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Field Equations (field + equation)
Selected AbstractsQuantitative Phase Field Modeling of Precipitation Processes,ADVANCED ENGINEERING MATERIALS, Issue 12 2006Q. Bronchard Phase Field modelling of microstructural evolution in alloys has already a long and successful history. One of the basics of the theory is the introduction of continuous fields (concentration, long-range order parameters) that describe the local state of the alloy. These fields have a meaning only at a mesoscopic scale. One consequence is that we can treat much larger systems than with microscopic methods such as Monte Carlo or molecular dynamics simulations. The aim of this work is to precisely analyse the status of the mesoscopic free energy densities that are used in Phase Field theories and, simultaneously, to clarify the form that the Phase Field equations should adopt. [source] Role of the one-body Jastrow factor in the transcorrelated self-consistent field equationINTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 7 2006Naoto Umezawa Abstract The one-body Jastrow factor has been introduced into the transcorrelated variational Monte Carlo (TC-VMC) method. The principal role of the one-body Jastrow factor in the Jastrow,Slater-type wave function is to prevent an unfavorable effect of the two-body Jastrow factor that alters the charge density. In the TC-VMC method, since the one-body orbitals are optimized by the transcorrelated self-consistent field (TC-SCF) equations, which take into account the electron,electron correlation interactions originating from the two-body Jastrow factor, the unfavorable effect of altering charge density can be avoided without introducing the one-body Jastrow factor. However, it is found that it is still better to incorporate a one-body Jastrow factor into the TC-VMC method for the practical effect of reducing numerical errors caused by the Monte Carlo sampling and the re-weighting calculations in solving the TC-SCF equations. Moreover, since the one-body Jastrow function adopted in the present work is constructed from the two-body Jastrow factor without increasing any variational parameter, the computational cost is not significantly increased. The preferable effect of the use of the one-body Jastrow factor in the TC-VMC calculation is demonstrated for atoms. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006 [source] The dual curvature tensors and dynamics of gravitomagnetic matterANNALEN DER PHYSIK, Issue 9 2004J.Q. Shen Abstract Gravitomagnetic charge that can also be referred to as the dual mass or magnetic mass is the topological charge in gravity theory. A gravitomagnetic monopole at rest can produce a stationary gravitomagnetic field. Due to the topological nature of gravitomagnetic charge, the metric of spacetime where the gravitomagnetic matter is present will be nonanalytic. In this paper both the dual curvature tensors (which can characterize the dynamics of gravitational charge/monopoles) and the antisymmetric gravitational field equation of gravitomagnetic matter are presented. We consider and discuss the mathematical formulation and physical properties of the dual curvature tensors and scalar, antisymmetric source tensors, dual spin connection (including the low-motion weak-field approximation), dual vierbein field as well as dual current densities of gravitomagnetic charge. It is shown that the dynamics of gravitomagnetic charge can be founded within the framework of the above dual quantities. In addition, the duality relationship in the dynamical theories between the gravitomagnetic charge (dual mass) and the gravitoelectric charge (mass) is also taken into account in the present paper. [source] Optimal separation times for electrical field flow fractionation with Couette flowsELECTROPHORESIS, Issue 20 2008Jennifer Pascal Abstract The prediction of optimal times of separation as a function of the applied electrical field and cation valence have been studied for the case of field flow fractionation [Martin M., Giddings J. C., J. Phys. Chem. 1981, 85, 727] with charged solutes. These predictions can be very useful to a priori design or identify optimal operating conditions for a Couette-based device for field flow fractionation when the orthogonal field is an electrical field. Mathematically friendly relationships are obtained by applying the method of spatial averaging to the solute species continuity equation; this is accomplished after the role of the capillary geometrical dimensions on the applied electrical field equations has been assessed [Oyanader M. A., Arce P., Electrophoresis 2005; 26, 2857]. Moreover, explicit analytical expressions are derived for the effective parameters, i.e. diffusivity and convective velocity as functions of the applied (orthogonal) electrical field. These effective transport parameters are used to study the effect of the cation valence of the solutes and of the magnitude of the applied orthogonal electrical field on the values of the optimal time of separation. These parameters play a significant role in controlling the optimal separation time, leading to a family of minimum values, for particular magnitudes of the applied orthogonal electrical field. [source] Role of geometrical dimensions in electrophoresis applications with orthogonal fieldsELECTROPHORESIS, Issue 15 2005Mario A. Oyanader Abstract The role of geometrical dimensions in electrophoresis applications with axial and orthogonal (secondary) electric fields is investigated using