Kirkwood superposition approximation calculations

Kuzovkov and Kotomin [89-91] (see also [92, 93]) were the first to use the complete Kirkwood superposition approximation (2.3.62) in the kinetic calculations for bimolecular reaction in condensed media. This approximation allows us to cut off the infinite hierarchy of equations for the correlation functions describing spatial distribution of particles of the two kinds and to restrict ourselves to the treatment of minimal set of the kinetic equations which realistically could be handled (Fig. 2.21). In earlier studies [82, 84, 91, 94-97] a shortened superposition approximation was widely used... 

In fact, the latter is a functional of the correlation function of dissimilar particles, i.e., to calculate K(t) we need to know either Y(r, t) or p. In its turn, equation (4.1.16) demonstrates that these latter are coupled with three-point densities etc. Therefore, to solve the problem, we have to cut off the infinite equation hierarchy, thus only approximately describing the fluctuation spectrum. Usually it is done by means of the complete Kirkwood superposition approximation, equations (2.3.62) and (2.3.63), or the shortened approximation, equations (2.3.64) and (2.3.65). 

The single-species A + A —> A reaction allows us also to test the applicability of the shortened and the complete Kirkwood superposition approximations, (2.3.62) and (2.3.64) [98], The calculated quantities of the saturation concentrations... 

Calculation of the second-order term in Eq. (3.5.4) and the first-order term in Eq. (3.5.5) requires knowledge of the triplet distribution function in the reference fluid which is usually replaced by the Kirkwood superposition approximation. Following Smith, we will refer to the approach as a whole as the reference averaged Mayer (RAM) function theory. Another choice of reference system based upon a division of the Mayer function is that of hard spheres with a diameter chosen so that the first-order term in the free energy vanishes. This gives rise to the so called blip function theory. ... 

We can now turn to the question of matrix correlations, which arises because of the application of two conflicting approximations in the course of the calculation the Gaussian approximation for large and the continuinn approximation for small number densities of the matrix units. A reasonable way out of this dilemma is to retain the Gaussian approximation and to introduce matrix correlations. Here the only correlation effect considered is the principle that two matrix units cannot be located at the same position. This means that the factorization used in Eq. 2 cannot be applied. Matrix correlations can be accounted for, within the Ifamework of the stochastic model, by introducing a three-particle distribution function g2 R,R ) [16-18]. Applying the Kirkwood superposition approximation [19],... 

Incorporation of the superposition approximation leads inevitably to a closed set of several non-linear integro-differential equations. Their nonlinearity excludes the use of analytical methods, except for several cases of asymptotical automodel-like solutions at long reaction time. The kinetic equations derived are solved mainly by means of computers and this imposes limits on the approximations used. For instance, we could derive the kinetic equations for the A + B — C reaction employing the higher-order superposition approximation with mo = 3,4,... rather than mo = 2 for the Kirkwood one. (How to realize this for the simple reaction A + B —> B will be shown in Chapter 6.) However, even computer calculations involve great practical difficulties due to numerous coordinate variables entering these non-linear partial equations. 

If we succeeded in calculating the series in equation (7.1.17), the accumulation kinetics problem under question would be solved. However, an infinite set of coupled equations for pm turns out to be too complicated and thus we restrict ourselves to its cut-off by means of Kirkwood s superposition approximation, in order to get a closed set of non-linear equations for macriscopic densities n (t) and nB(t), as well as for the three joint correlation functions XA(r,t),XB(r,t) and Y(r,t). 

In most of the previous calculations the terms AU3, AC/4,... have been neglected in comparison with U2- This is crdled the binary cluster approximation, and is equivalent to Kirkwood s superposition approximation in the theory ofliquids. As another approximation it is reasonable to assume that uij depends only on r,j, i.e., Uij = w(rjj). Wth these approximations we obtain... 

Relaxation and electrophoretic effects are calculable with the help of Kirkwood s superposition approximation for the three-particle correlation function gikm-... 

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