Kinetic Energy Tensor Research Articles

A local tensor that unifies kinetic energy density and vorticity in density functional theory

We present a kinetic energy tensor that unifies a scalar kinetic energy density commonly used in meta-generalized gradient approximation functionals and the vorticity density that appears in paramagnetic current-density-functional theory. Both types of functionals can thus be subsumed as special cases of a novel functional form that is naturally placed on the third rung of Jacob's ladder. Moreover, the kinetic energy tensor is related to the exchange hole curvature, is gauge invariant, and has very clearcut N-representability conditions. The latter conditions enable the definition of an effective number of non-negligible orbitals. Whereas quantities such as the electron localization function can discriminate effective one-orbital regions from other regions, the present kinetic energy tensor can discriminate between one-, two-, three-, and four-or-more orbital regions.

Study of the kinetic energy densities of electrons as applied to quantum dots in a magnetic field

There are three expressions for the kinetic energy density t(r) expressed in terms of its quantal source, the single‐particle density matrix: t A(r), the integrand of the kinetic energy expectation value; t B(r), the trace of the kinetic energy tensor; t C(r), a virial form in terms of the ‘classical’ kinetic field. These kinetic energy densities are studied by application to ‘artificial atoms‘ or quantum dots in a magnetic field in a ground and excited singlet state. A comparison with the densities for natural atoms and molecules in their ground state is made. The near nucleus structure of these densities for natural atoms is explained. We suggest that in theoretical frameworks which employ the kinetic energy density such as molecular fragmentation, density functional theory, and information‐entropic theories, one use all three expressions on application to quantum dots, and the virial expression for natural atoms and molecules. New physics could thereby be gleaned.

Correction to "Direct Measurements of Quantum Kinetic Energy Tensor in Stable and Metastable Water near the Triple Point: An Experimental Benchmark".

This study presents the first direct and quantitative measurement of the nuclear momentum distribution anisotropy and the quantum kinetic energy tensor in stable and metastable (supercooled) water near its triple point, using deep inelastic neutron scattering (DINS). From the experimental spectra, accurate line shapes of the hydrogen momentum distributions are derived using an anisotropic Gaussian and a model-independent framework. The experimental results, benchmarked with those obtained for the solid phase, provide the state of the art directional values of the hydrogen mean kinetic energy in metastable water. The determinations of the direction kinetic energies in the supercooled phase, provide accurate and quantitative measurements of these dynamical observables in metastable and stable phases, that is, key insight in the physical mechanisms of the hydrogen quantum state in both disordered and polycrystalline systems. The remarkable findings of this study establish novel insight into further expand th...

On some controversy regarding νOH assignments in CsH2PO4

Spectroscopic methods such as Raman scattering and IR absorption, as well as structural methods such as x-ray and n-diffraction, offer rather ambiguous interpretation of the high-frequency phonon spectrum of CsH2PO4 (CDP), especially regarding the proper assignment of proton vibrational modes. Empirical lattice dynamics (LD) simulations of the proton vibrational density of states ((H-VDOS)) in CDP also reveal a discrepancy in the frequency assignments of the OH stretching modes of its two non-equivalent hydrogen bonds. This may be resolved by accounting for the LD simulated directional (H-VDOS) along the three Cartesian axes of CDP, from which the corresponding anisotropy of the proton kinetic energy tensor can be deduced. The results may then be tested by a simple deep inelastic neutron scattering (DINS) measurement on a single crystal of para-electric CDP at room temperature.

Nonzero Ideal Gas Contribution to the Surface Tension of Water.

Surface tension, the tendency of fluid interfaces to behave elastically and minimize their surface, is routinely calculated as the difference between the lateral and normal components of the pressure or, invoking isotropy in momentum space, of the virial tensor. Here we show that the anisotropy of the kinetic energy tensor close to a liquid-vapor interface can be responsible for a large part of its surface tension (about 15% for water, independent from temperature).

Direct Measurements of Quantum Kinetic Energy Tensor in Stable and Metastable Water near the Triple Point: An Experimental Benchmark.

