Trajectory construction of Dirac evolution

We extend our programme of representing the quantum state through exact stand-alone trajectory models to the Dirac equation. We show that the free Dirac equation in the angular coordinate representation is a continuity equation for which the real and imaginary parts of the wave function, angular versions of Majorana spinors, define conserved densities. We hence deduce an exact formula for the propagation of the Dirac spinor derived from the self-contained first-order dynamics of two sets of trajectories in 3-space together with a mass-dependent evolution operator. The Lorentz covariance of the trajectory equations is established by invoking the ‘relativity of the trajectory label'. We show how these results extend to the inclusion of external potentials. We further show that the angular version of Dirac's equation implies continuity equations for currents with non-negative densities, for which the Dirac current defines the mean flow. This provides an alternative trajectory construction of free evolution. Finally, we examine the polar representation of the Dirac equation, which also implies a non-negative conserved density but does not map into a stand-alone trajectory theory. It reveals how the quantum potential is tacit in the Dirac equation.

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Energy-consistent entrainment relations for jets and plumes

Journal of Fluid Mechanics ◽

10.1017/jfm.2015.534 ◽

2015 ◽

Vol 782 ◽

pp. 333-355 ◽

Cited By ~ 37

Author(s):

Maarten van Reeuwijk ◽

John Craske

Keyword(s):

Kinetic Energy ◽

Richardson Number ◽

Turbulence Kinetic Energy ◽

Mean Flow ◽

Unified Framework ◽

First Order ◽

Entrainment Coefficient ◽

The Mean ◽

Flow Turbulence ◽

Pure Plume

We discuss energetic restrictions on the entrainment coefficient${\it\alpha}$for axisymmetric jets and plumes. The resulting entrainment relation includes contributions from the mean flow, turbulence and pressure, fundamentally linking${\it\alpha}$to the production of turbulence kinetic energy, the plume Richardson number$\mathit{Ri}$and the profile coefficients associated with the shape of the buoyancy and velocity profiles. This entrainment relation generalises the work by Kaminskiet al. (J. Fluid Mech., vol. 526, 2005, pp. 361–376) and Fox (J. Geophys. Res., vol. 75, 1970, pp. 6818–6835). The energetic viewpoint provides a unified framework with which to analyse the classical entrainment models implied by the plume theories of Mortonet al.(Proc. R. Soc. Lond.A, vol. 234, 1955, pp. 1–23) and Priestley & Ball (Q. J. R. Meteorol. Soc., vol. 81, 1954, pp. 144–157). Data for pure jets and plumes in unstratified environments indicate that to first order the physics is captured by the Priestley and Ball entrainment model, implying that (1) the profile coefficient associated with the production of turbulence kinetic energy has approximately the same value for pure plumes and jets, (2) the value of${\it\alpha}$for a pure plume is roughly a factor of$5/3$larger than for a jet and (3) the enhanced entrainment coefficient in plumes is primarily associated with the behaviour of the mean flow and not with buoyancy-enhanced turbulence. Theoretical suggestions are made on how entrainment can be systematically studied by creating constant-$\mathit{Ri}$flows in a numerical simulation or laboratory experiment.

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Quantization of κ-deformed Dirac equation

International Journal of Modern Physics A ◽

10.1142/s0217751x2050147x ◽

2020 ◽

Vol 35 (25) ◽

pp. 2050147

Author(s):

E. Harikumar ◽

Vishnu Rajagopal

Keyword(s):

Dirac Equation ◽

Particle Physics ◽

Equations Of Motion ◽

Space Time ◽

Dirac Field ◽

Oscillator Algebra ◽

First Order ◽

Deformed Oscillator Algebra ◽

Mass Dependent ◽

Conserved Currents

In this paper, we study the quantization of Dirac field theory in the [Formula: see text]-deformed space–time. We adopt a quantization method that uses only equations of motion for quantizing the field. Starting from [Formula: see text]-deformed Dirac equation, valid up to first order in the deformation parameter [Formula: see text], we derive deformed unequal time anticommutation relation between deformed field and its adjoint, leading to undeformed oscillator algebra. Exploiting the freedom of imposing a deformed unequal time anticommutation relations between [Formula: see text]-deformed spinor and its adjoint, we also derive a deformed oscillator algebra. We show that deformed number operator is the conserved charge corresponding to global phase transformation symmetry. We construct the [Formula: see text]-deformed conserved currents, valid up to first order in [Formula: see text], corresponding to parity and time-reversal symmetries of [Formula: see text]-deformed Dirac equation also. We show that these conserved currents and charges have a mass-dependent correction, valid up to first order in [Formula: see text]. This novel feature is expected to have experimental significance in particle physics. We also show that it is not possible to construct a conserved current associated with charge conjugation, showing that the Dirac particle and its antiparticle satisfy different equations in [Formula: see text] space–time.

