Kirkwood shortened

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... 

The applications of the many-particle densities will be demonstrated on a full scale in further Chapters. It should be only said here that the many-particle density formalism being combined with the shortened Kirkwood superposition approximation, equation (2.3.64), results in the well-known equations of the standard kinetics for both neutral [83] and charged particles [100] giving just another way of their derivation. On the other hand, the use of the full-scale (complete) Kirkwood s approximation, equation (2.3.62), permits us to take into account the many-particle (cooperative) effects [81, 91, 99-102] we are studying in this book. 

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). 

In this Section we consider namely the latter. Under the shortened Kirkwood s approximation equation (2.3.64) is expressed through Y(r,t) only, therefore the substitution of equation (2.3.64) into (2.3.16) results in a closed equation uniquely defining Y(r, t). This equation is linear in Y (see the title of this Section) and equations (4.1.14), (4.1.15) are not used any longer. Substituting equation (2.3.64) into (4.1.16), using the relative coordinate r = r i - r [ and after some manipulations... 

The shortened Kirkwood superposition approximation (2.3.64) differs from the complete one, equation (2.3.63), by the additional condition imposed on the correlation function of similar particles Xv(r,t) = 1 at any time t. Its substitution into equation (5.1.4) and taking into account equation (5.1.5) leads, as one could expect, to the linearized equation (4.1.23) for the correlation dynamics. Therefore, the applicability range of the linearized kinetic... 

Lastly, in conclusion of the Chapter 5 a study of the effect of the shortened and complete Kirkwood superposition approximations on asymptotic kinetics of a number of single-species reaction should be noted [112, 113]. The general conclusion was drawn that for irreversible reactions the complete superposition approximation, (2.3.62), gives better results than its shortened... 

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... 

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