Two component equation

Equations (2.15) or (2.16) are the so-called Stefan-Maxwell relations for multicomponent diffusion, and we have seen that they are an almost obvious generalization of the corresponding result (2.13) for two components, once the right hand side of this has been identified physically as an inter-molecular momentum transfer rate. In the case of two components equation (2.16) degenerates to... 

Consider the union of only two components (equation 2.7-28), where x, and are each distributed as Figure 2.7-2. If there were only one bar in each histogram, the values Xj and x would be added and the probabilities multiplied as the probability of joint occurrence (equation 2.7-29). 

Instead of a two-component equation as in the non-relativistic case, for fully relativistic calculations one has to solve a four-component equation. Conceptually, fully relativistic calculations are no more complicated than non-relativistic calculations, hut they are computationally demanding, in particular, for correlated molecular relativistic calculations. Unless taken care of at the outset, spurious solutions can occur in variational four-component relativistic calculations. In practice, this problem is handled by employing kinetically balanced basis sets. The kinetic balance relation is... 

Any powder can be considered to be made up of two components, the fraction above and below a certain size and assumptions made as to the weights of the individual grains in each of the two components. Equation (2.2) may then be used to determine the sampling accuracy of a single powder. Furthermore, if the particles are counted instead of weighed, a more general equation is applicable [1] ... 

Many important applications of equations-of-state involve the study of the PVT behavior of mixtures. For example, in the general formulation of an equation-of-state for a mixture of two components, Equation 3.21 can be rewritten in the form of Equation 3.25 for the mixed system ... 

The central idea of all elimination methods for the small component is to employ relations (13) and (14) and to substitute in Eq. (5) by an expression for the large component only. This yields a two-component equation for the latter only, which can be written as... 

The Dirac equation with four spinor components demands large computational efforts to solve. Relativistic effects in electronic structure calculations are therefore usually considered by means of approximate one- or two-component equations. The approximate relativistic (also called quasi-relativistic) Hamiltonians consist of the nonrelativistic Hamiltonian augmented with additional... 

The transformation of the Dirac equation into two-component equations are also discussed in Chapter 11, whereas Chapter 12 deals with the relativistic direct perturbation theory. 

With equation (20) as starting point, the first approximation one can make in order to derive quasi-relativistic two-component equations is to assume that the upper (0 ) and the lower (0y) components are identical. Note that the ansatz... 

We have arrived at the system of three rigorous equations which allow us to solve the Dirac equation in the two-component form to an arbitrary degree of accuracy. Starting from the two-component equation... 

General three-component diffusion equations may be reduced in two ways to concern only two chemically different components. One of these ways leads to the ordinary two-component equation presented above. The other leads to equations for self diffusion of a component in a mixture with a second component (Lamm > > ). The former component is split in two parts, (ideally) labelled by the isotope tracer procedure, which form a diffusion gradient. The latter component is assumed to have a constant concentration during the self-diffusion experiment (the more general case is of minor interest). We will mainly reproduce here the result which has a bearing upon the (relative) constancy of the resistivities. Let the chemically different components be a and b. The former is composed of two isotopically different, but with respect to diffusion properties identical, substances (a) 1 and (a)2 c = -(- c. In view of what has been stated... 

The experimental data were fitted with two-component equation (Perkins and Batchelor 2012) ... 

As described above, the nonrelativistic Pauli wave function consists of two component spinors. It would be of interest to see if the Dirac equation can be reduced to two-component equations since we are specifically interested in the solution to the electronic problem, particularly given that the small component of the Dirac functions are on the order of Z/2c. Furthermore, computational effort is much smaller in two-component solutions than in four-component solutions. The following derivation is due to Greiner. ... 

New transformation formalisms for obtaining two component equations are currently being investigated and being applied to atomic systems. Such studies and others together with implementation of the Douglas-Kroll spin-orbit Hamiltonian, give special importance to this active field of relativistic quantum chemistry. 

A word of caution is due regarding the interpretation of the value of the peak current. It will be remembered from the discussion of the effects of the electrical double layer on electrode kinetics that there is a capacitance effect at an electrode-electrolyte interface. Consequently the true electrode potential is modified by the capacitance effect as it is also by the ohmic resistance of the solution. Equation (2.41) should really be written in a form which described these two components. Equation (2.44) shows such a modification. 

Substituting Eq. (2.74) for the a matrices and Eq. (2.79) for the four-component wavefunction, the Dirac equation can alternatively be written as two coupled two-component equations... 

Inserting this expression in Eq. (2.83) we obtain a single two-component equation for the large component... 

The required two-component equations and details of the calculation procedure are given by Hogsett and Mazur (1983). 

An alternative route to the calculation of relativistic effects is the systematic development of a perturbation operator for the relativistic effects. Historically, perturbation approaches were connected with the attempt to reduce the four-component form of the Dirac equation to two pairs of two-component equations. This approach caused problems with singular operators, which will be discussed in the next section. 

On a planar waveguide, the refractive-index profile n(x) depends only on x, so that each positon (x, z) on the ray path is determined by the two component equations of Eq. (1-18) in the x- and z-directions... 

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