a rectangular capillary channel. In particular, the role of the applied orthogonal electrical field in controlling key parameters involved in the effective diffusivity and effective (axial) velocity of the solute is identified. Such mathematically friendly relationships are obtained by applying the method of spatial averaging to the solute species continuity equation; this is accomplished after the role of the capillary geometrical dimensions on the applied electrical field equations has been studied. Moreover, explicit analytical expressions are derived for the effective parameters, i.e., diffusivity and convective velocity as functions of the applied (orthogonal) electric field. Previous attempts (see Sauer et al., 1995) have only led to equations for these parameters that require numerical solution and, therefore, limited the use of such results to practical applications. These may include, for example, the design of separation processes as well as environmental applications such as soil reclamation and wastewater treatment. An illustration of how a secondary electrical field can aid in reducing the optimal separation time is included. [source] Coherent state path integral and super-symmetry for condensates composed of bosonic and fermionic atomsFORTSCHRITTE DER PHYSIK/PROGRESS OF PHYSICS, Issue 9-10 2007B. Mieck Abstract A super-symmetric coherent state path integral on the Keldysh time contour is considered for bosonic and fermionic atoms which interact among each other with a common short-ranged two-body potential. We investigate the symmetries of Bose-Einstein condensation for the equivalent bosonic and fermionic constituents with the same interaction potential so that a super-symmetry results between the bosonic and fermionic components of super-fields. Apart from the super-unitary invariance U(L | S) of the density terms, we specialize on the examination of super-symmetries for pair condensate terms. Effective equations are derived for anomalous terms which are related to the molecular- and BCS- condensate pairs. A Hubbard-Stratonovich transformation from ,Nambu'-doubled super-fields leads to a generating function with super-matrices for the self-energy whose manifold is given by the orthosympletic super-group Osp(S,S | 2L). A nonlinear sigma model follows from the spontaneous breaking of the ortho-symplectic super-group Osp(S,S | 2L) to the coset decomposition Osp(S,S | 2L) \ U(L | S), U(L | S). The invariant subgroup U(L | S) for the vacuum or background fields is represented by the density terms in the self-energy whereas the super-matrices on the coset space Osp(S,S | 2L) \ U(L | S) describe the anomalous molecular and BCS- pair condensate terms. A change of integration measure is performed for the coset decomposition Osp(S,S | 2L) \ U(L | S) , U(L | S), including a separation of density and anomalous parts of the self-energy with a gradient expansion for the Goldstone modes. The independent anomalous fields in the actions can be transformed by the inverse square root of the metric tensor of Osp(S,S | 2L) \ U(L | S) so that the non-Euclidean integration measure with super-Jacobi-determinant can be removed from the coherent state path integral and Gaussian-like integrations remain. The variations of the independent coset fields in the effective actions result in classical field equations for a nonlinear sigma model with the anomalous terms. The dynamics of the eigenvalues of the coset matrices is determined by Sine-Gordon equations which have a similar meaning for the dynamics of the molecular- and BCS-pair condensates as the Gross-Pitaevskii equation for the coherent wave function in BEC phenomena. [source] The spinorial method of classifying supersymmetric backgroundsFORTSCHRITTE DER PHYSIK/PROGRESS OF PHYSICS, Issue 5-6 2006U. Gran Abstract We review how the classification of all supersymmetric backgrounds of IIB supergravity can be reduced to the evaluation of the Killing spinor equations and their integrability conditions, which contain the field equations, on five types of spinors. This is an extension of the work [hep-th/0503046] to IIB supergravity. By using the explicit expressions for the Killing spinor equations evaluated on the five types of spinors the Killing spinor equations become a linear system in terms of the fluxes, the geometry and the spacetime derivatives of the functions that determine the Killing spinors. This system can be solved to express the fluxes in terms of the geometry and to determine the conditions on the geometry of any supersymmetric background. Similarly, the integrability conditions of the Killing spinor equations are turned into a linear system. This can be used to determine the field equations that are implied by the Killing spinor equations for any supersymmetric background. These linear systems simplify for generic backgrounds with maximal and half-maximal number of H -invariant Killing spinors, H , Spin(9,1). In the maximal case, the Killing spinor equations factorise, whereas in the half-maximal case they do not. [source] M-theory and gauged supergravities,FORTSCHRITTE DER PHYSIK/PROGRESS OF PHYSICS, Issue 2 2005D. Roest Abstract We present a pedagogical discussion of the emergence of gauged supergravities from M-theory. First, a review of maximal supergravity and its global symmetries and supersymmetric solutions is given. Next, different procedures of dimensional reduction are explained: reductions over a torus, a group manifold and a coset manifold and reductions with a twist. Emphasis is placed