This study presents the first direct and quantitative measurement of the nuclear momentum distribution anisotropy and the quantum kinetic energy tensor in stable and metastable (supercooled) water near its triple point, using deep inelastic neutron scattering (DINS). From the experimental spectra, accurate line shapes of the hydrogen momentum distributions are derived using an anisotropic Gaussian and a model-independent framework. The experimental results, benchmarked with those obtained for the solid phase, provide the state of the art directional values of the hydrogen mean kinetic energy in metastable water. The determinations of the direction kinetic energies in the supercooled phase, provide accurate and quantitative measurements of these dynamical observables in metastable and stable phases, that is, key insight in the physical mechanisms of the hydrogen quantum state in both disordered and polycrystalline systems. The remarkable findings of this study establish novel insight into further expand the capacity and accuracy of DINS investigations of the nuclear quantum effects in water and represent reference experimental values for theoretical investigations.

The Harmonic Picture of Nuclear Mean Kinetic Energies in Heavy Water

This paper presents a study of the total mean kinetic energy, ⟨EK⟩, and of individual projections along a given molecular axis, ⟨EK⟩α, for D and O nuclei in D2O, derived using a harmonic model. Our theoretical approach assumes decoupling amongst translational, rotational and vibrational modes. Resulting values of these dynamical quantities are discussed in terms of the anisotropy of the quantum kinetic energy tensor, its relation to the local potential, and deviations from the hypothesis of harmonicity and mode decoupling. Results are compared with corresponding quantities obtained from Deep Inelastic Neutron Scattering experiments performed on liquid and solid D2O, where the short-time dynamics and local environment of D and O atoms were probed. The present study confirms an overall picture where even small changes in the short-range environment of D and O nuclei have a strong influence on the quantum behaviour of heavy water.

Direct Measurement of Competing Quantum Effects on the Kinetic Energy of Heavy Water upon Melting

Even at room temperature, quantum mechanics plays a major role in determining the quantitative behavior of light nuclei, changing significantly the values of physical properties such as the heat capacity. However, other observables appear to be only weakly affected by nuclear quantum effects (NQEs); for instance, the melting temperatures of light and heavy water differ by less than 4 K. Recent theoretical work has attributed this to a competition between intra- and intermolecular NQEs, which can be separated by computing the anisotropy of the quantum kinetic energy tensor. The principal values of this tensor change in opposite directions when ice melts, leading to a very small net quantum mechanical effect on the melting point. This Letter presents the first direct experimental observation of this phenomenon, achieved by measuring the deuterium momentum distributions n(p) in heavy water and ice using deep inelastic neutron scattering (DINS) and resolving their anisotropy. Results from the experiments, supplemented by a theoretical analysis, show that the anisotropy of the quantum kinetic energy tensor can also be captured for heavier atoms such as oxygen. (The iceberg image in the Table of Contents and Abstract graphics was used with permission of the NOAA’s National Ocean Service, 2012 (http://commons.wikimedia.org/wiki/File:Iceberg_-_NOAA.jpg).)

Competition between fusion-evaporation and multifragmentation in central collisions in 58Ni + 48Ca reaction at 25 AMeV

The experimental data concerning the 58Ni+48Ca reaction at Elab(Ni)=25A MeV, collected by the CHIMERA 4π device, have been analyzed in order to investigate the competition among different reaction mechanisms for central collisions in the Fermi energy domain. The method adopted to select such central collisions, by means of construction of the kinetic energy tensor and, consequently, the evaluation of the "flow angle" event by event is presented. Some global variables, able to characterize the pattern of central collisions, have been constructed, and the main features of the reaction products are analyzed in terms of some observable, like mass and velocity distributions, as well as their correlations. Much emphasis was devoted to the competition between fusion-evaporation processes with consequent identification of a heavy residue and multifragmentation of a well defined (if any) transient nuclear system formed in central collisions.

