Asymptotic approximation applications

Considerable progress has recently been made in developing the theoretical background necessary for the application of the above method of transient kinetic analysis. An important step in this direction was the use of WKB asymptotics to derive approximate analytical expressions for short- and long-time transient sorption and permeation in membranes characterized by concentration-independent continuous S(X) and Dt(X) functions 150-154). The earlier papers dealing with this subject152 154) are referred to in a recent review 9). The more recent articles 1S0 1S1) provide the correct asymptotic expressions applicable to all kinetic regimes listed above the usefulness... 

Per-Olov Lowdin had a long and lasting interest in the analytical methods of quantum mechanics and my tribute to his legacy involves an application of the Wentzel-Kramers-Brillouin (WKB) asymptotic approximation method. It was the subject of a contribution(l) by Lowdin to the Solid State and Molecular Theory Group created by John C. Slater at the Massachusetts Institute of Technology. 

In the previous section we demonstrated the application of asymptotic expansion techniques to obtain the high- and low-frequency limits of the velocity field for flow in a circular tube driven by an oscillatory pressure gradient. In the process, we introduced such fundamental notions as the difference between a regular and a singular asymptotic expansion and, in the latter case, the concept of matching of the asymptotic approximations that are valid in different parts of the domain. However, all of the presentation was ad hoc, without the benefit of any formal introduction to the properties of asymptotic expansions. The present section is intended to provide at least a partial remedy for that shortcoming. We note, however,... 

The three chapters in this volume deal with various aspects of singular perturbations and their numerical solution. The first chapter is concerned with the analysis of some singular perturbation problems that arise in chemical kinetics. In it the matching method is applied to find asymptotic solutions of some dynamical systems of ordinary differential equations whose solutions have multiscale time dependence. The second chapter contains a comprehensive overview of the theory and application of asymptotic approximations for many different kinds of problems in chemical physics, with boundary and interior layers governed by either ordinary or partial differential equations. In the final chapter the numerical difficulties arising in the solution of the problems described in the previous chapters are discussed. In addition, rigorous criteria are proposed for... 

The variational principle has not been widely used in diffusion kinetic problems. Nevertheless, it is such a powerful technique that it is suitable for discussing the many-body problems which have still to be tackled. Wherever approximate methods are necessary, the variational principle should be considered. The trial function(s) should be chosen with care, based on a good idea of the nature of the trial function from its behaviour in certain asymptotic limits. The only application known to the author of the variation principle to a numerical study of a diffusion kinetic problem on a molecular system is that of Delair et al. [377]. They used the variational principle to generate an implicit finite difference scheme for solving the Debye—Smoluchowski equation. Interesting comments have been made by Brykalski and Krason more in the context of heat diffusion [510]. 

Another important test of the accuracy of the superposition approximation is the diffusion-controlled A + B — 0 reaction. For the first time it was computer-simulated by Toussaint and Wilczek [27]. They confirmed existence of new asymptotic reaction laws but did not test different approximations used in the diffusion-controlled theories. Their findings were used in [28] to discuss divergence in the linear and the superposition approximations. Since analytical calculations [28] were performed for other sets of parameters as used in [27], their comparison was only qualitative. It was Schnorer et al. [29] who first performed detailed study of the applicability of the superposition approximation. 

We will carry out our program in two steps. In this section we will derive the two-particle density operator Fn in a three-particle collision approximation for the application in the collision integral of Fl. As compared with Section II.2, the main difference will be the occurrence of bound states and, especially, the generalization of the asymptotic condition, which now has to account for bound states too. For the purpose of the application in the kinetic equation of the atoms (bound states) we need an approximation of the next-higher-order density matrix, that is, F 23 This quantity will be determined under inclusion of certain four-particle interaction. 

S A was also applied to the reversible reactions considered in Section XII.C.4. Since the results were not satisfactory, an extended superposition approach (ESA) was developed, then linearized, and later known as LESA [241]. Independently, a similar linearization over deviations from equilibrium was also made in Ref. 242. Although the asymptotic description of the quenching kinetics is improved, it was recognized [242] that LESA is not valid with a large equilibrium constant K because the superposition approach worsens when K increases [241]. This is especially true at earlier times when the deviations from equilibrium are not small. However, the authors who constructed LESA claimed that it is applicable at all times [241]. Therefore, it was taken for comparison with other approximations. In the irreversible limit (K —> oo), the kernels obtained in both works [241,242] coincide with that listed as LESA in Table V. 

The main advantage of this approximation is that it is exact for two-electron systems (if the correct kf(r) = 0 is utilised in (4.15) before performing the functional differentiation (3.17) required for its application) and also correctly accounts for the self-interaction energies of individual closed shells if a shellpartitioning scheme is used [71]. Furthermore, the RWDA reproduces the asymptotic r proportionality of the exact x-only potential (although with the incorrect prefactor of 1/2 [103]). 

It has been seen from the viewpoint of asymptotic analysis that when a one-step approximation is applicable for the chemical kinetics, the burning... 

The necessary application of activation-energy asymptotics parallels that given in Sections 9.2, 5.3.6, and 8.2.1 and has been developed for both one-reactant [172], [173] and two-reactant [174], [175] systems. For most problems (for example, for stability analyses), gradients of the total enthalpy and of the mixture ratio of the reactants are negligible in the first approximation within the reaction sheet, so that in the scaled variables appropriate to the reaction zone, both 6 -h Y Q/iY qLq ) and Y -h y2Lej/(v2Le2) are constant. Evaluation of these constants is aided by the further result that Yi = 0 (and hence dYJd = 0) downstream from the reaction zone, at least in the first approximations, so that for the reaction-zone analysis, the expressions... 

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