on the consistency of the truncations, the resulting gaugings and the possibility to generate field equations without an action. Using these techniques, we construct a number of gauged maximal supergravities in diverse dimensions with a string or M-theory origin. One class consists of the CSO gaugings, which comprise the analytic continuations and group contractions of SO(n) gaugings. We construct the corresponding half-supersymmetric domain walls and discuss their uplift to D- and M-brane distributions. Furthermore, a number of gauged maximal supergravities are constructed that do not have an action. [source] Time-domain approach to linearized rotational response of a three-dimensional viscoelastic earth model induced by glacial-isostatic adjustment: I. Inertia-tensor perturbationsGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 2 2005k Martinec SUMMARY For a spherically symmetric viscoelastic earth model, the movement of the rotation vector due to surface and internal mass redistribution during the Pleistocene glaciation cycle has conventionally been computed in the Laplace-transform domain. The method involves multiplication of the Laplace transforms of the second-degree surface-load and tidal-load Love numbers with the time evolution of the surface load followed by inverse Laplace transformation into the time domain. The recently developed spectral finite-element method solves the field equations governing glacial-isostatic adjustment (GIA) directly in the time domain and, thus, eliminates the need of applying the Laplace-domain method. The new method offers the possibility to model the GIA-induced rotational response of the Earth by time integration of the linearized Liouville equation. The theory presented here derives the temporal perturbation of the inertia tensor, required to be specified in the Liouville equation, from time variations of the second-degree gravitational-potential coefficients by the MacCullagh's formulae. This extends the conventional approach based on the second-degree load Love numbers to general 3-D viscoelastic earth models. The verification of the theory of the GIA-induced rotational response of the Earth is performed by using two alternative approaches of computing the perturbation of the inertia tensor: a direct numerical integration and the Laplace-domain method. The time-domain solution of both the GIA and the induced rotational response of the Earth is readily combined with a time-domain solution of the sea level equation with a time-varying shoreline geometry. In a follow-up paper, we derive the theory for the case when GIA-induced perturbations in the centrifugal force affect not only the distribution of sea water, but also deformations and gravitational-potential perturbations of the Earth. [source] Hierarchical model of the population dynamics of hippocampal dentate granule cellsHIPPOCAMPUS, Issue 5 2002G.A. Chauvet Abstract A hierarchical modeling approach is used as the basis for a mathematical representation of the population activity of hippocampal dentate granule cells. Using neural field equations, the variation in time and space of dentate granule cell activity is derived from the summed synaptic potential and summed action potential responses of a population of granule cells evoked by monosynaptic excitatory input from entorhinal cortical afferents. In this formulation of the problem, we have considered a two-level hierarchy: the synapses of entorhinal cortical axons define the first level of organization, and dentate granule cells, which include these synapses, define the second, higher level of organization. The model is specified by two state field variables, for membrane potential and for synaptic efficacy, respectively, with both evolving according to different time scales. The two state field variables introduce new parameters, physiological and anatomical, which characterize the dentate from the point of view of neuronal and synaptic populations: (1) a set of geometrical constraints corresponding to the morphological properties of granule cells and anatomical characteristics of entorhinal-dentate connections; and (2) a set of neuronal parameters corresponding to physiological mechanisms. Assuming no interaction between granule cells, i.e., neither ephaptic nor synaptic coupling, the model is shown to be mathematically tractable and allows solution of the field equations leading to the determination of activity. This treatment leads to the definition of two state variables, volume of stimulated synapses and firing time, which describe observed activity. Numerical simulations are used to investigate the populational characterization of the dentate by individual parameters: (1) the relationship between the conditions of stimulation of active perforant path fibers, e.g., stimulating intensity, and activity in the granule cell layer; and (2) the influence of geometry on the generation of activity, i.e., the influence of neuron density and synaptic density-connectivity. As an example application of the model, the granule cell population spike is reconstructed and compared with experimental data. Hippocampus 2002;12:698,712. © 2002 Wiley-Liss, Inc. [source] Solid,liquid,air coupling in multiphase porous mediaINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 3 2003Lyesse Laloui Abstract This paper addresses various issues concerning the modelling of solid,liquid,air coupling in multiphase porous media with an application to unsaturated soils. General considerations based on thermodynamics