On chaos in the oscillations of galaxies

This paper describes a search for chaotic behaviour in a low-dimensional model of the oscillations of a galaxy. The governing equations of the model are the complete set of moment equations of the second order associated with the collisionless Boltzmann equation. The moment equations reduce to the tensor virial equations and equations governing the evolution of the kinetic energy tensor of the system. The classical integrals of motion of a gravitational many-body system supplement the governing equations. Closure of the system of moment equations requires a specification of the density distribution in the system and the velocity field describing the mean motions of the stars. In this paper, we close the system for a galaxy possessing triplanar symmetry in the case that the planes of symmetry remain fixed in an inertial frame of reference. We model the galaxy as a heterogeneous ellipsoid with an arbitrary stratification of the density distribution, and we represent the mean motions of the stars in terms of a velocity field that sustains that density distribution consistently with the equation of continuity. The evolution of the model is quasi-homologous in the sense that the semi-axes of an ellipsoid characterizing the size and shape of the galaxy are functions of the time, whereas the stratification of the density does not change. With the aid of additional, non-classical integrals of the motion, which were discovered by Som Sunder & Kochhar, we eliminate the kinetic energy tensor from the virial equations and reduce the virial equations to a closed system of equations of motion that govern the semi-axes of the characteristic ellipsoid. Those equations of motion represent the dynamics of a galaxy as the dynamics of an oscillator with just three degrees of freedom, and they admit of a Hamiltonian formulation. We investigate chaotic behaviour in the model for axisymmetric galaxies. The oscillator that represents the dynamics of an axisymmetric galaxy has only two degrees of freedom, and the constancy of the energy integral enables an investigation of the motion with the aid of Poincare return maps. Our criteria for chaotic behaviour are dissolution of invariant curves in the return maps and, more generally, sensitivity of the motion to small changes in the initial conditions. We survey axisymmetric oscillations of a spherically symmetric model and of oblate and prolate, axisymmetric models with the geometries of an elliptical galaxy of Hubble type E5. Oscillations with excitation energies up to the order of 40 per cent of the binding energy of an equilibrium configuration are found to be regular. Substantial families of chaotic oscillations appear at excitation energies of the order of 40-50 per cent of the binding energy. Lyapunov times of these chaotic oscillations are generally in the range of 50-150 dynamical units of time. For galaxies with typical sizes and masses, the Lyapunov times would be of the order of 1.5-4.5 Gyr.

How Ambiguous Is the Local Kinetic Energy?†

The local kinetic energy and the closely related local electronic stress tensor are commonly used to elucidate chemical bonding patterns, especially for covalent bonds. We use three different approaches-transformation properties of the stress tensor, quasiprobability distributions, and the virial theorem from density-functional theory-to clarify the inherent ambiguity in these quantities, discussing the implications for analyses based on the local kinetic energy and stress tensor. An expansive-but not universal-family of local kinetic energy forms that includes the most common choices and is suitable for both chemical-bonding and atoms-in-molecule analysis is derived. A family of local electronic stress tensors is also derived. Several local kinetic energy functions that are mathematically justified, but unlikely to be conceptually useful, are derived. The implications of these forms for atoms-in-molecule analysis are discussed.

Implicit density-functional theory

A fermion ground state energy functional is set up in terms of particle density, relative pair density, and kinetic energy tensor density. It satisfies a minimum principle if constrained by a complete set of compatibility conditions. A partial set, which thereby results in a lower bound energy under minimization, is obtained from the solution of model systems, as well as a small number of exact sum rules. Prototypical application is made to several one-dimensional spinless noninteracting models. The effectiveness of ``atomic'' constraints on model ``molecules'' is observed, as well as the structure of systems with only finitely many bound states.

Partial stresses in heterogeneous media by a direct statistical approach

The total stress tensor in a structured or non-structured medium can be obtained by a direct statistical approach using the generalized virial theorem, without any reference to a potential function, as soon as positions, velocities and interactions of the particles are given by Molecular Dynamics. However, as shown here, it would be wrong to apply these results to a given class of particles in an heterogeneous medium without adding a cross internal virial tensor to the self internal virial tensor and the partial kinetic energy tensor relative to this class of particles. To cite this article: P. Jouanna, L. Pèdesseau, C. R. Mecanique 332 (2004).

Effect of Allowance for Small-Scale Perturbations of the Carrier Phase on Properties of the System of Equations of a Two-Fluid Flow with Incompressible Phases

The effect of the fluctuating components of kinetic energy and stress tensor of the carrier phase, which were previously obtained by the cell technique, on the properties of the system of equations of a gas–liquid flow with incompressible phases is considered. It is shown that the characteristic properties of this system and also the possibility of modeling the Zuber–Findlay empirical relation are determined by the tensor of fluctuating stresses of the carrier phase.

