Superposition approximation

The analogy just mentioned with the BBGKI set of equations being quite prominent still needs more detailed specification. To cut off an infinite hierarchy of coupled equations for many-particle densities, methods developed in the statistical theory of dense gases and liquids could be good candidates to be applied. However, one has to take into account that a number of the 

Standard approximate methods, e.g., the Percus-Yevick or hyper-chain approximations, are applicable for systems with the Gibbs distribution and are based on the distinctive Boltzmann factor like exp —U r)/ ksT)), where U(r) is the potential energy of interacting particles. The basic kinetic equation (2.3.53) has nothing to do with the Gibbs distribution. The only approximate method neutral with respect to the ensemble averaging is the Kirkwood approximation [76, 77, 87]. 

All superposition approximations mentioned above are based on the idea of multiplicative expansion, when m-reactant (m-point) distribution functions pmi, - im), with arguments being the generalised coordinates, are expressed through the correlation forms  

Pm with any m could be expressed through pm, m mo and we thus arrive at the superposition approximations. It is assumed that the correlation forms are small as compared to the but no distinctive small 

As was demonstrated by Kikuchi and Brush [88], using the Ising model as an example, an increase of mo in the expansion in the form secures the monotonic approach of the calculated critical parameters to exact results, except for the critical exponents which cannot be reproduced by algebraic expressions. It is important to note here that the superposition approximation permits exact (or asymptotically exact) solutions to be obtained for models revealing the critical point but not the phase transition. This should be kept in mind when interpreting the results of the bimolecular reaction kinetics obtained using approximate methods. 

---

Kirkwood derived an analogous equation that also relates two- and tlnee-particle correlation fiinctions but an approximation is necessary to uncouple them. The superposition approximation mentioned earlier is one such approximation, but unfortunately it is not very accurate. It is equivalent to the assumption that the potential of average force of tlnee or more particles is pairwise additive, which is not the case even if the total potential is pair decomposable. The YBG equation for n = 1, however, is a convenient starting point for perturbation theories of inliomogeneous fluids in an external field. 

In order to determine the force between plates as a function of their separation, one would have to perform a series of simulations with different wall separations and with the chemical potential of the fluid fixed at the bulk value. This is technically feasible, but very computationally intensive [42]. The qualitative behavior of the force law can, however, be estimated from the density profile of a fluid at a single wall using the wall sum rule and a superposition approximation [31,43]. The basic idea is that the density profile [denoted pH(z)] of a fluid between two walls at a separation H can be obtained from the density profile [denoted pj (z) of the same fluid at a single wall using... 

The formula for specialized distribution functions makes no such assumptions and hence involves g<3). It also involves g(4), g(5),.. . since correction is made for the possibility of a defect having two, three,. . . other partners simultaneously. Using Eq. (171) and the superposition approximation one finds that for the sodium chloride type lattice... 

The interfacial pair correlation functions are difficult to compute using statistical mechanical theories, and what is usually done is to assume that they are equal to the bulk correlation function times the singlet densities (the Kirkwood superposition approximation). This can be then used to determine the singlet densities (the density and the orientational profile). Molecular dynamics computer simulations can in... 

This is sometimes referred to as the superposition approximation. It is not, however, the superposition approximation used in the theory of liquids, first because Eq. (7.2.34) is exact (in the limit m —> o°), and second because the superposition approximation [as introduced by Kirkwood (1935) and used extensively in the theory of liquids] has the form... 

This equation for the doublet density if involves the triplet density f. It is a typical problem in statistical mechanics. To make progress, the hierarchy must be broken and this is usually done with a superposition approximation. The manner by which this is done is discussed in fee next sub-section. 

A similar situation exists in the molecular-distribution function theory of liquids and one usually resorts to a superposition approximation. This amounts to assuming that, e.g., = 2 or something similar. It will be seen shortly that, contrary to unimolecular reactions, for bi-molecular reactions the stochastic mean is not the same as the classical kinetic expression for the concentration. 

If the latter is dominant, then molecules in which all of the atoms are included in the it system would be better choices with which to test the LCAO superposition approximation. 

Chapter 5 deals with derivation of the basic equations of the fluctuation-controlled kinetics, applied mainly to the particular bimolecular A + B 0 reaction. The transition to the simplified treatment of the density fluctuation spectrum is achieved by means of the Kirkwood superposition approximation. Its accuracy is estimated by means of a comparison of analytical results for some test problems of the chemical kinetics with the relevant computer simulations. Their good agreement permits us to establish in the next Chapters the range of the applicability of the traditional Waite-Leibfried approach. 

Unfortunately, this expansion cannot be used as a basis for the development of approximate methods since - unlike the superposition approximation -in the case of considerable spatial correlation, neglect of the forms b(m m > mo leads to the correlation functions not satisfying the proper boundary conditions and increase of mo does not lead to the convergence of results. A comparison of the two kinds of expansion of the many-particle distribution function demonstrates that the superposition approximation even for small mo corresponds to the choice in the additive expansion of b 0 with any m. Therefore, in terms of the latter expansion the many-particle correlation forms are not neglected in the superposition approximations but are no longer independent. 

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. 

A general expression for the superposition approximation (2.3.55) has to be specified for a reaction under study. For instance, let us do it for the actual case of the bimolecular reaction employing many-particle densities Pm,m Single-particle densities are nothing but macroscopic concentrations... 

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... 

Fig. 2.21. The idea of the cut off of the infinite hierarchy of equations for the correlation functions by means of the Kirkwood superposition approximation.

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). 

As we have mentioned in Chapter 2, the accuracy of the kinetic equations derived using the superposition approximation cannot be checked up in the framework of the same theory. It is the analysis of the limiting case of the infinitely diluted system, no —> 0, which nevertheless permits us to compare approximate results obtained in the linearized approximation with the exact solution of the two-particle problem (Chapter 3). 

As no —> 0, the predominant term here is of the order of n0. Therefore, the shortcoming of equation (4.1.23) arises due to the incorrect use of the superposition approximation in a situation of very particular (strongly correlated) particle distribution. Strictly speaking, the correct treatment of the recombination process with arbitrary initial distribution requires the usage of the complete set of correlation functions. At the joint correlation level, such description yields reasonable results only for the particle distribution close to a random equation (4.1.12). For the infinitely diluted system the correlation function (4.1.10) of dissimilar closely spaced particles reveals... 

In this Section we consider again the kinetics of bimolecular A + B -A 0 recombination but instead of the linearized approximation discussed above, the complete Kirkwood superposition approximation, equation (2.3.62) is used which results in emergence of two new joint correlation functions for similar particles, Xu(r,t), v = A,B. The extended set of the correlation functions, nA(f),nB(f),Xfi,(r,t),Xa(r,t) and Y(r,t) is believed to be able now to describe the intermediate order in the particle spatial distribution. 

Substitution of equation (2.3.62) into a set of equations (4.1.13) to (4.1.16) for noncharged (neutral) particles (Uvil r) = 0) does not affect equations (4.1.18) and (4.1.19) whereas the linear equation (4.1.23) describing the correlation dynamics splits now into three integro-differential equations. Main stages of the passage from general equations (4.1.14)—(4.1.16) for the joint densities to those for the joint correlation functions have been demonstrated earlier, see (4.1.20) and (4.1.21). Therefore let us consider only those terms which are affected by the use of superposition approximation. Hereafter we use the relative coordinates f=f — f(, f = r 2 — r[ and... 

---