numerical solution of the master equations using finite element method. It is concluded that a homogeneous magnetic field parallel to the electrode surface will induce Lorentz force, in perpendicular to both the magnetic field and the current density vector, normal to the electrode surface. The Lorentz force acts on the molten salt and enforces the molten salt to circulate around the electrodes. For comparison, the averaged kinetic energy induced solely by the magnetic field of 1 T (without stirring) at 0.754 J·m−3 is higher than that induced solely by stirring of 300 rpm (without magnetic field) at 0.595 J·m−3. The averaged kinetic energy induced by the combination of the magnetic field and the stirring at 1.118 J·m−3 is similar to the sum of that induced, respectively, by magnetic field and by stirring. If the circular electrodes are replaced with rectangular electrodes, the averaged kinetic energy reaches an even higher value of 1.278 J·m−3. Finally, it is concluded that the magnetically enhanced convective flow accelerates the mass-transfer of uranium cations by reduction of the thickness of boundary layer and thus of the concentration gradient around the electrodes.