Experimental techniques for highly resolved measurements of rotation, strain and dissipation-rate tensors in turbulent flows

A new hot-wire probe with nine independently operating wires was designed, developed and validated in a two-dimensional turbulent boundary layer. The probe is capable of measuring all the elements of the gradient velocity tensor, the rate-of-strain tensor, the rate-of-rotation tensor/vorticity vector and the rate-of-dissipation of kinetic energy tensor with a spatial resolution of seven Kolmogorov viscous length scales. Measurements obtained with the probe were extensively compared with other experimental data and direct numerical simulation results. Scaling of all results with inner wall variables was unsuccessful in collapsing all the data onto one curve. Several of the invariants considered exhibit a very strong intermittent behaviour which is characterized by large amplitude bursts which can be of the order of ten RMS values. It appears that the fluctuating strain field dominates pressure fluctuations over enstrophy.

Dynamical model for turbulence. III. Numerical results

We present the numerical solutions of the equations for turbulence developed in papers I and II. (1) The model predicts the Kolmogorov law and Ko=5/3, in accord with recent data; (2) in the inertial-conductive regime, the model predicts the Corrsin spectrum for the temperature variance and the Batchelor constant Ba=σt Ko, where σt=0.72 is the turbulent Prandtl number; (3) the predicted energy spectrum in the dissipation region is in agreement with recent laboratory measurements; (4) in the inertial-convective region, the temperature variance spectrum is closer to the spectrum (−11/3) obtained by LES when the velocity field is rapidly stirred at all scales than(−17/3), which holds when the velocity field is frozen in time and has a Gaussian statistics; (5) for freely decaying turbulence, the power law spectra for energy and temperature variance, as well as the velocity and temperature integral scales, agree with the most recent LES data; (6) after a few evolutionary times, the skewness S reaches S=0.5, in accord with a variety of data; (7) for shear-driven flows, the Reynolds stress spectrum E12(k) has an inertial regime with a power −7/3, in accord with recent data; (8) for two shear-driven flows, plane strain and axisymmetric contraction, turbulent kinetic energy, Reynolds stress tensor, and dissipation rate εij versus time compare very well with DNS data; (9) the slow and rapid parts of the pressure–strain correlation tensor compare with DNS data better than with the three most widely used phenomenological models. The rapid parts are also in excellent agreement with the DNS data; (10) for homogeneous shear, turbulent kinetic energy and Reynolds stress tensor versus time match quite closely LES data. We recall that the model does not contain any free parameters.

Exact theorems concerning noninteracting kinetic energy density functional in D dimensions and their implications for gradient expansions

AbstractThe focal point of the present work is the single‐particle kinetic energy density tensor in D dimensions. This quantity enters both differential and various integral forms of the virial theorems, which are again set up in D dimensions. Major new results lie in (i) demonstrating that, by one‐dimensional quadrature, it is possible to construct the Pauli potential directly from the kinetic energy tensor, without the need for functional differentiation and (ii) generating the gradient expansion for the kinetic energy tensor, in D dimensions. © 1995 John Wiley & Sons, Inc.

Slater Sum and Kinetic Energy Tensor in Some Simple Inhomogeneous Electron Liquids

Starting from the Bloch equation for the canonical density matrix, the differential equation satisfied by the Slater sum is derived for some simple inhomogeneous electron liquids. The relation of this derivation to the differential form of the virial theorem, and in particular to the kinetic energy tensor, is finally discussed.

Momentum-space properties of N2.

Properties that depend on the anisotropy of the electron momentum distribution of ${\mathrm{N}}_{2}$(X ${\mathrm{}}^{1}$${\mathrm{\ensuremath{\Sigma}}}_{\mathrm{g}}^{+}$) have not been determined accurately even by a fairly sophisticated calculation [Phys. Rev. A 34, 4695 (1986)] using a configuration-interaction wave function. This paper reports calculations based on a series of wave functions designed to assess the sensitivity of one-electron momentum-space properties of ${\mathrm{N}}_{2}$ to variations in the treatment of electron correlation and the choice of basis sets of Slater-type functions. The most elaborate of these wave functions, which consists of 169 650 configurations, provides satisfactorily converged predictions of a wide variety of isotropic momentum-space properties as well as directional Compton profiles. However, the anisotropy of the kinetic energy tensor turns out to be so small in magnitude that even its sign remains uncertain.

The Dynamical Evolution of a Star Cluster

Using the second order tensor virial equations and the equation for the rate of change of the kinetic energy tensor, we follow the dynamical evolution of a star cluster with anisotropic velocity distribution. We show that the cluster executes finite amplitude oscillations both in size and eccentricity. However, unlike in the isotropic case, the amplitude now depends on the initial eccentricity.