permit the derivation and discussion of the general form of field equations; two cases are considered: a three phase porous material with solid, liquid and gas, and a two phase porous material with solid, liquid and empty space. Emphasis is placed on the presentation of differences in the formulation and on the role of the gas phase. The finite element method is used for the discrete approximation of the partial differential equations governing the problem. The two formulations are then analysed with respect to a documented drainage experiment carried out by the authors. The merits and shortcomings of the two approaches are shown. Copyright © 2003 John Wiley & Sons, Ltd. [source] Generating potentials via difference equationsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2006S. D. Maharaj Abstract The condition for pressure isotropy, for spherically symmetric gravitational fields with charged and uncharged matter, is reduced to a recurrence equation with variable, rational coefficients. This difference equation is solved, in general, using mathematical induction leading to an exact solution to the Einstein field equations which extends the isotropic model of John and Maharaj. The metric functions, energy density and pressure are well behaved, which suggests that this model could be used to describe a relativistic sphere. The model admits a barotropic equation of state, which approximates a polytrope close to the stellar centre. Copyright © 2006 John Wiley & Sons, Ltd. [source] New anisotropic models from isotropic solutionsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 1 2006S. D. Maharaj Abstract We establish an algorithm that produces a new solution to the Einstein field equations, with an anisotropic matter distribution, from a given seed isotropic solution. The new solution is expressed in terms of integrals of known functions, and the integration can be completed in principle. The applicability of this technique is demonstrated by generating anisotropic isothermal spheres and anisotropic constant density Schwarzschild spheres. Both of these solutions are expressed in closed form in terms of elementary functions, and this facilitates physical analysis. Copyright © 2005 John Wiley & Sons, Ltd. [source] A critical look at the kinematic-wave theory for sedimentation,consolidation processes in closed vesselsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2001R. Bürger Abstract The two-phase flow of a flocculated suspension in a closed settling vessel with inclined walls is investigated within a consistent extension of the kinematic wave theory to sedimentation processes with compression. Wall boundary conditions are used to spatially derive one-dimensional field equations for planar flows and flows which are symmetric with respect to the vertical axis. We analyse the special cases of a conical vessel and a roof-shaped vessel. The case of a small initial time and a large time for the final consolidation state leads to explicit expressions for the flow fields, which constitute an important test of the theory. The resulting initial-boundary value problems are well posed and can be solved numerically by a simple adaptation of one of the newly developed numerical schemes for strongly degenerate convection-diffusion problems. However, from a physical point of view, both the analytical and numerical results reveal a deficiency of the general field equations. In particular, the strongly reduced form of the linear momentum balance turns out to be an oversimplification. Included in our discussion as a special case are the Kynch theory and the well-known analyses of sedimentation in vessels with inclined walls within the framework of kinematic waves, which exhibit the same shortcomings. In order to formulate consistent boundary conditions for both phases in a closed vessel and in order to predict boundary layers in the presence of inclined walls, viscosity terms should be taken into account. Copyright © 2001 John Wiley & Sons, Ltd. [source] An efficient state-space ADI-PML algorithm for truncating DNG metamaterial FDTD domainsMICROWAVE AND OPTICAL TECHNOLOGY LETTERS, Issue 2 2007Omar Ramadan Abstract Efficient and unconditionally stable formulations of the anisotropic perfectly matched layer are presented for truncating double negative (DNG) metamaterial finite difference time domain (FDTD) grids. In the proposed formulations, the state-space equations are employed in the alternating direction implicit FDTD algorithm to obtain update equations for the field equations in the DNG metamaterial domains. Numerical example carried out in one-dimensional Lorentzian type DNG metamaterial domain is included to show the validity of the proposed formulations. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 494,498, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22162 [source] Five-dimensional relativity and extended elementary particlesANNALEN DER PHYSIK, Issue 6 2009G. Lessner Abstract Five-dimensional relativity, based on the 5-dimensional vacuum equations R,, = 0, is interpreted as a general relativistic field theory describing the inner structure of extended elementary particles. The most general} spherically symmetric, purely electric and asymptotically flat solution of the field equations is calculated. The mass formula of this solution depends on two parameters h > 0 and q > 0, and the mass surface over the (h-q)-plane shows two slits at h= and . These slits can be tentatively identified as hadron branch for h= and lepton branch for . In the slits hadrons and leptons with their masses and extensions can be settled in local minima. [source] On the pre-metric foundations of wave mechanics I: massless wavesANNALEN DER PHYSIK, Issue 4 2009D.H. Delphenich Abstract The mechanics of wave motion in a medium are founded in conservation laws for the physical quantities that the waves carry, combined with the constitutive laws of the medium, and define Lorentzian structures only in degenerate cases of the dispersion laws that follow from the field equations. It is suggested that the transition from wave motion to point motion is best factored into an intermediate step of extended matter motion, which then makes the dimension-codimension duality of waves and trajectories a natural consequence of the bicharacteristic (geodesic) foliation associated with the dispersion law. This process is illustrated in the conventional case of quadratic dispersion laws, as well as quartic ones, which include the Heisenberg,Euler dispersion law. It is suggested that the contributions to geodesic motion from the non-quadratic nature of a dispersion law might represent another source of quantum fluctuations about classical extremals, in addition to the diffraction effects that are left out by the geometrical optics approximation. [source] The Einstein-Elko system , Can dark matter drive inflation?ANNALEN DER PHYSIK, Issue 5-6 2007C.G. Böhmer Abstract Recently, a spin one half matter field with mass dimension one was discovered, called Elko spinors. The present work shows how to introduce these fields into a curved spacetime by the standard covariantisation scheme. After formulating the coupled Einstein-Elko field equations, the spacetime is assumed to be homogeneous and isotropic in order to simplify the resulting field equations. Analytical ghost Elko solutions are constructed which have vanishing energy-momentum tensor without and with cosmological constant. The cosmological Elko theory is finally related to the standard scalar field theory with self interaction that gives rise to inflation and it is pointed out that the Elko spinors are not only prime dark matter candidates but also prime candidates for inflation. [source] The conformal status of o = -3/2 Brans-Dicke cosmologyANNALEN DER PHYSIK, Issue 4 2007M.P. Da, browski Abstract Following recent fit of supernovae data to Brans-Dicke theory which favours the model with o = - 3/2 [1] we discuss the status of this special case of Brans-Dicke cosmology in both isotropic and anisotropic framework. It emerges that the limit o = -3/2 is consistent only with the vacuum field equations and it makes such a Brans-Dicke theory conformally invariant. Then it is an example of the conformal relativity theory which allows the invariance with respect to conformal transformations of the metric. Besides, Brans-Dicke theory with o = -3/2 gives a border between a standard scalar field model and a ghost/phantom model. In this paper we show that in o = -3/2 Brans-Dicke theory, i.e., in the conformal relativity there are no isotropic Friedmann solutions of non-zero spatial curvature except for k=-1 case. Further we show that this k=-1 case, after the conformal transformation into the Einstein frame, is just the Milne universe and, as such, it is equivalent to Minkowski spacetime. It generally means that only flat models are fully consistent with the field equations. On the other hand, it is shown explicitly that the anisotropic non-zero spatial curvature models of Kantowski-Sachs type are admissible in o = -3/2 Brans-Dicke theory. It then seems that an additional scale factor which appears in anisotropic models gives an extra deegre of freedom and makes it less restrictive than in an isotropic Friedmann case. [source] Quadratic metric-affine gravityANNALEN DER PHYSIK, Issue 4 2005D. Vassiliev Abstract We consider spacetime to be a connected real 4-manifold equipped with a Lorentzian metric and an affine connection. The 10 independent components of the (symmetric) metric tensor and the 64 connection coefficients are the unknowns of our theory. We introduce an action which is (purely) quadratic in curvature and study the resulting system of Euler,Lagrange equations. In the first part of the paper we look for Riemannian solutions, i.e. solutions whose connection is Levi-Civita. We find two classes of Riemannian solutions: 1) Einstein spaces, and 2) spacetimes with pp-wave metric of parallel Ricci curvature. We prove that for a generic quadratic action these are the only Riemannian solutions. In the second part of the paper we look for non-Riemannian solutions. We define the notion of a "Weyl pseudoinstanton" (metric compatible spacetime whose curvature is purely of Weyl type) and prove that a Weyl pseudoinstanton is a solution of our field equations. Using the pseudoinstanton approach we construct explicitly a non-Riemannian solution which is a wave of torsion in a spacetime with Minkowski metric. We discuss the possibility of using this non-Riemannian solution as a mathematical model for the neutrino. [source] Vacuum electrodynamics of accelerated systems: Nonlocal Maxwell's equationsANNALEN DER PHYSIK, Issue 10 2003B. Mashhoon Abstract The nonlocal electrodynamics of accelerated systems is discussed in connection with the development of Lorentz-invariant nonlocal field equations. Nonlocal Maxwell's equations are presented explicitly for certain linearly accelerated systems. In general, the field equations remain nonlocal even after accelerated motion has ceased. [source] | ||||||||||||||||