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Lagrangian averaging with geodesic mean

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2017.0558 ◽

2017 ◽

Vol 473 (2207) ◽

pp. 20170558 ◽

Cited By ~ 3

Author(s):

Marcel Oliver

Keyword(s):

Vector Field ◽

Kinetic Energy ◽

Additional Assumption ◽

Mean Flow ◽

First Order ◽

The Mean ◽

Definition Of ◽

Flow Map ◽

Intrinsic Definition ◽

Volume Preserving

This paper revisits the derivation of the Lagrangian averaged Euler (LAE), or Euler- α equations in the light of an intrinsic definition of the averaged flow map as the geodesic mean on the volume-preserving diffeomorphism group. Under the additional assumption that first-order fluctuations are statistically isotropic and transported by the mean flow as a vector field, averaging of the kinetic energy Lagrangian of an ideal fluid yields the LAE Lagrangian. The derivation presented here assumes a Euclidean spatial domain without boundaries.

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An Approach for Multi-Stage Calculations Incorporating Unsteadiness

Volume 1: Turbomachinery ◽

10.1115/92-gt-282 ◽

1992 ◽

Cited By ~ 21

Author(s):

Michael Giles

Keyword(s):

Unsteady Flow ◽

Linear Perturbation ◽

Mean Flow ◽

Flow Equations ◽

First Order ◽

Multi Stage ◽

Blade Row ◽

Asymptotic Parameter ◽

The Mean ◽

Asymptotic Formulation

This paper describes a mathematical approach to the calculation of unsteady flow in multi-stage turbomachinery. An asymptotic formulation is used, with the small asymptotic parameter being the level of unsteadiness in each blade row. The baseline flow is the nonlinear steady flow that is computed by many existing multi-stage calculation methods. The first order linear perturbation is the unsteady flow field arising from stator/rotor interactions between neighboring blade rows. The second order correction contains the information about the time-averaged effect of unsteadiness on the mean flow. The advantage of this asymptotic approach is that it leads to a set of equations which can be solved numerically very much more cheaply than the full nonlinear unsteady flow equations, while still retaining the key features of the flow.

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Eddy forcing of the mean flow in the Southern Ocean

Journal of Geophysical Research Atmospheres ◽

10.1029/1999jc900332 ◽

2001 ◽

Vol 106 (C2) ◽

pp. 2713-2722 ◽

Cited By ~ 3

Author(s):

Chris W. Hughes

Keyword(s):

Southern Ocean ◽

Mean Flow ◽

The Mean

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Simple explicit model of liquid mean flow in region above axial impeller

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19852396 ◽

1985 ◽

Vol 50 (11) ◽

pp. 2396-2410

Author(s):

Miloslav Hošťálek ◽

Ivan Fořt

Keyword(s):

Vortex Flow ◽

Flow Model ◽

Quantitative Agreement ◽

Mean Flow ◽

Flat Bottom ◽

Proposed Model ◽

Minimum Number ◽

The Mean ◽

Model Equations ◽

Radial Circulation

The study describes a method of modelling axial-radial circulation in a tank with an axial impeller and radial baffles. The proposed model is based on the analytical solution of the equation for vortex transport in the mean flow of turbulent liquid. The obtained vortex flow model is tested by the results of experiments carried out in a tank of diameter 1 m and with the bottom in the shape of truncated cone as well as by the data published for the vessel of diameter 0.29 m with flat bottom. Though the model equations are expressed in a simple form, good qualitative and even quantitative agreement of the model with reality is stated. Apart from its simplicity, the model has other advantages: minimum number of experimental data necessary for the completion of boundary conditions and integral nature of these data.

