The superposition approximation

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. 

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... 

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... 

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. 

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... 

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... 

To consider a role of dynamical reactant interactions, let us treat the Coulomb one as a distinctive example. Similarly to the case of an equilibrium system of charged particles, the superposition approximation does not permit... 

The approximations of the superposition-type like equation (2.3.54), are used in those problems of theoreticals physics when other-kind expansions (e.g., in powers of a small parameter) cannot be employed. First of all, we mean physics of phase transitions and critical phenomena [4, 13-15] where there are no small parameters at all. Neglect of the higher correlation forms a(ml in (2.3.54) introduces into solution errors which cannot be, in fact, estimated within the framework of the method used. That is, accuracy of the superposition-like approximations could be obtained by a comparison with either simplest explicitly solvable models (like the Ising model in the theory of phase transitions) or with results of direct computer simulations. Note, first of all, several distinctive features of the superposition approximations. 

All the above-said demonstrates well that there are arguments for and against applicability of the superposition approximation in the kinetics of bimolecular reactions. Because of the absence of exactly solvable problems, it is computer simulation only which can give a final answer. Note at once some peculiarities of such computer simulations. The largest deviations from the standard chemical kinetics could be expected at long t (large ). Unlike computer simulations of equilibrium phenomena [4] where the particle density is constant, in the kinetics problems particle density n(t) decays in time which puts natural limits on time of reaction. An increase of the standard deviation at small values of N(t) = (N) when calculating the mean concentration in computer simulations compel us to interrupt simulations at the reaction depth r = Io 3, where... 

The same is true for low-dimensional systems, d — 1 and 2. The point is not only that for such systems the better statistics could be achieved accompanied with reasonable computational time spent for it. Another circumstance is that we can expect here that the superposition approximation gives greater errors. For example, for one-dimensional contact recombination the so-called bus effect is known [17] given particles A and B can react only after particles separating them disappear during reaction. This topological effect is not foreseen by the superposition approximation but can affect considerably the reaction rate. 

Let us consider accuracy of the superposition approximation for the two quite different classes of problems a long-range reaction of immobile particles and a diffusion-controlled one, where the diffusion length Id arises. 

Computer simulations of bimolecular reactions for a system of immobile particles (incorporating their production) has a long history see, e.g., [18-22]. For the first time computer simulation as a test of analytical methods in the reaction kinetics was carried out by Zhdanov [23, 24] for d, = 3. Despite the fact that his simulations were performed up to rather small reaction depths, To < 1, it was established that of all empirical equations presented for the tunnelling recombination kinetics (those of linear approximation - (4.1.42) or (4.1.43)) turned out to be mostly correct (note that equations (5.1.14) to (5.1.16) of the complete superposition approximation were not considered.) On the other hand, irrespective of the initial reactant densities and space dimension d for reaction depths T To his theoretical curves deviate from those computer simulated by 10%. Accuracy of the superposition approximation in d = 3 case was first questioned by Kuzovkov [25], it was also... 

Fig. 5.2. The kinetics of tunnelling recombination n(t) for d= 1. The initial concentrations t.a(0) = ib(0) = 0.5 (N = No = No = 104 at t = 0) on the chain from 10s sites a - computer simulations (standard deviations are indicated), b - the superposition approximation, c - linear approximation, d - the complete neglect of all correlations, see...

The correlation functions can serve as an additional test of the correctness of the superposition approximation. A comparison of computer simulations and this approximation are presented in Figs 5.5 and 5.6. Since due to insufficient statistics the correlation functions begin to oscillate at short distances, they were additionally smoothed through averaging over interval Ar 10ro. 

Analysis of the correlation functions demonstrates also impressive general agreement between the superposition approximation and computer simulations. Note, however certain overestimate of the similar particle correlations, X r,t), at small r, especially for d = 1. In its turn the correlation function of dissimilar particles, Y(r,t), demonstrates complete agreement with the statistical simulations. Since the time development of concentrations is defined entirely by Y(r, t), Figs 5.2 and 5.3 serve as an additional evidence for the reliability of the superposition approximation. An estimate of the small distances here at which the function Y (r, t) is no longer zero corresponds quite well to the earlier introduced correlation length o, equation (5.1.47) as one can see in fact that at moment t there are no AB pairs separated by r < o-... 

For unequal concentrations, n (t) < ns(t), the reaction depth r < T0 = 3 reached in computer simulations is not enough for finding asymptotic laws but still permits to estimate qualitatively the accuracy of the superposition approximation. In Figs 5.7 and 5.8 numerical solution of the relevant kinetic equations is compared with computer simulations. To make situation more transparent, the linear approximation results are plotted in curves (d) for a single choice of initial concentrations only. 

Fig. 5.7. The kinetics of the tunnelling recombination n t) for d = 1. Full curves a to c are averages over 10 realisations, standard deviations are shown. Broken lines - the superposition approximation. Initial concentrations tia(O) = 0.5 whereas tib(0) = 0.5 (a), 0.75 (b), 1.00 (c). Curve d is the linear approximation with parameters of c.

An increase of the standard deviation at r 3 due to small number of survived particles, demonstrates a limited possibility of the direct statistical simulations for a system with a variable number of particles. However, certain conclusions could be drawn even for such limited statistical information. Say, if for equal concentrations the analytical theory based on the superposition approximation seems to be quite adequate, for unequal concentrations its deviation from the computer simulations greatly increases in time. The superposition approximation gives the lower bound estimate of the actual kinetic curves tia( ) but if for d = 2 shown in Fig. 5.8 the deviation is considerable, for d, = 1 (Fig. 5.7) it is not observed, at least for the reaction depths considered. 

More concrete conclusions could be drawn for the linear approximation applicability it is adequate for small reaction depths r < 1, whereas at r > 1 it is in serious error. In its turn, errors of the superposition approximation are essentially less, the relevant lower bound estimate is quite acceptable to fit theoretical parameters to the experimental curves. 

Fig. 5.9. The correlation functions of similar X, Xn) and dissimilar (F) particles for d = 2. Full (t = 10) and broken (t = 103) lines are results of the superposition approximation. Symbols show computer simulations. Initial concentrations tia(0) = 0.25 and ne(0) = 0.3125 as in curve b, Fig. 5.8. Curves are averaged over 10 realisations and additionally over intervals...

Quantitative deviations are seen also from the correlation shown in Fig. 5.9. The correlation functions of dissimilar particles Y (r, t) are in good agreement with simulations, which results also in a reliable reproduction of the decay kinetics for nA(t) - unlike behaviour of the correlation functions of the similar particles Xv r,t) which is very well pronounced for XA(r,t). Positive correlations, Xu(r,t) > 1 as r < , argue for the similar particle aggregation, and the superposition approximation tends to overestimate their density. The obtained results permit to conclude that the approximation (2.3.63) of the three-particle correlation function could be in a serious error for the excess of one kind of reactants. 

The multipole interaction of immobile particles (4.1.44) is an additional way to check up advantages of the superposition approximation [8]. The reason is that the tunnelling recombination (3.1.2) serves better as an example of short-range reaction. Indeed, the distinctive scale tq characterizing distant (non-contact) interaction could be defined as... 

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