Fields, scalar

The earliest appearance of the nonrelativistic continuity equation is due to Schrodinger himself [2,319], obtained from his time-dependent wave equation. A relativistic continuity equation (appropriate to a scalar field and formulated in terms of the field amplitudes) was found by Gordon [320]. The continuity equation for an electron in the relativistic Dirac theory [134,321] has the well-known form [322] ... 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nomelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that ate nomrally below the relativistic scale, the Berry phase obtained from the Schrddinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

The representation of molecular properties on molecular surfaces is only possible with values based on scalar fields. If vector fields, such as the electric fields of molecules, or potential directions of hydrogen bridge bonding, need to be visualized, other methods of representation must be applied. Generally, directed properties are displayed by spatially oriented cones or by field lines. 

Suppose that f(x, y, z) is a scalar field, and we wish to investigate how / changes between the points r and r + dr. Here... 

Just to remind you, the electron density and therefore the exchange potential are both scalar fields they vary depending on the position in space r. We often refer to models that make use of such exchange potentials as local density models. The disagreement between Slater s and Dirac s numerical coefficients was quickly resolved, and authors began to write the exchange potential as... 

Scalar Fields Consider a continuous field theory in Euclidean space with action... 

Lee next generalizes discrete quantum mechanics to the case of a massless scalar field interacting with an arbitrary external current J ([tdlee85a], [tdlee85b]). 

The solution of a pure convection equation for a scalar field O ... 

The electron density of a molecule in the presence of electric perturbation is a scalar field with perturbation expansion [6], [11]... 

As was pointed out in the Chapter 1, it is very useful to study a scalar field with the help of equipotential or level surfaces. At each point of such a surface the potential is constant, and correspondingly its equation is... 

From the physical point of view it is obvious that there is a relationship between the distribution of density of a fluid and the geometric properties of the scalar field [/.To illustrate this we will proceed from the equation of motion... 

Equation (3.6) defines a scalar field, the value of which at any point in the simulation box is the curvature of the stream line passing through this point and can also be reformulated to avoid the use of the parameter s ... 

The model system is a periodic box of arbitrary unit side length. A linear cutoff N = 8 in the frequency spectrum of the Fourier decomposition corresponds to a minimal characteristic length A = 0.125 for the scalar fields investigated systems have goal curvatures Co chosen from the set 0.1,0.2,0.5,1,5,10. ... 

Volume rendering is a technique for displaying a sampled 3D scalar field directly, without first fitting geometric primitives to the samples. It is a recon-... 

The term scalar field is used to describe a region of space in which a scalar function is associated with each point. If there is a vector quantity specified at each point, the points and vectors constitute a vector field. 

The Laplacian is constructed from second partial derivatives, so it is essentially a measure of the curvature of the function in three dimensions (Chapter 6). The Laplacian of any scalar field shows where the field is locally concentrated or depleted. The Laplacian has a negative value wherever the scalar field is locally concentrated and a positive value where it is locally depleted. The Laplacian of the electron density, p, shows where the electron density is locally concentrated or depleted. To understand this, we first look carefully at a onedimensional function and its first and second derivatives. 

Quantum mechanics allows the determination of the probability of finding an electron in an infinitesimal volume surrounding any particular point in space (x,j,z) that is, the probability density at this point. Since we can assign a probability density to any point in space, the probability density defines a scalar field, which is known as the probability density distribution. When the probability density distribution is multiplied by the total number of electrons in the molecule,... 

The main idea of the simplex decomposition method is to divide the lattice, on which the scalar field is specified, into small subunits and approximate the... 

Producing a reasonably good accuracy for analytically defined surfaces, this scheme of calculation is very inaccurate when the field is specified by the discrete set of values (the lattice scalar field). The surface in this case is located between the lattice sites of different signs. The first, second, and mixed derivatives can be evaluated numerically by using some finite difference schemes, which normally results in poor accuracy for discrete lattices. In addition, the triangulation of the surface is necessary in order to compute the integral in Eq. (8) or calculate the total surface area S. That makes this method very inefficient on a lattice in comparison to the other methods. 

In order to characterize quantitatively the polydisperse morphology, the shape and the size distribution functions are constructed. The size distribution function gives the probability to find a droplet of a given area (or volume), while the shape distribution function specified the probability to find a droplet of given compactness. The separation of the disconnected objects has to be performed in order to collect the data for such statistics. It is sometimes convenient to use the quantity v1/3 = [Kiropiet/ ]1 3 as a dimensionless measure of the droplet size. Each droplet itself can be further analyzed by calculating the mass center and principal inertia momenta from the scalar field distribution inside the droplet [110]. These data describe the droplet anisotropy. 

In order to study the connectivity of the anisotropic morphology, the 2D cuts of the 3D scalar field can be made at different angles to, for example, an XY plane. The 2D Euler characteristic can be calculated for each angle of the sectioning plan, and the anisotropic connectivity can be specified in this way. 

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