Numerical implementation

The present paper is devoted to the theoretical formulation and numerical implementation of the NDCPA. The dynamical CPA is a one-site approximation in which variation of a site local environment (due to the presence, for example, of phonons with dispersion) is ignored. It is known from the coherent potential theory for disordered solids [21], that one can account in some extension the variation of a site local environment through an introduction of a nonlocal cohcn-cnt potential which depends on the difference between site... 

The simplest model of polymers comprises random and self-avoiding walks on lattices [11,45,46]. These models are used in analytical studies [2,4], in particular in the numerical implementation of the self-consistent field theory [4] and in studies of adsorption of polymers [35,47-50] and melts confined between walls [24,51,52]. 

With the new density matrices known up to time t, it is possible to advance the nuclear positions and momenta by integrating their Hamilton equations. This completes a cycle which can be repeated to advance to a later time t2-This sequence based on relaxing the density matrix for fixed nuclei and then correcting it to account for nuclear motions has been called the relax-and-drive procedure, and has been numerically implemented in several applications.(16, 15, 21, 46, 35-37)... 

The preceding equations form a set of algebraic and ordinary differential equations which were integrated numerically using the Gear algorithm (21) because of their nonlinearity and stiffness. The computation time on the CRAY X-MP supercomputer for a typical case in this paper was about 5 min. Further details on the numerical implementation of the algorithm are provided in (Richards, J. R. et al. J. ApdI. Polv. Sci.. in press). 

In principle, one may combine equilibrium and critical data in one database for the parameter estimation. From a numerical implementation point of view this can easily be done with the proposed estimation methods. However, it was not done because it puts a tremendous demand in the correlational ability of the EoS to describe all the data and it will be simply a computational exercise. 

A detailed numerical implementation of this method is discussed in [106]. W is the statistical weight of a trajectory, and the averages are taken over the ensemble of trajectories. In the unbiased case, W = exp -(3Wt), while in the biased case an additional factor must be included to account for the skewed momentum distribution W = exp(-/ Wt)w(p). Such simulations can be shown to increase accuracy in the reconstruction using the skewed momenta method because of the increase in the likelihood of generating low work values. For such reconstructions and other applications, e.g., to estimate free energy barriers and rate constants, we refer the reader to [117]. 

There are many varieties of density functional theories depending on the choice of ideal systems and approximations for the excess free energy functional. In the study of non-uniform polymers, density functional theories have been more popular than integral equations for a variety of reasons. A survey of various theories can be found in the proceedings of a symposium on chemical applications of density functional methods [102]. This section reviews the basic concepts and tools in these theoretical methods including techniques for numerical implementation. 

Because x appears as a parameter in the flamelet model, in numerical implementations a flamelet library (Pitsch and Peters 1998 Peters 2000) is constructed that stores T( , x ) forO < < 1 in a lookup table parameterized by ( ), (4, 2),and x - Based on the definition of a flamelet, at any point in the flow the reaction zone is assumed to be isolated so that no interaction occurs between individual flamelets. In order for this to be true, the probabilities of finding f = 0 and f = 1 must both be non-zero. 

In a numerical implementation of the FP model, Sp is found by replacing eigenvalues which are smaller than some minimum value with one, and there is no need to put the eigenvalues/eigenvectors in descending order. 

The simplest numerical implementation of inter-cell transport in an Eulerian PDF code consists of the following steps. 

We will thus restrict our attention to velocity, composition codes written in terms of u. The reader should keep in mind that velocity, composition codes based on u have only appeared relatively recently in the literature. Advances in the numerical implementation and new applications will most likely be reported after the appearance of this book. Readers interested in applying these methods are thus advised to consult the recent literature. 

In summary, the bi-variate case is handled in essentially the same manner as the univariate case. The numerical implementation will require an algorithm for choosing linearly... 

Generalization and Numerical Implementation of Pseudo-Time Schroedinger Equation for Quantum Scattering Calculations. 

Equation (130) is often useful for a straightforward numerical implementation of a metabolic system. Explicit examples of the parameter matrices are given in Section VIII.C. 

A numerical implementation of this approach can be generalised to include the polydispersity of the polymer. As polydispersity is increased the power law of — 0.8181... reduces. The onset of shear thinning, where Tn y 1 — 1, results in a slightly lower viscosity for polydisperse systems at this rate. So far we have neglected the origin of the characteristic time for the system which we would like to describe in terms of the chemical... 

For the numerical implementation of the QCL equation, the Liouville operator S is decomposed into a zero-order part Sq which is easy to evaluate and a nonadiabatic transition part if whose evaluation is difficult. The splitting suggests that we (a) employ a short-time expansion of the full exponential by use of a first-order Trotter formula... 

Let us now consider the stochastic realization of the off-diagonal operator which represents the main difficulty of the numerical implementation. 

The procedure just described is physically easily understandable but not suitable for numerical implementation. We now briefly describe its numerically efficient version based on the immittance matrix approach. 

Equation (12.17) represents the required boundary condition. It should be emphasized that it is essentially nonlocal both in space and time. In general, the numerical implementation of the operator in the right hand side of Eq. (12.17) is a nontrivial task. 

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