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First passage time for the state of exact chemical equilibrium

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19800777 ◽

1980 ◽

Vol 45 (3) ◽

pp. 777-782 ◽

Cited By ~ 1

Author(s):

Milan Šolc

Keyword(s):

Chemical Equilibrium ◽

First Passage Time ◽

Passage Time ◽

The State ◽

Recurrence Time ◽

First Passage ◽

First Order ◽

The Mean ◽

Passage Times ◽

Number Of Particles

The establishment of chemical equilibrium in a system with a reversible first order reaction is characterized in terms of the distribution of first passage times for the state of exact chemical equilibrium. The mean first passage time of this state is a linear function of the logarithm of the total number of particles in the system. The equilibrium fluctuations of composition in the system are characterized by the distribution of the recurrence times for the state of exact chemical equilibrium. The mean recurrence time is inversely proportional to the square root of the total number of particles in the system.

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The Structure and Photochromism of 2,4,4,6-Tetraphenyl-4H-thiopyran

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19921326 ◽

1992 ◽

Vol 57 (6) ◽

pp. 1326-1334 ◽

Cited By ~ 11

Author(s):

Jaroslav Vojtěchovský ◽

Jindřich Hašek ◽

Stanislav Nešpůrek ◽

Mojmír Adamec

Keyword(s):

Solid State ◽

Uv Light ◽

Order Reaction ◽

Boat Conformation ◽

X Rays ◽

Exponential Time ◽

First Order ◽

Double Plane ◽

The Mean ◽

Independent Molecules

2,4,4,6-Tetraphenyl-4H-thiopyran, C29H22S, orthorhombic, Pna21, a = 17.980(4), b = 6.956(2), c = 34.562(11) Å, V = 4323(2) Å3, Z = 8, Dx = 1.237 g cm-3, F(000) = 1696, λ(CuKα) = 1.54184 A, μ = 1.372 mm-2, T = 294 K. The final R was 0.050 for the unique set of 3103 observed reflections. The central 4H-thiopyran ring forms a boat conformation for both symmetrically independent molecules with average boat angles 4.4(3) and 6.8(3)° at S and C(sp3), respectively. The mean planes of phenyls at the position 2 and 6 are turned from the double plane of 4H-thiopyran by 42.5(5) and 35.8(3)°, respectively. The investigated material undergoes a photochromic change in the solid state after irradiation with UV light or X-rays. The maximum of the new absorption band is situated at 564 nm. The non-exponential time dependence of photochromic bleaching is analysed in terms of a dispersive first-order reaction.

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Technical note: Inherent benchmark or not? Comparing Nash–Sutcliffe and Kling–Gupta efficiency scores

Hydrology and Earth System Sciences ◽

10.5194/hess-23-4323-2019 ◽

2019 ◽

Vol 23 (10) ◽

pp. 4323-4331 ◽

Cited By ~ 60

Author(s):

Wouter J. M. Knoben ◽

Jim E. Freer ◽

Ross A. Woods

Keyword(s):

Time Series ◽

Coefficient Of Variation ◽

Ad Hoc ◽

Model Performance ◽

Technical Note ◽

Mean Flow ◽

Model Adequacy ◽

Robust Model ◽

The Mean ◽

Adequacy Assessment

Abstract. A traditional metric used in hydrology to summarize model performance is the Nash–Sutcliffe efficiency (NSE). Increasingly an alternative metric, the Kling–Gupta efficiency (KGE), is used instead. When NSE is used, NSE = 0 corresponds to using the mean flow as a benchmark predictor. The same reasoning is applied in various studies that use KGE as a metric: negative KGE values are viewed as bad model performance, and only positive values are seen as good model performance. Here we show that using the mean flow as a predictor does not result in KGE = 0, but instead KGE =1-√2≈-0.41. Thus, KGE values greater than −0.41 indicate that a model improves upon the mean flow benchmark – even if the model's KGE value is negative. NSE and KGE values cannot be directly compared, because their relationship is non-unique and depends in part on the coefficient of variation of the observed time series. Therefore, modellers who use the KGE metric should not let their understanding of NSE values guide them in interpreting KGE values and instead develop new understanding based on the constitutive parts of the KGE metric and the explicit use of benchmark values to compare KGE scores against. More generally, a strong case can be made for moving away from ad hoc use of aggregated efficiency metrics and towards a framework based on purpose-dependent evaluation metrics and benchmarks that allows for more robust model adequacy assessment.

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